\documentclass[twoside]{article}
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\markboth{\hfil B.V.P. with cubic-like nonlinearities \hfil EJDE--2000/52}
{EJDE--2000/52\hfil Idris Addou \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.~{\bf 2000}(2000), No.~52, pp.~1--42. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
  Multiplicity results for classes of one-dimensional p-Laplacian
  boundary-value problems with cubic-like nonlinearities 
\thanks{ {\em Mathematics Subject Classifications:} 34B15.
\hfil\break\indent
{\em Key words:}  p-Laplacian, time-maps, multiplicity results, 
     cubic-like nonlinearities.
\hfil\break\indent
\copyright 2000 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Submitted April 16, 1999. Revised May 1, 2000. Published July 3, 2000.} }
\date{}
%
\author{ Idris Addou }
\maketitle

\begin{abstract} 
 We study boundary-value problems of the type
 $$\displaylines{
 -(\varphi_{p}( u') ) ' =\lambda f( u) ,\hbox{ in }(0,1)  \cr
 u( 0)  =u( 1) =0,
 }$$
 where $p>1$, $\varphi_{p}( x) =\left| x\right| ^{p-2}x$, and
 $\lambda >0$. We provide multiplicity results when $f$ behaves like
 a cubic with three distinct roots, at which it satisfies Lipschitz-type
 conditions involving a parameter $q>1$. We shall show how changes in the
 position of $q$ with respect to $p$ lead to different behavior of the
 solution set. When dealing with sign-changing solutions, we assume
 that $f$ is {\it half-odd}; a condition generalizing the usual oddness.
 We use a quadrature method.
\end{abstract}
\maketitle

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

\label{sec.1} We consider a quasilinear Dirichlet boundary-value
problem of the type
\begin{equation} \gathered
-( \varphi_p( u') ) ' =\lambda f( u) ,\text{in }( 0, 1)  \label{p1} \\
u( 0) =u( 1) =0,
\endgathered\end{equation}
where $p>1$, $\varphi_p( x) =\left| x\right| ^{p-2}x$, $\lambda
>0$, and $f\in C({\mathbb R},{\mathbb R})$ is a cubic-like nonlinearity to be
specified below. We study non-existence, existence, and
multiplicity results. In some cases, the exact number of
solutions to (\ref{p1}) is given.

Several studies related to Dirichlet problems with cubic and
cubic-like nonlinearities are reviewed in Section \ref{sec.2}.
The purpose of this work is to study the solution set of Problem
(\ref{p1}) when $f$ is cubic-like but \textit{not necessarily an
odd function}. We take $f$ \thinspace to be a nonlinearity
satisfying:
\begin{gather}
f\in C(\mathbb R, \mathbb R) \text{ and }f( \alpha_-) =f(
\alpha_+) =f( 0) =0\text{ for constants $\alpha_-<0<\alpha_+$ }
 \label{pq1} \\
f( x) >0\text{\ \ \ for }x\in ( -\infty ,\alpha _-) \cup (
0,\alpha_+) ;
 \label{pq2}  \\
f( x) <0\text{\ \ \ for }x\in ( \alpha_-,0) \cup (
\alpha_+,+\infty ) ;
\label{pq3} \\
\lim_{s\rightarrow 0}\frac{f( s) }{\varphi_q( s) } =a_0>0\text{
for some }q>1 ;  \label{pq4}
\end{gather}
there exist $\delta >0$, $m_{\pm }>0$  and $M_{\pm }>0$ such that
\begin{equation} \gathered
-m_+ \geq \frac{f( \alpha_+) -f( \xi ) }{ \varphi_q( \alpha_+-\xi
) }\geq -M_+\text{,\ for all }\xi
\in ( \alpha_+-\delta ,\alpha_+) ,   \\
-m_- \geq \frac{f( \alpha_-) -f( \xi ) }{ \varphi_q( \alpha_--\xi
) }\geq -M_-\text{,\ for all }\xi \in ( \alpha_-,\alpha_-+\delta
) .   \endgathered \label{pq5}
\end{equation}

It is well known that when $f$ is \emph{odd}, the conditions
above are sufficient for studying constant sign solutions as well
as sign-changing solutions to (\ref{p1}). However, when $f$ is not
necessarily odd, only constant sign solutions can be handled with
these conditions. In order to study sign-changing solutions with
$f$ not necessarily odd, we introduce some functions generalizing
odd ones.

Let us assign to each function $f$ defined on ${\mathbb R}$, the
function $h_{f}$ defined on $[ 0,+\infty ) $ by $h_{f}( x) =f( x)
+f( -x) $, for all $x\in [ 0,+\infty ) $. Notice that the oddness
of a function $f$ on ${\mathbb R} $ may be characterized by the
condition: $h_{f}\equiv 0$ on $[ 0,+\infty ) $. Therefore, if
$I\subset [ 0,+\infty ) $ is a non-empty set, we shall say that:

\begin{itemize}
\item  $f$ is \textit{positively half-odd} (\textit{p.h.o.}, for brevity) on
$I\cup ( -I) $, if
\begin{equation}
h_{f}( x) \geq 0,\text{ for all }x\in I,  \label{Arizona}
\end{equation}

\item  $f$ is \textit{negatively half-odd} (\textit{n.h.o.}, for brevity) on
$I\cup ( -I) $, if
\begin{equation}
h_{f}( x) \leq 0,\text{ for all }x\in I.  \label{Florida}
\end{equation}
\end{itemize}

Also, we shall say that $f$ is \textit{strictly positively half-odd} (%
\textit{s.p.h.o.}, for brevity) on $I\cup ( -I) $ (resp. \textit{%
strictly negatively half-odd} (\textit{s.n.h.o.}, for brevity) on
$I\cup ( -I) $) if the strict inequality holds in (\ref
{Arizona}) (resp. in (\ref{Florida})).

\textit{Half-even} functions may be defined analogously. Assign
to each function $f\colon {\mathbb R}\to{\mathbb R}$, $g_f$
defined on $[ 0,+\infty )$ by $g_{f}( x) =f( x) -f( -x)$. Note
that $f$ is even if and only if $g_{f}( x)=0$ for all $x \in [
0,+\infty )$. If $I\subset [ 0,+\infty )$ is a non-empty set, we
say that $f$ is positively (resp., negatively) half-even on
$I\cup ( -I)$ if $g_f(x)\ge 0$ (resp., $g_f(x)\le 0$) for all
$x\in I$. (Note that the criteria for $f$ to be half-odd (resp.,
half-even) on $I\cup ( -I) $ involve only behavior of $h_{f}$
(resp., $g_f$) on $I\subset [ 0,+\infty )$, not on $I\cup ( -I)$.)


When dealing with sign-changing solutions to (\ref{p1}) we shall
assume that
\begin{equation}
f\text{ \ is\ \textit{p.h.o.}\ \ \ on }[ \alpha_-,-\alpha _-] .
\label{pho}
\end{equation}
Note that condition (\ref{pho}) implies that the function
\begin{equation}
u\mapsto F( u) :=\int_{0}^{u}f( \xi ) d\xi \text{ \ \ is\
\textit{p.h.e.}\ \ on\ }[ \alpha_-,-\alpha_-] . \label{phe}
\end{equation}
Indeed, let $h( \xi ) :=F( \xi ) -F( -\xi ) $ for all $\xi \in [
0,-\alpha_-] $. One has $h(0) =0$ and $h'( \xi ) =f( \xi )+f(
-\xi ) \geq 0$ for all $\xi \in [ 0,-\alpha _-] $. Thus $F( \xi )
-F( -\xi ) \geq 0$ for all $\xi \in [ 0,-\alpha_-] $.

Also, notice that condition (\ref{pho}) together with (\ref{pq1}), (\ref{pq3}%
) imply that
\begin{equation}
0<-\alpha_-\leq \alpha_+.  \label{jzehjf}
\end{equation}
Indeed, if $\alpha_+<-\alpha_-$, $f( -\alpha_-) <0$ from
(\ref{pq3}). But $f( -\alpha_-) =f( -\alpha_-) +f( \alpha_-) \geq
0$. A contradiction.

On the other hand, (\ref{pho}) together with (\ref{pq2}),
(\ref{pq3}) imply that
\begin{gather}
\text{for all }x\in [ \alpha_-,0) \text{ there exists a unique
}y( x) \in ( 0,-x] \text{ such that } \label{12kwdj} \\ \text{for
all } t\in (0, -x], \text{ }  F( t) =F( x) \text{ if and only if
} t=y(x) .
 \notag
\end{gather}
Indeed, the function $k( t) :=F( t) -F( x) $ for all $t\in [
0,-x] $, satisfies $k'( t) =f( t) >0$ for all $t\in ( 0,-x) $,
from (\ref{pq2}), $k( 0) =-F( x) <0$, from (\ref{pq3}) and $k(
-x) =F( -x) -F( -( -x) ) \geq 0$.

To prove exact multiplicity results, we shall need in the case
where $1<p<q $, the condition
\begin{equation}
f\in C^2( I_{\pm }( \alpha_{\pm }) ) \text{\ \ and }\pm ( ( p-2)
( p-1) f( x) -x^2f''( x) ) >0,  \label{Miami}
\end{equation}
for all $x\in I_{\pm }( \alpha_{\pm }) $, where, for all $z>0$
(resp. $z<0$), $I_+( z) $ (resp. $I_-( z) $) designates the open
interval $( 0,z) $ (resp. $( z,0) $). Notice that (\ref{Miami})
holds if, for example, $f$ satisfies (\ref{pq1})-(\ref{pq3}) and
\begin{equation}
f\in C^2( I_{\pm }( \alpha_{\pm }) ) ,\text{ \ \ }%
p\geq 2\text{\ \ and\ \ }\pm f''<0\text{ \ \ in }I_{\pm }(
\alpha_{\pm }) .  \label{jfghnb}
\end{equation}
In the case where $1<q\leq p$ we shall make some assumptions
concerning the variations of the function $x\mapsto H( x)
:=p\int_0^xf( t) dt-xf( x) $. Namely, we use the condition,
\begin{equation}
\pm H( \cdot ) \text{ is strictly increasing in }I_{\pm }(
\alpha_{\pm }) .  \label{njk}
\end{equation}

\textbf{A convention.} For all integer $k\geq 1$ and $\kappa
=+,-$, we shall say that $f$ satisfies $(\mathbf{H})_k^\kappa $
if $f$ satisfies

\begin{itemize}
\item  $(\ref{pq5})_+$ in the case where $k=1$ and $\kappa =+$, or,

\item  $(\ref{pq5})_-$ in the case where $k=1$ and $\kappa =-$, or,

\item  $(\ref{pho})$ and $(\ref{pq5})_-$ in the case where $k\geq 2$ and
$\kappa \in \left\{ +,-\right\} $.
\end{itemize}

Also, we shall say that $f$ satisfies $(\mathbf{K})_k^\kappa $ if
$f$ satisfies

\begin{itemize}
\item  $(\ref{njk})_+$ in the case where $k=1$ and
$\kappa =+$, or,

\item  $(\ref{njk})_-$ in the case where $k=1$ and
$\kappa =-$, or,

\item  $(\ref{njk})_+$ and  $(\ref{njk})_-$ in the case where
$k\geq 2$ and $\kappa \in \left\{ +,-\right\} $.
\end{itemize}

Recall \cite{Otani} that the first eigenvalue of
\begin{eqnarray*}
-( \left| u'\right| ^{p-2}u') '
&=&\lambda \left| u\right| ^{p-2}u\text{ \ in }( 0, 1) \\
u( 0) &=&u( 1) =0,
\end{eqnarray*}
is given by $\lambda_1( p) =( p-1) [ 2\int_0^1\left\{ 1-\xi
^p\right\} ^{-\frac 1p}d\xi ] ^p=( p-1) ( 2\pi /p\sin ( \frac \pi
p) ) ^p$ and the other eigenvalues constitute the sequence
$\lambda_1( p) <\lambda_2( p) <\cdots <\lambda_k( p) <\cdots $,
$\lambda_n( p) =n^p\lambda_1( p) $.

The method we use is the quadrature method. This one enable us to
look for solutions of (\ref{p1}) in some prescribed subsets of
$C^1( [ 0, 1] ) $. For any $k\in {\mathbb N^*}$,  let
\begin{equation*}
S_k^+=\left\{
\begin{array}{r}
u\in C^1( [ \alpha ,\beta ] ) :u\text{ admits exactly }( k-1)
\text{ zeros in }( \alpha ,\beta )
\\
\text{all simple, }u( \alpha ) =u( \beta ) =0\text{ and }u'(
\alpha ) >0
\end{array}
\right\} ,
\end{equation*}
$S_k^{-}=-S_k^+$ and $S_k=S_k^+\cup S_k^{-}$.

\textbf{Definition. }\textit{Let }$u\in C( [ \alpha ,\beta ] )
$\textit{\ be a function with two consecutive zeros }
$x_{1}<x_{2}.$\textit{\ We call the I-hump of }$u$\textit{\ the
restriction of }$u$\textit{\ to the open interval }$I=(
x_{1},x_{2}) . $\textit{\ When there is no confusion we refer to
a hump of }$u$.

\bigskip Observe that each function in $S_k^+$ has exactly $k$ humps such
that the first one is positive, the second is negative, and so on
with alternations. Let $A_k^+$ $( k\geq 1) $ be the subset of
$S_k^+$ consisting of the functions $u$ satisfying:

\begin{itemize}
\item  Every hump of $u$ is symmetrical about the center of the interval of
its definition.

\item  Every positive (resp. negative) hump of $u$ can be obtained by
translating the first positive (resp. negative) hump.

\item  The derivative of each hump of $u$ vanishes once and only once.
\end{itemize}

Let $A_k^{-}=-A_k^+$ and $A_k=A_k^+\cup A_k^{-}.$\ Now we are
ready to state the main results. The first one concerns the case
where $1<q<p$.

\begin{theorem}
\label{theorem1}Let $1<q<p$.
\begin{description}
\item[(i)]  If (\ref{pq1})-(\ref{pq4}) and $(\ref{pq5})_{\pm }$ hold, there
exists $J_{\pm }>0,$\ such that Problem (\ref{p1}) admits at least
a solution in $A_{1}^{\pm }$ for all $\lambda \in (0,J_{\pm }]$.
Moreover, if $(\ref{njk})_{\pm }$ holds then,

\begin{itemize}
\item  If $0<\lambda \leq J_{\pm }$, Problem (\ref{p1}) admits a unique
solution in $A_{1}^{\pm }$.

\item  If $\lambda >J_{\pm }$, Problem (\ref{p1}) admits no solution in
$A_{1}^{\pm }$.
\end{itemize}

\item[(ii)]  If (\ref{pq1})-(\ref{pq4}), $(\ref{pq5})_-$, and (\ref{pho})
hold, there exists (beside $J_-$) a positive number $J_{*}>0$
such that, for all integer $n\in {\mathbb N} ^{*}$,

\begin{description}
\item[(a)]  If $0<\lambda \leq (nJ_-+nJ_{*})^{p}$, Problem (\ref{p1})
admits at least a solution in $A_{2n}^{\kappa }$, for all $\kappa
\in \left\{ -,+\right\} $.

\item[(b)]  If $0<\lambda \leq (nJ_-+(n+1)J_{*})^{p}$, Problem (\ref{p1})
admits at least a solution in $A_{2n+1}^+$.

\item[(c)]  If $0<\lambda \leq ((n+1)J_-+nJ_{*})^{p}$, Problem (\ref{p1})
admits at least a solution in $A_{2n+1}^{-}$.
\end{description}

Moreover, if both $(\ref{njk})_-$ and $(\ref{njk})_+$ hold, it
follows that

\begin{description}
\item[(a1)]  If $0<\lambda \leq (nJ_-+nJ_{*})^{p}$, Problem (\ref{p1})
admits a unique solution in $A_{2n}^{\kappa }$, for all $\kappa
\in \left\{ -,+\right\} $.

\item[(a2)]  If$\;\lambda >(nJ_-+nJ_{*})^{p}$, Problem (\ref{p1}) admits
no solution in $A_{2n}^{\kappa }$, for all $\kappa \in \left\{
-,+\right\} $.

\item[(b1)]  If $0<\lambda \leq (nJ_-+(n+1)J_{*})^{p}$, Problem (\ref{p1})
admits a unique solution in $A_{2n+1}^+$.

\item[(b2)]  If $\lambda >(nJ_-+(n+1)J_{*})^{p}$, Problem (\ref{p1})
admits no solution in $A_{2n+1}^+$.

\item[(c1)]  If $0<\lambda \leq ((n+1)J_-+nJ_{*})^{p}$, Problem (\ref{p1})
admits a unique solution in $A_{2n+1}^{-}$.

\item[(c2)]  If $\lambda >((n+1)J_-+nJ_{*})^{p}$, Problem (\ref{p1})
admits no solution in $A_{2n+1}^{-}$.
\end{description}
\end{description}
\end{theorem}

\begin{theorem}
\label{theorem2}Let $1<p=q$. For all $k\geq 1$ and $\kappa =+,-$,
assume that (\ref{pq1})-(\ref{pq4}), and
$(\mathbf{H})_{k}^{\kappa }$ hold. Then, Problem (\ref{p1}) admits
at least a solution in $A_{k}^{\kappa }$ for all $\lambda
>\lambda_{k}/a_{0}$. Moreover, if $(\mathbf{K})_{k}^{\kappa }$
holds, it follows that

%\begin{description}
\begin{description}
\item[(a)]  If $\lambda >\lambda_{k}/a_{0}$, Problem (\ref{p1}) admits a
unique solution in $A_{k}^{\kappa }$.

\item[(b)]  If $\lambda \leq \lambda_{k}/a_{0}$, Problem (\ref{p1}) admits
no solution in $A_{k}^{\kappa }$.
\end{description}
%\end{description}
\end{theorem}

\begin{theorem}
\label{theorem3}Let $1<p<q$.

\begin{description}
\item[(i)]  If (\ref{pq1})-(\ref{pq4}), and $(\ref{pq5})_{\pm }$ hold, then
there exists a real number $\mu_{1}^{\pm }>0$ such that

\begin{itemize}
\item  If $\lambda <\mu_{1}^{\pm }$, Problem (\ref{p1}) admits no solution
in $A_{1}^{\pm }$.

\item  If $\lambda =\mu_{1}^{\pm }$, Problem (\ref{p1}) admits at least a
solution in $A_{1}^{\pm }$.

\item  If $\lambda >\mu_{1}^{\pm }$, Problem (\ref{p1}) admits at least two
solutions in $A_{1}^{\pm }$.
\end{itemize}

Moreover, if $(\ref{Miami})_{\pm }$ holds, it follows that

\begin{itemize}
\item  If $\lambda <\mu_{1}^{\pm }$, Problem (\ref{p1}) admits no solution
in $A_{1}^{\pm }$.

\item  If $\lambda =\mu_{1}^{\pm }$, Problem (\ref{p1}) admits a unique
solution in $A_{1}^{\pm }$.

\item  If $\lambda >\mu_{1}^{\pm }$, Problem (\ref{p1}) admits exactly two
solutions in $A_{1}^{\pm }$.
\end{itemize}

\item[(ii)]  If (\ref{pq1})-(\ref{pq4}), $(\ref{pq5})_-$ and (\ref{pho})
hold, then there exist two strictly increasing sequences
$(\mu_{k})_{k\geq 2}$ and $(\nu_{k})_{k\geq 2}$ such that
\begin{equation*}
\mu_{k}>\nu_{k}>0\;\;\;\text{for \ all\ \ \ }k\geq 2\text{,\ \ \ and\ \ \ }%
\lim_{k\rightarrow +\infty }\mu_{k}=\lim_{k\rightarrow +\infty
}\nu _{k}=+\infty ,
\end{equation*}
and such that, for all $k\geq 2$,

\begin{itemize}
\item  If $\lambda >\mu_{k}$, Problem (\ref{p1}) admits at least two
solutions in $A_{k}^{\pm }$.

\item  If $0<\lambda <\nu_{k}$, Problem (\ref{p1}) admits no solution in
$A_{k}^{\pm }$.
\end{itemize}
\end{description}
\end{theorem}

Regarding the results described in Section \ref{sec.2}, Theorems
\ref {theorem1}, \ref{theorem2}, and \ref{theorem3} seem to be
new even when $p=2. $

\bigskip \ The paper is organized as follows. Section \ref{sec.2} is
dedicated to some review related to boundary-value problems with
cubic, and cubic-like nonlinearities. The quadrature method used
for proving our results is recalled in Section \ref{sec.3}. Some
preliminary lemmas are the aim of Section \ref{sec.4}. The main
results are proved in Section \ref {sec.5}. We close our study by
listing some open questions in Section \ref{sec.6}.

\section{Some known results}

\label{sec.2}In this section we shall present some results
concerning boundary-value problems with cubic, and cubic-like
nonlinearities. We shall not attempt to make a complete
historical review.

The classical paper by Smoller and Wasserman
\cite{SmollerWasserman} deals with semilinear problems when the
nonlinearity is cubic. They consider the boundary-value problem
\begin{equation} \gathered
-u''( y) =f( u( y) ) ,
\text{ \ }y\in ( -\lambda ,\lambda )  \label{Pm} \\
u( -\lambda ) =u( \lambda ) =0  \endgathered
\end{equation}
with the cubic nonlinearity $f( u) =-( u-a) ( u-b) ( u-c) $, and
$a<b<c$ are its real roots. They show that the solution set
depends strongly on the position of the roots of $f$.
Notice that the change of variable $y=\sqrt{\lambda }x$ transforms Problem (%
\ref{Pm}) to
\begin{equation} \gathered
-u''( x) =\lambda f( u( x)) ,\text{ \ }x\in ( -1,1)  \label{PM} \\
u( -1) =u( 1) =0.  \endgathered
\end{equation}
In the case where $0=a<b<c$, they show the existence of a
critical $\lambda _{0}>0$ such that for $0<\lambda <\lambda_{0}$,
Problem (\ref{PM}) admits no nontrivial positive solution, it has
exactly one positive solution at $\lambda =\lambda_{0}$, and it
has exactly two positive solutions for $\lambda >\lambda_{0}$.

Concerning the case $0<a<b<c$, their study was completed by Wang
\cite{Wang} who showed that, under an additional condition, the
behavior of the solution set of (\ref{PM}) is the same as that of
the case $0=a<b<c$.

Smoller and Wasserman \cite{SmollerWasserman} have also studied
some cases when $f$ \thinspace has one or two negative roots, and
they have studied the same equation with Neumann and periodic
boundary conditions.

Notice that in the two papers mentioned above, autonomous
boundary value problems with cubic nonlinearities were studied.

Korman and Ouyang \cite{KormanOuyang1} consider the
non-autonomous problem
\begin{equation} \gathered
-u'' =\lambda f( x,u) ,\;x\in (
-1,1) ,  \label{p2} \\
u( -1) =u( 1) =0  \endgathered
\end{equation}
where $\lambda >0$ is a real parameter, and $f$ is the cubic
nonlinearity
\begin{equation}
f( x,u) =a( x) u^2( 1-b( x) u) ,\;\;x\in ( -1,1) ,u\in  {\mathbb
R} ,  \label{Lasvegas}
\end{equation}
and $a( x) $, $b( x) $ are even functions, $a( x) \in C^1( -1,1)
\cap C^0[ -1,1] $, $b( x) \in C^2( -1,1) \cap C^0[ -1,1] $
satisfying
$$\gathered
a( x) ,b( x) >0\quad \text{ for }-1\leq x\leq 1.\\
xb'( x) >0\quad \text{\and}\quad xa'( x) <0
\quad \text{for } x\in ( -1,1) \backslash \left\{ 0\right\} \\
b''( x) b( x) -2( b') ^2( x) >0\quad\text{for }-1<x<1 \,.
\endgathered $$
They show that Problem (\ref{p2}) admits exactly two solutions for
large $\lambda $'s, admits no solution for small $\lambda $'s,
and admits finitely many solutions for the other values of
$\lambda $. More precisely they prove:

\begin{theorem}
\label{Thm4}\cite{KormanOuyang1}\ There exists a critical
$\lambda_{1}$, such that for $0<\lambda <\lambda_{1}$ Problem
(\ref{p2}) has no solution; it has at least one solution at
$\lambda =\lambda_{1};$ and it has at least two solutions for
$\lambda >\lambda_{1}$. All solutions lie on a single curve of
solutions, which is smooth in $\lambda $. For each $\lambda
>\lambda_{1}$ there are finitely many solutions, and different
solutions are strictly ordered on $( -1,1) $. Moreover, there
exists $\lambda_{2}\geq \lambda_{1}$, so that for $\lambda
>\lambda_{2}$ Problem (\ref{p2}) has exactly two solutions
denoted by $u^{-}( x,\lambda ) <u^+( x,\lambda ) $, with $u^+(
x,\lambda ) $ strictly monotone increasing in $\lambda $, $u^{-}(
0,\lambda ) $ strictly monotone decreasing in $\lambda $, and
$\lim_{\lambda \rightarrow \infty }u^+( x,\lambda ) =1/b( x) $,
$\lim_{\lambda \rightarrow \infty }u^{-}( x,\lambda ) =0$ for all
$x\in ( -1,1) $. (All solutions of (\ref{p2}) are positive by the
maximum principle.)
\end{theorem}

Remark that the nonlinearity (\ref{Lasvegas}) has a double root
$u_0=0$ and a simple positive root $u_1=1/b( x) $. A case where
the nonlinearity of the problem admits three simple roots was
also studied by Korman and Ouyang \cite{KormanOuyang1}. Indeed,
they consider the cubic nonlinearity
\begin{equation*}
f( x,u) =u( u-a( x) ) ( b-u), \quad x\in ( -1,1) ,\; u\in
{\mathbb R},
\end{equation*}
but this time, $b$ is a positive constant, and the function $a(
x) \in C^1[ -1,1] $ satisfies the following conditions:
$$\gathered
a( x) \geq a_0>0,a'( x) >0\text{ for } x\in ( 0, 1) \\
a( -x) =a( x) \text{\ for }x\in ( -1,1). \\
a( x) <\frac 12b\text{\ for all }x\in ( -1,1) .
\endgathered$$
By the maximum principle every solution satisfies $0<u<b$ in $(
-1,1) $. Thus, they prove:

\begin{theorem}
\label{Thm5}\cite{KormanOuyang1}\ There exists a critical
$\lambda_{1}$, such that for $0<\lambda <\lambda_{1}$ Problem
(\ref{p2}) has no solution; it has at least one solution at
$\lambda =\lambda_{1};$ and it has at least two solutions for
$\lambda >\lambda_{1}$. All solutions lie on a single smooth
curve of solutions. For each $\lambda >\lambda_{1}$ there are
finitely many solutions, and different solutions are strictly
ordered. Moreover, there exists $\lambda_{2}\geq \lambda_{1}$ so
that for $\lambda
>\lambda_{2}$ Problem (\ref{p2}) has exactly two solutions denoted by
$u^{-}( x,\lambda ) <u^+( x,\lambda ) $, and $\lim_{\lambda
\rightarrow \infty }u^+( x,\lambda ) =b$ for all $x\in ( -1,1) $.
Solution $u^{-}( x,\lambda ) $ develops a spike layer at $x=0$ as
$\lambda \rightarrow \infty $.
\end{theorem}

Notice that the cubic nonlinearity $f( x,u) $ in Theorems \ref
{Thm4} and \ref{Thm5} is such that
\begin{equation}
x\frac{\partial f}{\partial x}( x,u) <0\text{\ \ \ for\ \ }%
x\neq 0.  \label{sdgfh}
\end{equation}
Next, Korman and Ouyang \cite{KormanOuyang3} have studied Problem
(\ref{p2}) when the condition (\ref{sdgfh}) is violated. Indeed,
they consider
\begin{equation*}
f( x,u) =( u-a) ( u-b( x) ) ( c( x) -u) \text{,\ \ for }x\in (
-1,1) ,u\in  {\mathbb R} ,
\end{equation*}
$a$ is a constant, $b( x) $ and $c( x) $ are even functions and
of class $C^1( -1,1) \cap C^0[ -1,1] $, satisfy the following
conditions:
\begin{gather}
0<a<b( x) <c( x) \text{\ \ for\ all }x\in (-1,1) ,  \label{sdrt1} \\
c''( x) <0\text{\ \ for all }x\in (-1,1)  \label{sdrt2} \\
b'( x) +c'( x) \geq 0\text{\ \ for all }x\in ( 0, 1)  \label{sdrt3} \\
c'( x) <0\text{\ \ for all }x\in ( 0, 1). \label{sdrt4}
\end{gather}

First, they show that any solution satisfies
\begin{equation*}
0<u( x) <c( x) \text{\ \ for all }x\in (-1,1) ,
\end{equation*}
and prove the following

\begin{theorem}
\label{Thm6}\cite{KormanOuyang3} Assume that $a$, $b( x) $ and $c(
x) $ satisfy (\ref{sdrt1})-(\ref{sdrt4}). Assume in addition that
\begin{equation}
\int_{-1}^1F( x,a) dx<\int_{-1}^1F( x,c( x) ) dx.  \label{A34}
\end{equation}
All solutions of (\ref{p2}) lie on at most countably many
unbounded smooth solution curves. One of the curves, referred to
as the lower curve, starts at $\lambda =0$, $u=0$, it is strictly
increasing in $\lambda $, and $\lim_{\lambda \rightarrow \infty
}u( x,\lambda ) =a$ for all $x\in ( -1,1) $. Each upper curve has
two branches $u^{-}( x,\lambda ) <u^+( x,\lambda ) $, and as
$\lambda \rightarrow \infty $, $u^{-}( x,\lambda ) $ tends to $a$
for all $x\in ( -1,1) \backslash \left\{ 0\right\} $. For $u^+(
x,\lambda ) $ there is a $p\in ( 0, 1) $, such that as $\lambda
\rightarrow \infty $, $u^+( x,\lambda ) $ tends to $c( x) $ for
$x\in ( -p,p) $ and to $a$ for $x\in ( -1,1) \backslash ( -p,p)
$. The number $p$ is the same for all upper curves. Each upper
curve has at most finitely many turns for $\lambda $ belonging to
any bounded interval.
\end{theorem}

Concerning the asymptotic behavior of solutions on the upper
curve, they obtained a detailed information. They consider the
condition
\begin{equation}
F( x,a) <F( x,c( x) ) \text{\ \ for all }x\in ( -1,1) .
\label{A315}
\end{equation}
Denoting $r_1( x) <r_2( x) $ the roots of $\frac{%
\partial f}{\partial x}( x,u) $, they assume that
\begin{equation}
r_2( x) <c( 1) \text{ \ for all }x\in ( -1,1) .  \label{A316}
\end{equation}
They proved the following

\begin{theorem}
\label{Thm7}\cite{KormanOuyang3} Assume all conditions of Theorem
\ref {Thm6} hold with the condition (\ref{A34}) replaced by
(\ref{A315}) and assume additionally (\ref{A316}). Then all of
the conclusions of Theorem \ref {Thm6} hold and, in addition, the
upper curve is unique and it consists for $\lambda $ sufficiently
large of two branches, referred to as an upper and lower branch,
$u^+( x,\lambda ) >u^{-}( x,\lambda ) $ for all $x$, and
$\lim_{\lambda \rightarrow +\infty }u^+( x,\lambda ) =c( x) $ for
all $x\in ( -1,1) $, $\lim_{\lambda \rightarrow +\infty }u^{-}(
x,\lambda ) =a$ for all $x\in ( -1,1) \backslash \left\{
0\right\} $, and $u^{-}( 0,\lambda ) >b( 0) $ for all $\lambda $
(i.e. the lower branch approaches a spike-layer). In particular,
for sufficiently large $\lambda $ Problem (\ref{p2}) has exactly
three solutions.
\end{theorem}

Notice that Theorems \ref{Thm6} and \ref{Thm7} do not provide
exact multiplicity results \emph{for all} $\lambda >0$. Also, in
Theorems \ref {Thm4} and \ref{Thm5}, the exact number of
solutions for $\lambda_1\leq
\lambda \leq \lambda_2$ remains, in \cite{KormanOuyang1}, an open question%
\footnote{%
Concerning Theorem \ref{Thm4}, Korman and Ouyang
\cite{KormanOuyang1} believed, based on numerical evidence, that
at $\lambda =\lambda_1$ the solution is unique, while for
$\lambda >\lambda_1$ there are exactly two solutions.}.

In this direction, Korman and Ouyang \cite{KormanOuyang2} have
studied Problem (\ref{p2}) when the cubic nonlinearity is given by
\begin{equation*}
f( x,u) =u( u-a( x) ) ( b(
x) -u) ,\text{for }x\in ( -1,1) ,\text{and }%
u\in {\mathbb R} ,
\end{equation*}
that is, the nonlinearity has three distinct roots
\begin{equation*}
0<a( x) <b( x) ,\text{ for all }x\in ( -1,1) ,
\end{equation*}
and proved a distinguished result. Indeed they find the
\textit{exact} number of solutions to (\ref{p2}) \textit{for all}
$\lambda >0$. Notice that here the two positive roots may depend
on $x$. They assume that
\begin{equation*}
a( x) \text{ and }b( x) \text{ are even functions of class }C^2(
-1,1) \cap C^0[ -1,1] ,
\end{equation*}
and
\begin{equation}
b''( x) <0\text{\ \ \ \ for all }x\in ( -1,1) .  \label{Hadjera}
\end{equation}
Letting $\alpha ( x) =a( x) +b( x) $ and $\beta ( x) =a( x) b( x)
$, for all $x\in ( -1,1) $ they assume that these even functions
satisfy the conditions
\begin{gather}
\alpha '( x) <0\text{\ \ \ for }x\in ( 0, 1),  \label{Labelle} \\
\beta '( x) >0\text{\ \ \ for }x\in ( 0, 1) , \label{Lacolombe1} \\
\alpha ^{\prime \prime \prime }( x) \leq 0\text{ for all }x\in
( 0, 1) ,  \label{Lacolombe2} \\
\alpha ( x) -\sqrt{\alpha ^2( x) -3\beta (
x) }<\alpha ( 1) \text{\ for all }x\in ( 0, 1) , \label{Lacolombe3} \\
\frac 12\alpha ( 0) <\frac{\alpha ( x) +\sqrt{\alpha ^2( x)
-3\beta ( x) }}3\text{\ \ for all }x\in (0, 1) .
\label{Lacolombe4}
\end{gather}
Here again they noted that any nontrivial solution of (\ref{p2})
is positive by the maximum principle. First they prove an
alternative result:

\begin{theorem}
\label{Theorem3.1}\cite{KormanOuyang2}\ For Problem (\ref{p2})
assume that the conditions (\ref{Hadjera})-(\ref{Lacolombe4}) are
satisfied. Then only two possibilities can occur:

\begin{description}
\item[(A)]  Problem (\ref{p2}) has no nontrivial solution for any
$\lambda >0$.

\item[(B)]  There is a $\lambda_{0}>0$ so that Problem (\ref{p2}) has
either zero, one, or two solutions depending on whether $\lambda
<\lambda _{0}$, $\lambda =\lambda_{0}$, or $\lambda
>\lambda_{0}$, respectively. Moreover, all solutions are even
functions and lie on a single $\subset $-like curve. Solutions on
the lower branch tend to zero over $( -1,1) \backslash \left\{
0\right\} $, and moreover the maximum value of solutions on the
lower branch decreases monotonously.
\end{description}
\end{theorem}

Next, they give a condition ensuring existence of a positive solution of (%
\ref{p2}) for some $\lambda >0$, thus they obtained an exact
multiplicity result for all $\lambda >0$.

\begin{theorem}
\cite{KormanOuyang2}\ In addition to the conditions of Theorem
\ref {Theorem3.1} assume that $\int_{0}^1F( x,b( x) ) dx>0,$ where
$F( x,u) =\int_{0}^{u}f( x,t) dt$. Then, Possibility (B) of
Theorem \ref{Theorem3.1} holds. If moreover $ F( x,b( x) )
>0\text{\ \ for\ all }x\in ( -1,1) $ then the upper branch tends to $b( x) $ over $( -1,1) $\ as
$\lambda \rightarrow \infty $.
\end{theorem}

Observe that all the results described from the beginning of this
section are concerned by polynomial cubic nonlinearities. So, it
is interesting to have some description of the solution set of
(\ref{p2}) when $f$ behaves like a cubic nonlinearity but is not
being given by formula.

In this direction Korman et al. \cite{KormanEtAl} have studied
the solution set of (\ref{p2}) when $f$ is a cubic-like
nonlinearity in $u$ and they provide exact multiplicity results
for all $\lambda >0$.

First, they consider an autonomous case, and assume that $f$ has
three distinct roots $a<b<c$ and they provide two results. The
first one concerns the case where the least root $a$ is equal to
zero. They assume that $f=f( u) \in C^2({\mathbb R}) $ has the
following properties
\begin{gather}
f( 0) =f( b) =f( c) =0\ \ \text{for some } 0<b<c,  \label{Austine1} \\
f( x) >0\text{\ \ for }x\in ( -\infty ,0) \cup ( b,c)  \label{Austine2} \\
f( x) <0\text{\ \ for }x\in ( 0,b) \cup (c,+\infty )  \label{Austine3}\\
\int_0^cf( u) du>0  \label{Austine4} \\
f''( u) \text{ changes sign exactly once when }u>0, \label{Austine5} \\
f''( u) \text{ has exactly one positive root.} \label{Austine6}
\end{gather}
Thus, they prove

\begin{theorem}
\label {KormanShi12} \cite{KormanEtAl}\ Under the conditions
(\ref{Austine1})-(\ref{Austine6}) there is a critical
$\lambda_{0}>0$ such that for $\lambda <\lambda_{0}$ Problem
(\ref{p2}) has no nontrivial solutions, it has exactly one
nontrivial solution for $\lambda =\lambda_{0}$, and exactly two
nontrivial solutions for $\lambda >\lambda_{0}$. Moreover, all
solutions lie on a single curve, which for $\lambda >\lambda_{0}$
has two branches denoted by $u^{-}( x,\lambda ) <u^+( x,\lambda )
$, with $u^+( x,\lambda ) $ strictly monotone increasing in
$\lambda $, $u^{-}( 0,\lambda ) $ strictly monotone decreasing in
$\lambda $, and $\lim_{\lambda \rightarrow \infty }u^+( x,\lambda
) =c $, $\lim_{\lambda \rightarrow \infty }u^{-}( x,\lambda ) =0$
for $x\in ( -1,1) \backslash \left\{ 0\right\} $, while $u^{-}(
0,\lambda ) >b$ for all $\lambda >\lambda_{0}$.
\end{theorem}

The second result in the autonomous case concerns the case where
the least root $a$ is strictly positive. They consider the problem
\begin{equation} \gathered
-u'' =\lambda f( u-a) ,\text{\ \ in }(-1,1)   \\
u( -1) =u( 1) =0  \endgathered \label{KO}
\end{equation}
where $a$ is a positive constant and $f$ satisfies
(\ref{Austine1})-(\ref {Austine4}) and
\begin{gather}
\text{for }u>0\text{, }f''( u-a) \text{ changes
sign exactly once}  \label{Austine7} \\
\ \text{and has exactly one root.}  \notag
\end{gather}
Also, they assume an additionally condition which is
\begin{equation}
f( \beta ) \beta -2[ F( \beta ) -F( -a) ] \geq 0  \label{Austine8}
\end{equation}
where $\beta $ is the unique solution of $f'( \beta ) =\frac{f(
\beta ) }\beta $. Thus, they prove:

\begin{theorem}
\cite{KormanEtAl}\ Consider Problem (\ref{KO}) with $f( u) $
as described by (\ref{Austine1})-(\ref{Austine4}), (\ref{Austine7}), and (%
\ref{Austine8}). Then there exists a critical $\lambda_{0}$ such
that for Problem (\ref{KO}) there exists exactly one positive
solution for $0<\lambda <\lambda_{0}$, exactly two positive
solutions for $\lambda =\lambda_{0}$, and exactly three positive
solutions for $\lambda >\lambda _{0}$. Moreover, all solutions
lie on two smooth in $\lambda $ solution curves, all different
solutions of (\ref{KO}) at the same $\lambda $ are strictly
ordered on $( -1,1) $. One of the curves, referred to as the
lower curve, starts at $\lambda =0$, $u=0$, it is strictly
increasing in $\lambda $, and $\lim_{\lambda \rightarrow \infty
}u( x,\lambda ) =a$. The upper curve is a parabola-like curve,
consisting of two branches $u^{-}( x,\lambda ) <u^+( x,\lambda )
$. The upper branch is monotone increasing in $\lambda $ and
$\lim_{\lambda \rightarrow \infty }u^+( x,\lambda ) =a+c$ for all
$x\in ( -1,1) $. The lower branch approaches a spike-layer, namely
$\lim_{\lambda \rightarrow \infty }u^{-}( x,\lambda ) =a$ for all
$x\in ( -1,1) \backslash \left\{ 0\right\} $, while $u^{-}(
x,\lambda ) >a+b$ for all $\lambda >\lambda_{0}$.
\end{theorem}

In the same paper \cite{KormanEtAl} they provide an exact
multiplicity result for all $\lambda >0$ in an non-autonomous
case. They consider Problem (\ref{p2}) with
\begin{equation*}
f( x,u) =u^2( b( x) -u) ,\text{ for }%
x\in ( -1,1) ,u\in {\mathbb R},
\end{equation*}
and assume that the positive function $b( x) \in C^3[ -1,1] $
satisfies the following conditions:
\begin{gather}
b( -x) =b( x) \text{\ \ \ for all }x\in [-1,1] ,  \label{e32} \\
b'( x) <0\text{\ \ \ for all }x\in ( 0, 1] , \label{e33}\\
b''( x) <0\text{\ \ \ for all }x\in (0, 1] ,  \label{e34} \\
b^{\prime \prime \prime }( x) \leq 0\text{\ \ \ for all }x\in
( 0, 1] ,  \label{e35} \\
b( 1) \geq \frac 12b( 0) >0.  \label{e36}
\end{gather}
They provide an example of a function $b( x) $ satisfying
conditions (\ref{e32})-(\ref{e36}) by $b( x) =a-x^2$ with
constant $a\geq 3$ and they show that any nontrivial solution of
(\ref{p2}) satisfies $0<u( x) <b( x) \text{\ \ \ for all }x\in (
-1,1) $. Next, they prove:

\begin{theorem}
\cite{KormanEtAl}\ Under the conditions (\ref{e32})-(\ref{e36})
there is a critical $\lambda_{0}>0$
such that for $\lambda <\lambda_{0}$ Problem (%
\ref{p2}) has no nontrivial solutions, it has exactly one
nontrivial solution for $\lambda =\lambda_{0}$, and exactly two
solutions which for $\lambda >\lambda_{0}$ has two branches
denoted by $u^{-}( x,\lambda ) <u^+( x,\lambda ) $, with $u^+(
x,\lambda ) $ strictly monotone increasing in $\lambda $, $u^{-}(
x,\lambda ) $ strictly monotone decreasing in $\lambda $, and
$\lim_{\lambda \rightarrow \infty }u^+( x,\lambda ) =b( x) $,\
$\lim_{\lambda \rightarrow \infty }u^{-}( x,\lambda ) =0$\ for
all $x\in ( -1,1) $.
\end{theorem}

Korman and Shi  prove an exact multiplicity result which
generalize Theorem \ref{KormanShi12} by weakening the convexity
assumptions on $f$.


\begin{theorem}
\cite{KormanShi}\ Suppose $f\in C^{2}[0,\;\infty )$, $f\left(
0\right) =0,$ $f\left( x\right) <0$\ for $x\in \left( 0,b\right)
\cup \left( c,\infty \right) ,$ and $f\left( x\right) >0$\ for
$x\in \left( b,c\right) ,$ where $c>b>0$. Assume that for some
$c>\eta >\gamma
>b$ we have, $f^{\prime \prime }\left( u\right) >0\text{\ for }0<u<\gamma
,$ $f^{\prime \prime }\left( u\right) <0\text{\ for }\gamma
<u<\eta ,$ $2F(\eta )-\eta f(\eta )>0,$ $f(u)-uf^{\prime
}(u)>0\text{ for all }u>\eta .$ Then there is a critical $\lambda
_{0}>0$ such that for $0<\lambda <\lambda _{0}$ Problem (%
\ref{p2}) has no nontrivial solution, it has exactly one solution
for $\lambda =\lambda _{0}$, and exactly two solutions for
$\lambda >\lambda _{0}.$ Moreover, all solutions lie on a unique
smooth solution curve.
\end{theorem}


In an other paper, the same authors, Korman et al.
\cite{KormanEtAl2} have extended the previous result to the case
where the dimension space is two. They consider the problem
\begin{equation} \gathered
\Delta u+\lambda f( u) = 0\;\;\text{ in }\left| x\right| <R \\
u =0\text{\ \ on\ }\left| x\right| =R  \endgathered\label{atf11}
\end{equation}
on a ball in two dimensions, i.e. $x=( x_1,x_2) $. They assume
that $f\in C^2({\mathbb R}) $ has the following properties
\begin{gather}
f( 0) =f( b) =f( c) =0\text{\ \ for some constants }0<b<c,  \label{atf21} \\
f( u) <0\text{\ \ for }u\in ( 0,b) \cup (c,\infty )  \label{atf22} \\
f( u) >0\text{\ \ for }u\in ( -\infty ,0) \cup( b,c)  \notag \\
f'( 0) <0  \label{atf23} \\
\int_0^cf( u) du>0  \label{atf24}
\end{gather}
There exists $\alpha \in ( 0,c)$  such that
\begin{equation} \gathered
f''( u) >0\text{\ for }u\in ( 0,\alpha) \text{\ \ and}  \\
f''( u) <0\text{\ for }u\in ( \alpha ,c) .  \endgathered
\label{atf25}
\end{equation}
Also, letting
\begin{equation*}
g_\mu ( u) =\mu ( f'( u) u-f(u) ) -2f( u) ,\mu \in ( 0,\infty ) ,
\end{equation*}
they assume that
\begin{gather}
\text{The function }g_\mu ( s) \text{ can have at most one sign
change when}  \label{atf27} \\
s\in ( 0,c) \text{ for any value of the parameter }\mu \in (
0,\infty ).  \notag
\end{gather}
The final condition on the function $f( u) $ is
\begin{equation}
( f') ^2u-f'f-ff''u>0\text{\ \ for }b<u<\beta  \label{atf221}
\end{equation}
where $\beta$ is the unique solution of the equation $f'( \beta )
=f( \beta ) /\beta $.

Condition (\ref{atf22}) implies that $f( u) >0$ for $u<0$.
Therefore by the maximum principle, all solutions of
(\ref{atf11}) are positive, hence by a well-known result of
Gidas, Ni and Nirenberg \cite{GNN} they are radially symmetric.
Also, by a result of Lin and Ni \cite{LN} all solutions of the
linearized equation
\begin{equation} \gathered
\Delta w+\lambda f'( u) w =0\text{\ \ in\ }\left|
x\right| <R   \\
w =0\text{\ \ on\ }\left| x\right| =R  \endgathered\label{atfL}
\end{equation}
are also radially symmetric. Therefore, the authors were lead to
study the ODE version of (\ref{atf11}). Also, without lost of
generality, they take the unit ball; $R=1$, and consider in two
dimensions
\begin{gather}
u''( r) +\frac 1ru'( r) +\lambda f'( u) =0,r\in ( 0, 1)
\label{atf31} \\
u'( 0) =u( 1) =0  \notag
\end{gather}
and prove the following

\begin{theorem}
\label{Thm8}\cite{KormanEtAl2} Assume that $f( u) $ satisfies
assumption (\ref{atf27}) and the conditions (\ref{atf21}-\ref{atf25}) and (%
\ref{atf221}). Then there is a critical $\lambda_{0}>0$, such
that for $\lambda <\lambda_{0}$ Problem (\ref{atf31}) has no
nontrivial solution, it has exactly one nontrivial solution for
$\lambda =\lambda_{0} $, and exactly two nontrivial solutions for
$\lambda >\lambda_{0}$. Moreover, all solutions lie on a single
smooth solution curve, which for $\lambda >\lambda_{0}$ has two
branches denoted by $0<u^{-}( r,\lambda ) <u^+( r,\lambda ) $,
with $u^+( r,\lambda ) $ strictly monotone increasing in $\lambda
$ and $\lim_{\lambda \rightarrow \infty }u^+( r,\lambda ) =c$ for
$r\in [ 0, 1) $. For the lower branch, $\lim_{\lambda \rightarrow
\infty }u^{-}( r,\lambda ) =0$ for $r\neq 0$, while $u^{-}(
0,\lambda ) >\gamma $ for $\lambda >\lambda_{0}$, where $\gamma $
is the unique number $\in ( b,c) $ such that $\int_{0}^{\gamma
}f( u) du=0$. (Recall that any nontrivial solution is positive by
the maximum principle.)
\end{theorem}

Notice that Theorem \ref{Thm8} deals with the exact multiplicity
solutions of (\ref{atf31}) for all $\lambda >0$ but with the
restriction to two dimensions. Thus it would be interesting to
know what happens in higher dimensions. The main reason which
makes Theorem \ref{Thm8} holds only in two dimensions was proving
positivity of any nontrivial solution of the linearized Problem
(\ref{atfL}). This difficult task was recently overcome by Ouyang
and Shi \cite{OuyangShi} by using Pohozhaev type identity.
 Ouyang and Shi \cite{OuyangShi} consider
\begin{equation} \gathered
u^{\prime \prime }\left( r\right) +\frac{n-1}{r}u^{\prime }\left(
r\right) +\lambda f\left( u\right) =0,\;r\in \left( 0,1\right)
,\text{ }n\geq 1, \label{123} \\
u^{\prime }\left( 0\right)=u\left( 1\right) =0.
\endgathered\end{equation}
They assume that $f\in C^{2}\left( {\mathbb R}_{+}\right) $
satisfies the following properties:
\begin{equation}
f\left( 0\right) \leq 0,\text{ }f\left( b\right) =f\left(
c\right) =0\text{\ for some constants }0<b<c, \label{124}
\end{equation}
\begin{eqnarray}
f( u) &<&0\text{\ \ for }u\in ( 0,b) \cup (
c,\infty )  \label{125} \\
f( u) &>&0\text{\ \ for }u\in ( -\infty ,0) \cup ( b,c) ,  \notag
\end{eqnarray}
\begin{equation}
\int_0^cf( u) du>0,  \label{126}
\end{equation}
\begin{gather}
\text{There exists }\alpha \in ( 0,c) \text{, such that}
\label{127} \\
f''( u) >0\text{ for }u\in ( 0,\alpha ) \text{\ and }f''( u)
<0\text{ for }u\in ( \alpha ,c) .  \notag
\end{gather}
Let $\theta $ be the smallest positive number such that
$\int_0^\theta
f( s) ds=0$ and $\rho =\alpha -\frac{f( \alpha ) }{%
f'( \alpha ) }$. Clearly, $\theta \in ( b,c) $. Define $K( u)
=\dfrac{uf'( u) }{f( u) }$. If $\theta <\rho $, They assume that
\begin{gather}
K( u) >K( \theta ) \text{\ \ on }( b,\theta
)  \label{128} \\
K( u) \text{\ is non increasing on }( \theta ,\rho )
\notag \\
K( u) <K( \rho ) \text{ on }( \rho ,\alpha ) .  \notag
\end{gather}
(If $\theta \geq \rho $ this condition is empty.) Next, they
prove the following

\begin{theorem}
\cite{OuyangShi} Assume that $f( u) $ satisfies the conditions
listed above.

{\bf{(a)}} If $f(0)=0,$ there exists a critical $\lambda _{0}>0$,
such that for $0<\lambda <\lambda _{0}$ Problem (\ref{123}) has
no nontrivial solution, it has exactly one nontrivial solution
for $\lambda =\lambda _{0}$, and exactly two nontrivial solutions
for $\lambda >\lambda _{0}.$ Moreover, all solutions lie on a
single smooth solution curve $\Sigma $, which for $\lambda
>\lambda _{0}$ has two branches denoted by $\Sigma ^{+}$ (the
upper branch) and $\Sigma ^{-}$ (the lower branch); $\Sigma ^{+}$
continues to the right up to $(\infty ,c);$ $\Sigma ^{-}$
continues to the right down to $(\infty ,g)$ for some $g\geq
\beta ;$ there exists a unique turning point on the curve, the
curve bend to the right at the turning point. (See Fig 12 in
\cite{OuyangShi}.)

 {\bf{(b)}} If $f(0)<0,$ there exist
$\bar{\lambda}>\lambda _{0}>0$, such that for $0<\lambda <\lambda
_{0}$ Problem (\ref{123}) has no solution, it has exactly one
solution for $\lambda >\bar{\lambda}$ or $\lambda =\lambda _{0}$,
and exactly two solutions for $\lambda _{0}<\lambda \leq
\bar{\lambda}.$ Moreover, all solutions lie on a single smooth
solution curve $\Sigma $, which for $\lambda >\lambda _{0}$ has
two branches denoted by $\Sigma ^{+}$ (the upper branch) and
$\Sigma ^{-}$ (the lower branch); $\Sigma ^{+}$ continues to the
right up to $(\infty ,c);$ $\Sigma ^{-}$ continues to the right
down to $(\bar{\lambda},g)$ for some $g\geq \beta ;$ there exists
a unique turning point on the curve, the curve bend to the right
at the turning point. (See Fig. 10 in \cite{OuyangShi}.)
\end{theorem}


More recently Korman \cite{KormanPreprint} was able to avoid
having to prove this positivity condition, and replaced it by an
indirect argument. He shows that it is sufficient to prove that
any nontrivial solution of (\ref{atfL}) cannot vanish exactly
once. This way he considerably simplifies the proof of Ouyang and
Shi and make it more elegant.

 Korman \cite{KormanPreprint} considers Problem (\ref{atf11}) for $n\geq 1.$ He assumes that
$f\in C^2( \overline{ {\mathbb R}}_+) $, $f\left( 0\right) =0$
and satisfies (\ref{124})-(\ref{128}). Next, he proves the
following

\begin{theorem}
\cite{KormanPreprint} Assume that $f( u) $ satisfies the
conditions listed above. For Problem (\ref{atf11}) there is a
critical $\lambda_{0}>0$ such that Problem (\ref{atf11}) has
exactly $0$, $1$ or $2$ nontrivial solutions, depending on
whether $\lambda <\lambda_{0}$, $\lambda =\lambda_{0}$ or
$\lambda >\lambda_{0}$. Moreover, all solutions lie on a single
smooth solution curve, which for $\lambda >\lambda_{0}$ has two
branches denoted by $0<u^{-}( r,\lambda ) <u^+( r,\lambda ) $,
with $u^+( r,\lambda ) $ strictly monotone increasing in $\lambda
$ and $\lim_{\lambda \rightarrow \infty }u^+( \rho ,\lambda ) =c$
for $r\in [ 0, 1) $. For the lower branch $\lim_{\lambda
\rightarrow \infty }u^{-}( r,\lambda ) =0$ for $r\neq 0$, while
$u^{-}( 0,\lambda ) >b$ for all $\lambda >\lambda_{0}$.
\end{theorem}

Other related results are available in the literature. (See for
instance, Pimbley, \cite{Pimbley}, Wang and Kazarinoff
\cite{WangKaza1}, \cite {WangKaza2}, Schaaf \cite{Schaaf}, Korman
\cite{Korman97}, Shi and Shivaji \cite{ShiShivaji}). But as we
have indicated at the beginning of this section, we shall not
attempt to make a complete historical review. Thus, we apologize
to all authors whose results are close to cubic-like
nonlinearities and which have not been either described in this
section or listed in the references.

\section{{The method used}}

\label{sec.3}We shall make use of the quadrature method. Denote
by $g$ a nonlinearity and by $p$ a real parameter, and we assume,
\begin{equation}
g\in C({\mathbb R} ) \text{\ \ and\ \ \ }1<p<+\infty ,
\label{meth2}
\end{equation}
and consider the boundary value problem,
\begin{equation}
-( \left| u'\right| ^{p-2}u') ^{\prime }=g( u) \ \ \ \text{in\ }(
0, 1) ,\;u( 0) =u( 1) =0.  \label{meth3}
\end{equation}
Denote by $p'=p/( p-1) $ the conjugate exponent of $p$. Let $G(
s) =\int_{0}^{s}g( t) dt$. For any $E\geq 0$ and $\kappa =+,-$,
let,
\begin{equation*}
X_{\kappa }( E) =\left\{ s\in {\mathbb R} :\kappa s>0\text{ and
}E^{p}-p'G( \xi ) >0,\forall \,\xi ,0<\kappa \xi <\kappa
s\right\} \;\text{and,}
\end{equation*}
\begin{equation*}
r_{\kappa }( E) =0,\ \text{if }X_{\kappa }( E) =\emptyset \ \ \ \
\text{and}\ \ \ \ r_{\kappa }( E) =\kappa \sup ( \kappa X_{\kappa
}( E) ) \;\;\text{otherwise.}
\end{equation*}
Let
\begin{equation*}
\tilde{D}_{\kappa }=\left\{ E\geq 0:0<\left| r_{\kappa }( E)
\right| <+\infty \text{ and }\kappa \int_{0}^{r_{\kappa }( E) }(
E^{p}-p'G( t) ) ^{\frac{-1}{p}}dt<+\infty \right\}
\end{equation*}
and $\tilde{D}=\tilde{D}_+\cap \tilde{D}_-$. Also, let $\tilde{D}%
_{k}^{\kappa }:=\tilde{D}$ if $k\geq 2$, and $\tilde{D}_{1}^{\kappa }:=%
\tilde{D}_{\kappa }$. Define the following time-maps,
\begin{equation*}
T_{\kappa }( E) =\kappa \int_{0}^{r_{\kappa }( E) }( E^{p}-p'G(
t) ) ^{\frac{-1}{p}}dt,E\in \tilde{D}_{\kappa }.
\end{equation*}
\begin{equation*}
\begin{array}{rclcc}
T_{2n}^{\kappa }( E) & = & n( T_+( E)
+T_-( E) ) , & n\in  {\mathbb N}, & E\in \tilde{D}, \\
T_{2n+1}^{\kappa }( E) & = & T_{2n}^{\kappa }( E) +T_{\kappa }(
E) , & n\in  {\mathbb N}, & E\in \tilde{D}.
\end{array}
\end{equation*}

\begin{theorem}[Quadrature method]
\label{quad} Assume that (\ref{meth2}) holds. Let $E>0$, $\kappa
=+,-$. Then, Problem (\ref{meth3}) admits a solution
$u_{k}^{\kappa }\in A_{k}^{\kappa }$ satisfying $( u_{k}^{\kappa
}) '( 0) =\kappa E$ if and only if $E\in \tilde{D}_{k}^{\kappa
}-\left\{ 0\right\} $ and $T_{k}^{\kappa }( E) =(1/2),$\ and in
this case the solution is unique.
\end{theorem}

\begin{remark}
\label{Remark}In practice, to compute $\tilde{D}_{\kappa }$ we
first compute the set
\begin{equation*}
D_{\kappa }=\left\{ E>0:0<\left| r_{\kappa }( E) \right| <+\infty
\text{ and }\kappa g( r_{\kappa }( E) ) >0\right\}
\end{equation*}
and then we deduce $\tilde{D}_{\kappa }$ by observing that,
$D_{\kappa }\subset \tilde{D}_{\kappa }\subset
\overline{D}_{\kappa }$.
\end{remark}

\section{Some preliminary lemmas}

\label{sec.4}To apply Theorem \ref{quad} we have, first, to
determine the definition domains $\tilde{D}_+$ and $\tilde{D}_-$
of the time-maps $T_+$ and $T_-$ respectively. Lemma \ref{lemma1}
is used to. Next, we have to compute $\tilde{D}=\tilde{D}_+\cap
\tilde{D}_-$ which is the definition domain of the time-maps
$T_{k}^{\kappa }$ for all $k\geq 2$, $\kappa =+,-$. This is done
in Lemma \ref{lemma222}. Next, the aim of Lemma \ref{lemma5} is
to compare the maximum and minimum of any solution of our
problem. This comparison may be used subsequently. Next, we
define all the time-maps and make some useful transformations.
Lemmas \ref{lemma2}, \ref{lemma31}, and \ref{lemma32} are
dedicated to the computation of the limits of all the time-maps,
at each boundary point of their domains. In Lemma \ref{lemma4} we
show that under some appropriate conditions, the time-maps may be
monotonic increasing. The following step is crucial; when the
time maps are not monotonic, usually one
tries to compare there maximum and/or minimum with the real number $\frac{1}{%
2}$. This task is much simpler when $f$ is odd. Indeed, the
time-maps $T_+$ and $T_-$ are always equal if $f$ is odd. Thus;
(*) $T_{k}^{\pm }=kT_+$ for all $k\geq 2$. This way, to study the
maximum and/or minimum value of $T_{k}^{\pm }$ it suffices to
handle those of $T_+$ only. Unfortunately, in our \textit{p.h.o.}
case it seems that the identity (*) is not satisfied. To overcome
this difficulty we define two maps such that both $T_+$ and $T_-
$ are bounded from below by the first one, and from above by the
second one. Thus, $T_{k}^{\pm }$ is bounded from below by $k$
times the first map and from above by $k$ times the second one.
So, it suffices to study the two bounding maps. Moreover, these
estimates seem to be optimal in the sense that in the particular
odd case, they imply that $T_+$ and $T_- $ are equal!

In Lemma \ref{lemma6} we compare the time-maps $T_+$ and $T_-$
with the defined two maps. In Lemma \ref{lemma7} we deduces the
estimates of $T_{k}^{\pm }$ for $k\geq 2$. In Lemma \ref{lemma75}
we provide an identity which may be used in the sequel. This
identity seems to be interesting for its own right and motivates
us to ask a question in Section \ref{sec.6}. Lemma \ref{lemma8}
is dedicated to the limits of these two bounding maps.

Under appropriate conditions we provide, in Lemma \ref{lemma9},
estimates which are used to prove uniqueness of the minimum of
the time-maps $T_+$\ and $T_-$ respectively. This kind of
estimates was introduced by Smoller and Wasserman
\cite{SmollerWasserman} and was crucial in their study of
uniqueness of critical points.


\begin{lemma}
\label{lemma1}Consider the function defined in ${\mathbb R} ^{\pm
}$ by,
\begin{equation}
s\longmapsto G_{\pm }( \lambda ,E,s) :=E^{p}-p^{\prime }\lambda
F( s) ,  \label{Eq.1}
\end{equation}
where $E,\lambda >0$ and $p,q>1$ are real parameters, $F( s)
=\int_{0}^{s}f( t) dt$. Assume that (\ref{pq1})-(\ref{pq4}) hold.
Then,

\begin{description}
\item[(i)]  If $E>E_{*}^{\pm }( p,\lambda ) :=( p^{\prime
}\lambda F( \alpha_{\pm }) ) ^{1/p}$, the function $G_{\pm }(
\lambda ,E,\cdot ) $ is strictly positive in ${\mathbb R} ^{\pm
}$.

\item[(ii)]  If $E=E_{*}^{\pm }( p,\lambda ) $, the function
$G_{\pm }( \lambda ,E,\cdot ) $ is strictly positive in $(
0,\alpha_+) $ (resp. in $( \alpha_-,0) $) and vanishes at
$\alpha_{\pm }$.

\item[(iii)]  If $0<E<E_{*}^{\pm }( p,\lambda ) $, the function
$G_{\pm }( \lambda ,E,\cdot ) $ admits in the open interval $(
0,\alpha_+) $ (resp. $( \alpha_-,0) $) a unique zero $s_{\pm }(
\lambda ,E) $ and is strictly positive in the open interval $(
0,s_+( \lambda ,E) ) $ (resp. $(s_-( \lambda ,E) ,$\ $0)$).
Moreover,

\begin{description}
\item[(a)]  The function $E\mapsto s_{\pm }( \lambda ,E) $ is
$C^1$ in $( 0,E_{*}^{\pm }( p,\lambda ) ) $ and,
\begin{equation}
\pm \frac{\partial s_{\pm }}{\partial E}( \lambda ,E) =\frac{%
\pm ( p-1) E^{p-1}}{\lambda f( s_{\pm }( \lambda ,E) ) }>0,
\label{der}
\end{equation}
for all $E\in ( 0,E_{*}^{\pm }( p,\lambda ) ) $.

\item[(b)]  $\lim\limits_{E\rightarrow 0^+}s_{\pm }( \lambda
,E) =0$ \ and\ $\lim\limits_{E\rightarrow E_{*}^{\pm }}s_{\pm }(
\lambda ,E) =\alpha_{\pm }$,

\item[(c)]  $\lim\limits_{E\rightarrow 0^+}\left| s_{\pm }( \lambda
,E) \right| /E=\left\{
\begin{array}{lcc}
+\infty  & \text{if} & q-p>0 \\
( \dfrac{p-1}{\lambda a_{0}}) ^{1/p} & \text{if} & q-p=0 \\
0 & \text{if} & q-p<0,
\end{array}
\right. $

\item[(d)]  $\lim\limits_{E\rightarrow 0^+}F( \kappa s_{\eta }(
\lambda ,E) \xi ) /E^{p}=\xi ^{q}/( \lambda p^{\prime })
$,\newline for all $\xi >0$ and all $( \kappa ,\eta ) \in \left\{
+,-\right\} ^{2}$.
\end{description}
\end{description}
\end{lemma}

\paragraph{Proof.} For any fixed $p>1$ and $E\geq 0$, consider the function,
\begin{equation}
s\longmapsto G_{\pm }( \lambda ,E,s) :=E^{p}-p^{\prime }\lambda
F( s) ,  \label{Eq1}
\end{equation}
defined in ${\mathbb R}^{\pm }$. One has, $\frac{dG_{\pm }}{ds}(
\lambda ,E,s) =-p'\lambda f( s) $. Hence, according to
(\ref{pq2}), $G_+( \lambda ,E,\cdot )$ (resp. $G_-( \lambda
,E,\cdot )$) is strictly decreasing in $( 0,\alpha_+) $ (resp. in
$( -\infty ,\alpha _-) $) and according to (\ref{pq3}), it is
strictly increasing in $( \alpha_+,+\infty ) $ (resp. in $(
\alpha_-,0) $). Moreover, according to (\ref{pq1}), $
\frac{dG_{\pm }}{ds}( \lambda ,E,\alpha_{\pm }) =0.$ Therefore,
it follows that $G_{\pm }( \lambda ,E,\cdot ) $ is strictly
positive in ${\mathbb R}^{\pm }$ for all $E>E_{*}^{\pm }:=(
p'\lambda F( \alpha _{\pm }) ) ^{1/p}$, admits a unique positive
(resp. negative) zero, $\alpha_{\pm }$, and is strictly positive
in $( 0,\alpha +) $ (resp. in $( \alpha_-,0) $) at $E=E_{*}^{\pm
}$, and finally admits a first positive (resp. negative) zero
$s_{\pm }=s_{\pm }( \lambda ,E) $ and is strictly positive in $(
0,s_+) $ (resp. in $( s_-,0) $) for all $E:0<E<E_{*}^{\pm }$,
moreover $\left| s_{\pm }\right| <\left| \alpha_{\pm }\right| $.

\paragraph{Proof of (a)}
For any $p>1$ and $\lambda >0$, consider the real valued function,
\begin{equation*}
( E,s) \longmapsto G_{\pm }( E,s) :=E^{p}-p'\lambda F( s) ,
\end{equation*}
defined in $\Omega_+=( 0,E_{*}^+) \times ( 0,\alpha _+) $ (resp.
$\Omega_-=( 0,E_{*}^{-}) \times ( \alpha_-,0) $). One has $G_{\pm
}\in C^1( \Omega_{\pm }) $ and, $ \frac{\partial G_{\pm
}}{\partial s}( E,s) =-p'\lambda f( s) \text{\ \ in }\Omega_{\pm
}, $ hence, according to (\ref{pq2}) (resp. (\ref{pq3})), it
follows that, $ \pm \frac{\partial G_{\pm }}{\partial s}( E,s)
<0\text{\ \ \ in\ }\Omega_{\pm }, $ and one may observe that
$s_{\pm }( \lambda ,E) $ belongs to the open interval $(
0,\alpha_+) $ (resp. $( \alpha _-,0) $) and satisfies, from its
definition,
\begin{equation}
G_{\pm }( E,s_{\pm }( \lambda ,E) ) =0. \label{Eq.2}
\end{equation}
So, one can make use of the implicit function theorem to show
that the function $E\mapsto s_{\pm }( \lambda ,E) $ is $C^1(
( 0,E_{*}^{\pm }) ,%
{\mathbb R}
) $ and to obtain the expression of $\dfrac{\partial s_{\pm }}{%
\partial E}( \lambda ,E) $ given in \textbf{(a)}. Its sign is
given by (\ref{pq2}) (resp. (\ref{pq3}) together with the fact
that $s_{\pm }( \lambda ,E) $ belongs to $( 0,\alpha_+) $ (resp.
$( \alpha_-,0) $). Therefore, Assertion \textbf{(a)} is proved.

\paragraph{Proof of (b)}

For any fixed $p>1$ and $\lambda >0$, Assertion \textbf{(a)} of
the current lemma implies that the function defined in $(
0,E_{*}^{\pm }) $ by $E\mapsto s_{\pm }( \lambda ,E) $ is
strictly increasing (resp. strictly decreasing). It is bounded
from below by $0$ (resp. by $\alpha_-$) and from above by
$\alpha_+$ (resp. by $0$). Then, the limits
$\lim\limits_{E\rightarrow 0^+}s_{\pm }( \lambda ,E) =\ell_0^{\pm
}$ and $\lim\limits_{E\rightarrow E_{*}^{\pm }}s_{\pm }( \lambda
,E) =\ell_{*}^{\pm }$) exist as real numbers and moreover,
\begin{equation*}
\alpha_-\leq \ell_{*}^{-}<\ell_0^{-}\leq 0\leq \ell_0^+<\ell
_{*}^+\leq \alpha_+.
\end{equation*}

Let us observe that, for any fixed $p>1$ and $\lambda >0$, the
function, $( E,s) \mapsto G_{\pm }( E,s) $, is continuous in $[
0,E_{*}^+] \times [ 0,\alpha_+] $ (resp. in $[ 0,E_{*}^{-}]
\times [ \alpha_-,0] $) and the function $E\mapsto s_{\pm }(
\lambda ,E) $ is continuous in $( 0,E_{*}^{\pm }) $ and satisfies
(\ref{Eq.2}). So, by passing to the limit in (\ref{Eq.2}) as $E$
tends to $0^+$, one gets, $ 0=\lim_{E\rightarrow 0^+}G_{\pm }(
E,s_{\pm }( \lambda ,E) ) =G_{\pm }( 0,\ell_0^{\pm }) .  $ Hence,
$\ell_0^{\pm }$ is a zero, belonging to $[ 0,\alpha _+] $, (resp.
$[ \alpha_-,0] $) to the equation in $s, $ $ G_{\pm }( 0,s) =0. $
By resolving this equation one gets: $\ell_0^{\pm }=0$. Also, by
passing to the limit in (\ref{Eq.2}) as $E$ tends to $E_{*}^{\pm
}$, one gets,
\begin{equation*}
0=\lim_{E\rightarrow E_{*}^{\pm }}G_{\pm }( E,s_{\pm }( \lambda
,E) ) =G_{\pm }( E_{*}^{\pm },\ell_0^{\pm }) .
\end{equation*}
Hence, $\ell_{*}^{\pm }$ is a zero, belonging to $( 0,\alpha _+]
$, (resp. $[ \alpha_-,0) $) to the equation in $s,
$\begin{equation*} G_{\pm }( E_{*}^{\pm },s) =0.
\end{equation*}
By resolving this equation one gets: $\ell_{*}^{\pm }=\alpha_{\pm
}$. Therefore, Assertion \textbf{(b)} follows.

\paragraph{Proof of (c)}

Let $\Phi_{q}( s) =\int_{0}^{s}\varphi ( t) dt=(1/q)\left|
s\right| ^{q}$. Observe that from the definition of $s_{\pm }(
\lambda ,E) $ one has
\begin{equation}
E^{p}=p'\lambda F( s_{\pm }( \lambda ,E) ) , \label{dklg}
\end{equation}
hence, dividing by $\left| s_{\pm }( \lambda ,E) \right| ^{p}$,
using l'Hopital's rule and (\ref{pq4}) one gets,
\begin{equation*}
\begin{array}{ccl}
\lim\limits_{E\rightarrow 0^+}E^{p}/\left| s_{\pm }( \lambda
,E) \right| ^{p} & = & \lim\limits_{E\rightarrow 0^+}\dfrac{%
p'\lambda }{q}\left| s_{\pm }( \lambda ,E) \right|
^{q-p}\dfrac{F( s_{\pm }( \lambda ,E) ) }{\Phi
_{q}( s_{\pm }( \lambda ,E) ) } \\
& = & \dfrac{p'\lambda }{q}\lim\limits_{E\rightarrow 0^+}\left|
s_{\pm }( \lambda ,E) \right| ^{q-p}\lim\limits_{s\rightarrow 0}%
\dfrac{f( s) }{\varphi_{q}( s) } \\
& = & \dfrac{p'\lambda }{q}a_{0}\cdot \lim\limits_{E\rightarrow
0^+}\left| s_{\pm }( \lambda ,E) \right| ^{q-p}.
\end{array}
\end{equation*}
Therefore, Assertion \textbf{(c)} follows.

\paragraph{Proof of (d)}
Remark that for all $\xi >0$ and all $( \kappa ,\eta ) \in
\left\{ +,-\right\} ^{2}$ one has,
\begin{equation*}
\frac{F( \kappa s_{\eta }( \lambda ,E) \xi ) }{E^{p}}%
=\frac{F( \kappa s_{\eta }( \lambda ,E) \xi ) }{\Phi
_{q}( \kappa s_{\eta }( \lambda ,E) \xi ) }\frac{%
\Phi_{q}( \kappa s_{\eta }( \lambda ,E) \xi ) }{%
E^{p}}.
\end{equation*}
Using l'Hopital's rule and (\ref{pq4}) one gets,
\begin{equation*}
\lim_{E\rightarrow 0^+}\frac{F( \kappa s_{\eta }( \lambda ,E) \xi
) }{\Phi_{q}( \kappa s_{\eta }( \lambda ,E) \xi )
}=\lim_{E\rightarrow 0^+}\frac{f( \kappa s_{\eta }( \lambda ,E)
\xi ) }{\varphi_{q}( \kappa s_{\eta }( \lambda ,E) \xi ) }=a_{0}.
\end{equation*}
On the other hand, since $\Phi_{q}( \cdot ) $ is an odd function
then
\begin{equation*}
\frac{\Phi_{q}( \kappa s_{\eta }( \lambda ,E) \xi )
}{E^{p}}=\frac{\Phi_{q}( s_{\eta }( \lambda ,E) \xi ) }{E^{p}}.
\end{equation*}
Thus, for all $\xi >0$, using l'Hopital's rule, (\ref{der}) and
(\ref{pq4}) one gets,
\begin{equation*}
\begin{array}{ccl}
\lim\limits_{E\rightarrow 0^+}\dfrac{\Phi_{q}( s_{\eta }( \lambda
,E) \xi ) }{E^{p}} & = & \lim\limits_{E\rightarrow 0^+}\dfrac{\xi
s_{\eta }'( \lambda ,E) \varphi
_{q}( s_{\eta }( \lambda ,E) \xi ) }{pE^{p-1}} \\
& = & \lim\limits_{E\rightarrow 0^+}\dfrac{\xi ^{q}}{\lambda p'}%
\dfrac{\varphi_{q}( s_{\eta }( \lambda ,E) ) }{%
f( s_{\eta }( \lambda ,E) ) }=\dfrac{\xi ^{q}}{%
a_{0}\lambda p'}.
\end{array}
\end{equation*}
Hence, Assertion \textbf{(d)} follows. Therefore, Lemma
\ref{lemma1} is proved.  \hfill$\diamondsuit$

\paragraph{Remark.} We have used condition (\ref{pq4}) in, and only in, the
process of the proofs of assertions \textbf{(c)} and \textbf{(d)}.

\bigskip Now we are ready to compute $X_{\pm }( \lambda ,E) $
as defined in Section \ref{sec.3}, for any $E>0$ and $\lambda
>0$.\ In fact,
\begin{equation*}
X_+( \lambda ,E) =\left\{
\begin{array}{lcl}
( 0,+\infty ) & \text{if} & E>E_{*}^+ \\
( 0,\alpha_+) & \text{if} & E=E_{*}^+ \\
( 0,s_+( \lambda ,E) ) & \text{if} & 0<E<E_{*}^+,
\end{array}
\right.
\end{equation*}
\begin{equation*}
X_-( \lambda ,E) =\left\{
\begin{array}{lcl}
( -\infty ,0) & \text{if} & E>E_{*}^{-} \\
( \alpha_-,0) & \text{if} & E=E_{*}^{-} \\
( 0,s_-( \lambda ,E) ) & \text{if} & 0<E<E_{*}^{-},
\end{array}
\right.
\end{equation*}
where $s_{\pm }( \lambda ,E) $ is defined in Lemma \ref{lemma1}%
. Then
\begin{equation}
r_+( \lambda ,E) :=\sup X_+( \lambda ,E) =\left\{
\begin{array}{lcl}
+\infty & \text{if} & E>E_{*}^+ \\
\alpha_+ & \text{if} & E=E_{*}^+ \\
s_+( \lambda ,E) & \text{if} & 0<E<E_{*}^+,
\end{array}
\right.  \label{mfrkl1}
\end{equation}
\begin{equation}
r_-( \lambda ,E) :=\inf X_-( \lambda ,E) =\left\{
\begin{array}{lcl}
-\infty & \text{if} & E>E_{*}^{-} \\
\alpha_- & \text{if} & E=E_{*}^{-} \\
s_-( \lambda ,E) & \text{if} & 0<E<E_{*}^{-},
\end{array}
\right.  \label{mfrkl2}
\end{equation}
and one deduces from Lemma \ref{lemma1} the following limits
\begin{equation}
\pm \frac{\partial r_{\pm }}{\partial E}( \lambda ,E)
>0,\forall \lambda >0,E\in ( 0,E_{*}^{\pm }( p,\lambda
) ) ,  \label{eq11}
\end{equation}
\begin{equation}
\lim\limits_{E\rightarrow 0^+}r_{\pm }( \lambda ,E) =0\text{\ \ \
\ \ and \ \ \ }\lim\limits_{E\rightarrow E_{*}^{\pm }}r_{\pm }(
\lambda ,E) =\alpha_{\pm }.  \label{abhu}
\end{equation}
\begin{equation}
\lim\limits_{E\rightarrow 0^+}\left| r_{\pm }( \lambda ,E)
\right| /E=\left\{
\begin{array}{lcl}
+\infty & \text{if} & q-p>0 \\
( \dfrac{p-1}{\lambda a_0}) ^{1/p} & \text{if} & q-p=0 \\
0 & \text{if} & q-p<0,
\end{array}
\right.  \label{bbhu}
\end{equation}
\begin{equation}
\lim\limits_{E\rightarrow 0}F( \kappa r_\eta ( \lambda ,E) \xi )
/E^p=\xi ^q/( \lambda p') ,\;\;\text{ for all }\xi >0\text{ and
}( \kappa ,\eta ) \in \left\{ +,-\right\} ^2.  \label{bbhuu}
\end{equation}

On the other hand,
\begin{equation*}
0<\left| r_{\pm }( \lambda ,E) \right| <+\infty \text{ \ \ if and
only if \ }0<E\leq E_{*}^{\pm },
\end{equation*}
and,
\begin{equation*}
\pm \lambda f( r_{\pm }( \lambda ,E) ) >0\text{ \ \ if and only
if \ }0<E<E_{*}^{\pm }.
\end{equation*}
Then,

\begin{equation*}
\begin{array}{ccl}
D_{\pm }( \lambda ) & := & \left\{ E>0:0<\left| r_{\pm }(
p,\lambda ,E) \right| <+\infty \text{\ and }\pm \lambda f(
r_{\pm }( \lambda ,E) ) >0\right\} \\
& = & ( 0,E_{*}^{\pm }) ,
\end{array}
\end{equation*}
and
\begin{equation*}
D( \lambda ) :=D_+( \lambda ) \cap D_-( \lambda ) =( 0,\inf (
E_{*}^+( \lambda ) ,E_{*}^{-}( \lambda ) ) ) .
\end{equation*}

\begin{lemma}
\label{lemma222}If $f$ satisfies (\ref{pq1})-(\ref{pq3}) and
(\ref{pho}), then for all $\lambda >0:\;E_{*}^{-}( \lambda ) \leq
E_{*}^+( \lambda ) $. Therefore, $D( \lambda ) =( 0,E_{*}^{-}(
\lambda ) ) $.
\end{lemma}

\paragraph{Remark.}
For all $\lambda >0$, $k\geq 1$ and $\kappa =+,-$, define
\begin{equation*}
E_k^\kappa ( \lambda ) =\left\{
\begin{array}{l}
E_{*}^\kappa ( \lambda ) \text{\ if }k=1 \\
E_{*}^{-}( \lambda ) \text{\ if }k\geq 2\text{\ and }\kappa \text{
arbitrary.}
\end{array}
\right.
\end{equation*}
Therefore, for all $\lambda >0$, $k\geq 1$ and $\kappa =+,-$, one
has $D_k^\kappa ( \lambda ) =(0,E_k^\kappa ( \lambda ) )$, where
$D_k^\kappa $ is defined in Section \ref{sec.3}.

\paragraph{Proof of Lemma \ref{lemma222}.}
For all $\lambda >0$, one has $E_{*}^{-}( \lambda ) \leq E_{*}^+(
\lambda ) $ if and only if, $p'\lambda F( \alpha_-) \leq
p'\lambda F( \alpha _+) $, which is equivalent to,
\begin{equation*}
-\int_{0}^{-\alpha_-}f( -t) dt\leq \int_{0}^{\alpha _+}f( t) dt,
\end{equation*}
that is,
\begin{equation*}
0\leq \int_{0}^{-\alpha_-}( f( -t) +f( t) )
dt+\int_{-\alpha_-}^{\alpha_+}f( t) dt.
\end{equation*}
The first integral is positive from (\ref{pho}) and the second
one is too from (\ref{pq2}) and (\ref{jzehjf}) (Recall that
(\ref{jzehjf}) is a consequence of (\ref{pq1}), (\ref{pq3}) and
(\ref{pho}), see the Introduction). Therefore, Lemma
\ref{lemma222} is proved.
\hfill$\diamondsuit$


\begin{lemma}
\label{lemma5}Assume that (\ref{pq1})-(\ref{pq3}) and (\ref{pho})
hold. For all $\lambda >0$ and\newline $E\in ( 0,E_{*}^{-}(
\lambda ) ) $, one has
\begin{equation*}
y( r_-( \lambda ,E) ) =r_+( \lambda ,E)
\end{equation*}
where $y(\cdot )$ is defined in (\ref{12kwdj}). Explicitly, for
all $\lambda >0 $ and $E\in ( 0,E_{*}^{-}( \lambda ) ) $ one has,
\begin{equation*}
0<r_+( \lambda ,E) \leq -r_-( \lambda ,E) ,
\end{equation*}
and
\begin{equation*}
F( t) =F( r_-( \lambda ,E) ) \Longleftrightarrow t=r_+( \lambda
,E) ,\text{\ }\forall t\in ( 0,-r_-( \lambda ,E) ] .
\end{equation*}
\end{lemma}

\paragraph{Proof.} For all $\lambda >0$ and $E\in ( 0,E_{*}^{-}(
\lambda ) ) $, one has from the definition of $r_{\pm }( \lambda
,E) :E^{p}-p'\lambda F( r_{\pm }( \lambda ,E) ) =0$. Thus,
\begin{equation}
F( r_+( \lambda ,E) ) -F( r_-( \lambda ,E) ) =0.  \label{dyuk}
\end{equation}
On the other hand
\begin{equation*}
F( r_-( \lambda ,E) ) =\int_{0}^{r_-( \lambda ,E) }f( t)
dt=-\int_{0}^{-r_-( \lambda ,E) }f( -t) dt,
\end{equation*}
and
\begin{eqnarray*}
F( r_+( \lambda ,E) )  &=&\int_{0}^{r_+(
\lambda ,E) }f( t) dt \\
&=&\int_{0}^{-r_-( \lambda ,E) }f( t) dt+\int_{-r_-( \lambda ,E)
}^{r_+( \lambda ,E) }f( t) dt.
\end{eqnarray*}
Thus,
\begin{eqnarray}
F( r_+( \lambda ,E) ) -F( r_-( \lambda ,E) )  &=&\int_{0}^{-r_-(
\lambda ,E) }(
f( t) +f( -t) ) dt  \label{gqw!dlk} \\
&&+\int_{-r_-( \lambda ,E) }^{r_+( \lambda ,E) }f( t) dt.  \notag
\end{eqnarray}
Observe that by (\ref{eq11}) and (\ref{abhu}), $0<-r_-( \lambda
,E) <-\alpha_-$ and by (\ref{pho}) the function $t\mapsto ( f( t)
+f( -t) ) $ is positive in $( 0,-\alpha_-) $, thus the first
integral in (\ref{gqw!dlk}) is positive. Now, if we assume that
there exists at least a $\lambda_{0}>0$ and an $E_{0}\in (
0,E_{*}^{-}( \lambda_{0}) ) $ such that $r_+( \lambda_{0},E_{0})
+r_-( \lambda _{0},E_{0}) >0$, it follows that $0<-r_-( \lambda
_{0},E_{0}) <r_+( \lambda_{0},E_{0}) <\alpha_+$. Therefore, since
$f$ is strictly positive in $( 0,\alpha_+) $ it follows that the
second integral in (\ref{gqw!dlk}) is strictly positive and thus,
$F( r_+( \lambda_{0},E_{0}) ) -F( r_-( \lambda_{0},E_{0}) ) >0$
which is a contradiction with (\ref{dyuk}). Therefore, for all
$\lambda >0$ and $E\in ( 0,E_{*}^{-}( \lambda ) ) $ one has,
\begin{equation*}
0<r_+( \lambda ,E) \leq -r_-( \lambda ,E) .
\end{equation*}
Regarding (\ref{dyuk}), it remains to prove that for all $\lambda
>0$ and $E\in ( 0,E_{*}^{-}( \lambda ) ) $ one has,
\begin{equation*}
F( t) =F( r_-( \lambda ,E) ) \Longrightarrow t=r_+( \lambda ,E)
,\text{\ }\forall t\in ( 0,-r_-( \lambda ,E) ] .
\end{equation*}
But, this is immediate, since $F$ is strictly increasing in $(
0,-r_-( \lambda ,E) ] $. Therefore, Lemma \ref{lemma5} is proved.
%TCIMACRO{\TeXButton{End Proof}{\hfill$\diamondsuit$}}
%BeginExpansion
\hfill$\diamondsuit$%
%EndExpansion

At present, if $f$ satisfies (\ref{pq1})-(\ref{pq3}) we define,
for any $p,q>1$, $\lambda >0$ and $E\in D_{\pm }( \lambda ) $,
the time map $T_{\pm }$ \thinspace by
\begin{equation*}
T_{\pm }( \lambda ,E) :=\pm \int_{0}^{r_{\pm }( \lambda ,E) }(
E^{p}-p'\lambda F( \xi ) ) ^{-1/p}d\xi ,\;E\in D_{\pm }( \lambda
) =( 0,E_{*}^{\pm }) .
\end{equation*}
Actually, $T_{\pm }( \lambda ,E) $\ is defined for all $\lambda
>0$ and $E\in \tilde{D}_{\pm }( \lambda ) $ (see Remark \ref
{Remark}). A simple change of variables shows that,
\begin{equation}
T_{\pm }( \lambda ,E) =\left| r_{\pm }( \lambda ,E) \right|
\int_{0}^1( E^{p}-p'\lambda F( r_{\pm }( \lambda ,E) \xi ) )
^{-1/p}d\xi , \label{A}
\end{equation}
which can be written as,
\begin{equation}
T_{\pm }( \lambda ,E) =( \left| r_{\pm }( \lambda ,E) \right| /E)
\int_{0}^1( 1-p'\lambda F( r_{\pm }( \lambda ,E) \xi ) /E^{p})
^{-1/p}d\xi .  \label{qze}
\end{equation}
Also, observe that one has from (\ref{dklg}), (\ref{mfrkl1}) and
(\ref {mfrkl2}), $E^{p}=\lambda p'F( r_{\pm }( \lambda ,E) ) $,
so, (\ref{A}) may be written as,
\begin{equation}
T_{\pm }( \lambda ,E) =\pm ( \lambda p') ^{-1/p}\int_{0}^{r_{\pm
}( \lambda ,E) }( F( r_{\pm }( \lambda ,E) ) -F( r_{\pm }( \lambda
,E) \xi ) ) ^{-1/p}d\xi .  \label{a15}
\end{equation}
For any $p>1$ and $x\in [ 0,\alpha_+] $ (resp. $x\in [
\alpha_-,0] $ let us define $S_+( x) $ (resp. $S_-( x) $) by,
\begin{equation*}
S_{\pm }( x) :=\pm \int_{0}^{x}( F( x) -F( \xi ) ) ^{-1/p}d\xi
\in [ 0,+\infty ] .
\end{equation*}
Then, (\ref{a15}) may be written as,
\begin{equation}
T_{\pm }( \lambda ,E) =( \lambda p') ^{-1/p}S_{\pm }( r_{\pm }(
\lambda ,E) ) . \label{a16}
\end{equation}
On the other hand, we define for any $p>1$, $\lambda >0$ and
$E\in D( \lambda ) =( 0,E_{*}^{-}) $, the time maps,
\begin{equation}
T_{2n}^{\pm }( \lambda ,E) :=n( T_+( \lambda ,E) +T_-( \lambda
,E) ) ,\lambda >0,E\in ( 0,E_{*}^{-}) ,n\geq 0,  \label{a18}
\end{equation}
\begin{equation}
T_{2n+1}^{\pm }( \lambda ,E) :=T_{2n}^{\pm }( \lambda ,E) +T_{\pm
}( \lambda ,E) ,\lambda >0,E\in ( 0,E_{*}^{-}) ,n\geq 0.
\label{a19}
\end{equation}

The limits of these time maps are the aim of the following Lemmas.

\begin{lemma}
\label{lemma2}Assume that \textbf{(}\ref{pq1})-(\ref{pq3}) hold.

\begin{description}
\item[(i)]  If $(\ref{pq5})_{\pm }$ holds,
$\lim\limits_{E\rightarrow E_{*}^{\pm }}T_{\pm }( \lambda ,E) =(
\lambda p') ^{-1/p}S_{\pm }( \alpha_{\pm }) $ and\newline $S_{\pm
}( \alpha_{\pm }) <+\infty $ if and only if $q-p<0$.

\item[(ii)]  If (\ref{pq4}) holds, $\lim\limits_{E\rightarrow 0^+}T_{\pm
}( \lambda ,E) =\left\{
\begin{array}{lcc}
+\infty  & \text{if} & q-p>0 \\
\dfrac{1}{2}( \dfrac{\lambda_{1}}{\lambda a_{0}}) ^{1/p} & \text{%
if} & q-p=0 \\
0 & \text{if} & q-p<0.
\end{array}
\right. $

\item[(iii)]  If (\ref{pho}) holds, for all $\lambda >0$, $r_+(
\lambda ,E_{*}^{-}( \lambda ) ) =y( \alpha _-) $, where $y( \cdot
) $ is defined in (\ref{12kwdj}) and $T_+( \lambda ,E_{*}^{-}(
\lambda ) ) =( \lambda p') ^{-1/p}S_+( y( \alpha_-) ) $.
\end{description}
\end{lemma}


\paragraph{Proof of Lemma \ref{lemma2}}
\paragraph{Proof of (i)}
The value of the limit follows by passing to the limit in
(\ref{a16}) as $E$
tends to $E_{*}^{\pm }$. In order to show the second assertion of \textbf{(i)%
} one observes that
\begin{equation*}
S_{\pm }( \alpha_{\pm }) =\pm \int_0^{\alpha_{\pm }\mp \delta
}\left\{ F( \alpha_{\pm }) -F( \xi ) \right\} ^{-1/p}d\xi \pm
\int_{\alpha_{\pm }\mp \delta }^{\alpha_{\pm }}\left\{ \cdots
\right\} ^{-1/p}d\xi ,
\end{equation*}
where $\delta >0$ is given by (\ref{pq5}). The first integral
converges because the integrand function is continuous on the
compact interval whose extremities are $0$ and $\alpha_{\pm }\mp
\delta $. For the second one, it follows from $(\ref{pq5})_{\pm
}$ that, for all $\xi $, satisfying $( \pm \xi ) \in ( \pm
\alpha_{\pm }-\delta ,\pm \alpha_{\pm }) $, one has
\begin{equation*}
\pm m_{\pm }\varphi_q( \alpha_{\pm }-\xi ) \leq \pm f( \xi ) \leq
\pm M_{\pm }\varphi_q( \alpha_{\pm }-\xi ) ,
\end{equation*}
and since for any $E$ near $E_{*}^{\pm }$ one has $\pm r_{\pm }(
\lambda ,E) \in ( \pm \alpha_{\pm }-\delta ,\pm \alpha_{\pm }) $
then for all $\xi $, satisfying $( \pm \xi ) \in ( \pm
\alpha_{\pm }-\delta ,\pm r_{\pm }( \lambda ,E) ) $, one has
\begin{gather*}
\ m_{\pm }\int_\xi ^{r_{\pm }( \lambda ,E) }\varphi_q(
\alpha_{\pm }-x) dx\leq \int_\xi ^{r_{\pm }( \lambda ,E) }f( x)
dx  \leq M_{\pm }\int_\xi ^{r_{\pm }( \lambda ,E) }\varphi_q(
\alpha_{\pm }-x) dx
\end{gather*}
that is
\begin{equation*}
\frac{m_{\pm }}q\left\{ \left| \alpha_{\pm }-\xi \right|
^q-\left| \alpha _{\pm }-r_{\pm }( \lambda ,E) \right| ^q\right\}
\leq F( r_{\pm }( \lambda ,E) ) -F( \xi )
\end{equation*}
\begin{equation*}
\leq \frac{M_{\pm }}q\left\{ \left| \alpha_{\pm }-\xi \right|
^q-\left| \alpha_{\pm }-r_{\pm }( \lambda ,E) \right| ^q\right\}
\end{equation*}
then
\begin{equation*}
\pm ( \frac{M_{\pm }}q) ^{-1/p}\int_{\alpha_{\pm }\mp \delta
_{\pm }}^{r_{\pm }( \lambda ,E) }\left\{ \left| \alpha_{\pm }-\xi
\right| ^q-\left| \alpha_{\pm }-r_{\pm }( \lambda ,E) \right|
^q\right\} ^{-1/p}d\xi
\end{equation*}
\begin{equation*}
\leq \pm \int_{\alpha_{\pm }\mp \delta }^{r_{\pm }( \lambda ,E)
}\left\{ F( r_{\pm }( \lambda ,E) ) -F( \xi ) \right\} ^{-1/p}d\xi
\end{equation*}
\begin{eqnarray*}
\ \leq \pm ( \frac{m_{\pm }}q) ^{-1/p}\int_{\alpha_{\pm }\mp
\delta }^{r_{\pm }( \lambda ,E) }\left\{ \left| \alpha_{\pm }-\xi
\right| ^q-\left| \alpha_{\pm }-r_{\pm }( \lambda ,E) \right|
^q\right\} ^{-1/p}d\xi .
\end{eqnarray*}
By passing to the limit in these inequalities as $E$ tends to
$E_{*}^{\pm }$, one gets
\begin{equation*}
\pm ( \frac{M_{\pm }}q) ^{1/p}\int_{\alpha_{\pm }\mp \delta
}^{\alpha_{\pm }}\left| \alpha_{\pm }-\xi \right| ^{-q/p}d\xi
\leq \pm \int_{\alpha_{\pm }\mp \delta }^{\alpha_{\pm }}\left\{
F( \alpha _{\pm }) -F( \xi ) \right\} ^{-1/p}d\xi
\end{equation*}
\begin{equation*}
\leq \pm ( \frac{m_{\pm }}q) ^{-1/p}\int_{\alpha_{\pm }\mp \delta
}^{\alpha_{\pm }}\left| \alpha_{\pm }-\xi \right| ^{-q/p}d\xi ,
\end{equation*}
and from the well-known fact
\begin{equation*}
\pm \int_{\alpha_{\pm }\mp \delta }^{\alpha_{\pm }}\left|
\alpha_{\pm }-\xi \right| ^{-q/p}d\xi <+\infty \text{ if and only
if }p>q,
\end{equation*}
the second assertion of \textbf{(i)} follows.

\paragraph{Proof of (ii)}

By passing to the limit in (\ref{qze}) as $E$ tends to $0^+$, the
limit of $T_{\pm }( \lambda ,E) $ follows immediately from
(\ref{bbhu}), (\ref{bbhuu}) and the fact that,
\begin{equation*}
\int_0^1( 1-\xi ^q) ^{-1/p}=\frac 1qB( \frac 1q,1-\frac 1p) \in
{\mathbb R},
\end{equation*}
where $B( a,b) $ denotes the beta function. Remark that in the
particular case $q=p$ one has
\begin{equation*}
\int_0^1( 1-\xi ^p) ^{-1/p}=\frac 1pB( \frac 1p,1-\frac 1p) =\pi
/( p\sin ( \frac \pi p) ) =\frac 12( \lambda_1/( p-1) ) ^{1/p}.
\end{equation*}

Notice that condition (\ref{pq4}) was used implicitly in this
proof. In fact, to derive (\ref{bbhu}), (\ref{bbhuu}) we have
used (\ref{pq4}). See the remark located before the proof of
Lemma \ref{lemma1}.

\paragraph{Proof of (iii)}
By Lemma \ref{lemma5} it follows that for all $\lambda >0$, one
has $r_+( \lambda ,E_{*}^{-}( \lambda ) ) =y( r_-( \lambda
,E_{*}^{-}( \lambda ) ) ) $ and by (\ref{abhu}) it follows that
$r_-( \lambda ,E_{*}^{-}( \lambda ) ) =\alpha_-$. Thus, \\ $r_+(
\lambda ,E_{*}^{-}( \lambda ) ) =y( \alpha_-) $.

The formula of $T_+( \lambda ,E_{*}^{-}( \lambda ) ) $ follows
from a simple substitution in (\ref{a16}). Therefore, Lemma
\ref{lemma2} is proved.
%TCIMACRO{\TeXButton{End Proof}{\hfill$\diamondsuit$}}
%BeginExpansion
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%EndExpansion

\begin{lemma}
\label{lemma31}Assume that (\ref{pq1})-(\ref{pq4}) hold, $p,q>1$,
$\lambda
>0$ and $n\in  {\mathbb N}^{*}$. Then,
\begin{equation*}
\lim\limits_{E\rightarrow 0^+}T_{n}^{\pm }( \lambda ,E) =\left\{
\begin{array}{lcc}
+\infty  & \text{if} & q-p>0 \\
\dfrac{1}{2}( \dfrac{\lambda_{n}}{\lambda a_{0}}) ^{1/p} & \text{%
if} & q-p=0 \\
0 & \text{if} & q-p<0.
\end{array}
\right.
\end{equation*}
\end{lemma}

\begin{lemma}
\label{lemma32}Assume that (\ref{pq1})-(\ref{pq3}), $(\ref{pq5})_-$, and (%
\ref{pho}) hold, $p$, $q>1$, $\lambda >0$ and $n\in {\mathbb
N}^{*}$. Then,

\begin{itemize}
\item  In case $q-p\geq 0$, one has $\lim\limits_{E\rightarrow
E_{*}^{-}}T_{n}^{\pm }( \lambda ,E) =+\infty $.

\item  In case $q-p<0$, one has
\begin{equation*}
\begin{array}{lll}
\lim\limits_{E\rightarrow E_{*}^{-}}T_{2n}^{\pm }( \lambda ,E) &
= & ( \lambda p') ^{-1/p}( nS_-( \alpha
_-) +nS_+( \alpha_{*}) )  \\
\lim\limits_{E\rightarrow E_{*}^{-}}T_{2n+1}^+( \lambda ,E) & = &
( \lambda p') ^{-1/p}( nS_-( \alpha
_-) +( n+1) S_+( \alpha_{*}) )  \\
\lim\limits_{E\rightarrow E_{*}^{-}}T_{2n+1}^{-}( \lambda ,E) & =
& ( \lambda p') ^{-1/p}( ( n+1) S_-( \alpha_-) +nS_+( \alpha_{*})
) .
\end{array}
\end{equation*}
\end{itemize}
\end{lemma}

\paragraph{Proofs of Lemmas \ref{lemma31} and \ref{lemma32}.} These ones
follow from Lemma \ref{lemma2} and the definitions (\ref{a18})
and (\ref{a19}) of the time maps $T_k^{\pm }$. \hfill$\diamondsuit$

\begin{lemma}
\label{lemma4}For any $p$, $q>1$ and $\lambda >0$, assume that
$(\ref {njk})_{\pm }$ holds. Then,
\begin{equation*}
\frac{\partial T_{\pm }}{\partial E}( \lambda ,E)
>0,\;\forall E\in D_{\pm }( \lambda ) =( 0,E_{*}^{\pm
}( \lambda ) ) .
\end{equation*}
Therefore, if both $(\ref{njk})_+$ and $(\ref {njk})_-$ hold,
then, for all $k\geq 2$,
\begin{equation*}
\frac{\partial T_{k}^{\pm }}{\partial E}( \lambda ,E)
>0,\;\forall E\in D( \lambda ) =( 0,E_{*}^{-}(
\lambda ) ) .
\end{equation*}
\end{lemma}

\paragraph{Proof.} A simple computation shows that
\begin{eqnarray*}
\frac{\partial T_{\pm }}{\partial E}( \lambda ,E)  &=&\frac{1}{p%
}( p') ^{-1/p}( \pm \frac{\partial r_{\pm }}{%
\partial E}( \lambda ,E) ) \lambda ^{-1/p} \\
&&\times \int_{0}^1\frac{H( r_{\pm }( \lambda ,E) ) -H( r_{\pm }(
\lambda ,E) \xi ) }{( F( r_{\pm }( \lambda ,E) ) -F( r_{\pm }(
\lambda ,E) \xi ) ) ^{1+( 1/p) }}d\xi .
\end{eqnarray*}
where $H( x) =pF( x) -xf( x) $. Condition $(\ref{njk})_{\pm }$
implies that
\begin{equation*}
H( r_{\pm }( \lambda ,E) ) -H( r_{\pm }( \lambda ,E) \xi )
>0,\forall \xi \in ( 0, 1) .
\end{equation*}
Thus, the integral above is positive, and by (\ref{eq11}) it
follows that $\frac{\partial T_{\pm }}{\partial E}( \lambda ,E)
>0$, for all $E\in D_{\pm }( \lambda ) $. The last assertion is
immediate from the definition of $T_{k}^{\pm }( \lambda ,\cdot )
$. The proof of Lemma \ref{lemma4} is complete.
%TCIMACRO{\TeXButton{End Proof}{\hfill$\diamondsuit$}}
%BeginExpansion
\hfill$\diamondsuit$%
%EndExpansion

\bigskip \ When $f$ is odd, the time-maps $T_+(\lambda ,\cdot )$ and
$T_-(\lambda ,\cdot )$ are always equal. In our \emph{p.h.o.}
case, we show that both $T_+(\lambda ,\cdot )$ and $T_-(\lambda
,\cdot )$ are bounded from below and from above by a same
function respectively. These estimates imply in the particular
odd case that the two time-maps $T_+(\lambda ,\cdot )$ and
$T_-(\lambda ,\cdot )$ are equal. The following lemma is pioneer
in our analysis.

Let us define
\begin{equation*}
\Theta_+( x) =\int_{0}^{x}\{F( x) -F( -\xi ) \}^{-1/p}d\xi ,x\in
( 0,-\alpha_-) ,
\end{equation*}
and
\begin{eqnarray*}
\Theta_-( x) &=&\int_{x}^{-y( x) }\{F( x) -F( \xi ) \}^{-1/p}d\xi
+\int_{-y( x)
}^{0}\{F( x) -F( -\xi ) \}^{-1/p}d\xi , \\
\text{ for all }x &\in &( \alpha_-,0) ,\text{ where }y( x) \text{
is defined in (\ref{12kwdj}).}
\end{eqnarray*}

\begin{lemma}
\label{lemma6}Assume that (\ref{pq1})-(\ref{pq3}) and (\ref{pho})
hold. Then, for all $\lambda >0$ and $E\in ( 0,E_{*}^{-}( \lambda
) ) $, one has

\begin{description}
\item[(i)]  $\Theta_+( r_+( \lambda ,E) ) \leq
S_-( r_-( \lambda ,E) ) $,

\item[(ii)]  $\Theta_+( r_+( \lambda ,E) ) \leq
S_+( r_+( \lambda ,E) ) $,

\item[(iii)]  $S_-( r_-( \lambda ,E) ) \leq
\Theta_-( r_-( \lambda ,E) ) $,

\item[(iv)]  $S_+( r_+( \lambda ,E) ) \leq
\Theta_-( r_-( \lambda ,E) ) $.
\end{description}
\end{lemma}

\paragraph{Proof of Lemma \ref{lemma6}}
\paragraph{Proof of (i)}

For all $\lambda >0$ and $E\in ( 0,E_{*}^{-}( \lambda ) ) $, one
has
\begin{equation*}
S_-( r_-( \lambda ,E) ) =-\int_0^{r_-( \lambda ,E) }\{F( r_-(
\lambda ,E) ) -F( \xi ) \}^{-1/p}d\xi .
\end{equation*}
Using a simple change of variables one deduces,
\begin{equation}
S_-( r_-( \lambda ,E) ) =\int_0^{-r_-( \lambda ,E) }\{F( r_-(
\lambda ,E) ) -F( -\xi ) \}^{-1/p}d\xi .  \label{ehjkgr}
\end{equation}
Using (\ref{dyuk}) one gets
\begin{eqnarray*}
S_-( r_-( \lambda ,E) ) &=&\Theta_+(
r_+( \lambda ,E) ) \\
&&\ +\int_{r_+( \lambda ,E) }^{-r_-( \lambda ,E) }\{F( r_-(
\lambda ,E) ) -F( -\xi ) \}^{-1/p}d\xi
\end{eqnarray*}
and by Lemma \ref{lemma5} it follows that the integral above is
positive. Therefore, Assertion \textbf{(i)} is proved.

\paragraph{Proof of (ii)}
Recall that $F$ is \textit{p.h.e. }in $[ \alpha_-,-\alpha _-] $,
(see (\ref{phe})), hence, for all $\lambda >0$, $E\in (
0,E_{*}^{-}( \lambda ) ) $, and $0<\xi <r_+( \lambda ,E) \leq
-r_-( \lambda ,E) \leq -\alpha_-$, one has
\begin{equation}
\{F( r_-( \lambda ,E) ) -F( -\xi ) \}^{-1/p}\leq \{F( r_-(
\lambda ,E) ) -F( \xi ) \}^{-1/p},  \label{m5}
\end{equation}
Thus,
\begin{eqnarray*}&&
\int_0^{r_+( \lambda ,E) }\{F( r_-( \lambda ,E) ) -F( -\xi )
\}^{-1/p}d\xi \\ &\leq &\int_0^{r_+( \lambda ,E) }\{F( r_-(
\lambda ,E) ) -F( \xi ) \}^{-1/p}d\xi .
\end{eqnarray*}
Using (\ref{dyuk}), it follows that
\begin{eqnarray*}
\Theta_+( r_+( \lambda ,E) ) &=&\int_0^{r_+( \lambda ,E) }\{F(
r_+( \lambda
,E) ) -F( -\xi ) \}^{-1/p}d\xi \\
\ &\leq &\int_0^{r_+( \lambda ,E) }\{F( r_+(
\lambda ,E) ) -F( \xi ) \}^{-1/p}d\xi \\
\ &=&S_+( r_+( \lambda ,E) ) .
\end{eqnarray*}
Therefore, Assertion \textbf{(ii)} is proved.

\paragraph{Proof of (iii)}
Recall that $F$ is \textit{p.h.e. }in $[ \alpha_-,-\alpha _-] $,
(see (\ref{phe})), and for all $\lambda >0$ and $E\in (
0,E_{*}^{-}( \lambda ) ) $ one has $y( r_-( \lambda ,E) ) =r_+(
\lambda ,E) $. Thus, for $\xi \in ( -y( r_-( \lambda ,E) ) ,0) , $
\begin{equation*}
\{F( r_-( \lambda ,E) ) -F( \xi ) \}^{-1/p}\leq \{F( r_-( \lambda
,E) ) -F( -\xi ) \}^{-1/p}.
\end{equation*}
Therefore,
\begin{eqnarray*}
&&\int_{-y( r_-( \lambda ,E) ) }^0\{F(
r_-( \lambda ,E) ) -F( \xi ) \}^{-1/p}d\xi \\
\ &\leq &\int_{-y( r_-( \lambda ,E) ) }^0\{F( r_-( \lambda ,E) )
-F( -\xi ) \}^{-1/p}d\xi .
\end{eqnarray*}
So,
\begin{eqnarray*}
S_-( r_-( \lambda ,E) ) &=&\int_{r_-( \lambda ,E) }^0\{F( r_-(
\lambda ,E) )
-F( \xi ) \}^{-1/p}d\xi \\
&\leq &\int_{r_-( \lambda ,E) }^{-y( r_-( \lambda ,E) ) }\{F(
r_-( \lambda ,E) )
-F( \xi ) \}^{-1/p}d\xi \\
&&+\int_{-y( r_-( \lambda ,E) ) }^0\{F( r_-( \lambda ,E) ) -F(
-\xi ) \}^{-1/p}d\xi
\\
&=&\Theta_-( r_-( \lambda ,E) ) .
\end{eqnarray*}
Assertion \textbf{(iii)} is proved.

\paragraph{Proof of (iv)}

For all $\lambda >0$ and $E\in ( 0,E_{*}^{-}( \lambda ) ) $ one
has
\begin{eqnarray}
\Theta_-( r_-( \lambda ,E) ) &=&\int_{r_-( \lambda ,E) }^{-y(
r_-( \lambda ,E) ) }\{F( r_-( \lambda ,E) )
-F( \xi ) \}^{-1/p}d\xi  \label{Canada} \\
&&+\int_{-y( r_-( \lambda ,E) ) }^0\{F( r_-( \lambda ,E) ) -F(
-\xi ) \}^{-1/p}d\xi . \notag
\end{eqnarray}

Then by Lemma \ref{lemma5}, one has $-y( r_-( \lambda ,E) )
=-r_+( \lambda ,E) $ and $F( r_-( \lambda ,E) ) =F( r_+( \lambda
,E) ) $. Thus, using a simple change of variable, it follows that
\begin{equation*}
\int_{-y( r_-( \lambda ,E) ) }^0\{F( r_-( \lambda ,E) ) -F( -\xi
) \}^{-1/p}d\xi =S_+( r_+( \lambda ,E) ) .
\end{equation*}
On the other hand, the first integral in (\ref{Canada}) is
positive, since \newline $r_-( \lambda ,E) \leq -y( r_-( \lambda
,E) ) $. Thus, $\Theta_-( r_-( \lambda ,E) ) \geq S_+( r_+(
\lambda ,E) ) $.

Therefore, Assertion \textbf{(iv)} is proved which completes the
proof of Lemma \ref{lemma6}.
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\begin{remark}
It is well known that when $f$ is odd then
\begin{equation}
T_+(\lambda ,\cdot )=T_-(\lambda ,\cdot )\text{,\ for all }\lambda
>0.  \label{R1}
\end{equation}
The estimates in Lemma \ref{lemma6} imply (\ref{R1}) in the odd
case. In fact if $f$ is odd then $\alpha_+=-\alpha_-$, $F$ is
even,
\begin{equation*}
E_{*}^+(\lambda )=E_{*}^{-}(\lambda )\text{,\ \ for all }\lambda
>0
\end{equation*}
\begin{equation*}
r_+(\lambda ,E)=-r_-(\lambda ,E),\text{\ \ for all }\lambda >0.
\end{equation*}
Also, since for all $\lambda >0$ and $E\in (0,E_{*}^{\pm
}(\lambda ))$,
\begin{equation*}
\Theta_-( r_-( \lambda ,E) ) =\int_{r_-}^{-y( r_-) }\{F( r_-) -F(
\xi ) \}^{-1/p}d\xi +\int_{-y( r_-) }^{0}\{F( r_-) -F( -\xi )
\}^{-1/p}d\xi ,
\end{equation*}
then by Lemma \ref{lemma5}, it follows that
\begin{eqnarray*}
\Theta_-( r_-( \lambda ,E) ) &=&\int_{r_-}^{-r_+}\{F( r_+) -F(
\xi ) \}^{-1/p}d\xi +\int_{-r_+}^{0}\{F( r_+) -F( -\xi )
\}^{-1/p}d\xi , \\
&=&\int_{-r_+}^{0}\{F( r_+) -F( -\xi ) \}^{-1/p}d\xi .
\end{eqnarray*}
On the other hand, since the function defined on $(-r_+,r_+)$ by
$\xi \mapsto \{F( r_+) -F( -\xi ) \}^{-1/p}$ is even then
\begin{equation*}
\int_{-r_+}^{0}\{F( r_+) -F( -\xi ) \}^{-1/p}d\xi
=\int_{0}^{r_+}\{F( r_+) -F( -\xi ) \}^{-1/p}d\xi .
\end{equation*}
Therefore,
\begin{equation}
\Theta_-( r_-( \lambda ,E) ) =\Theta_+( r_+( \lambda ,E) )
\text{,\ for all }\lambda >0\text{ and all }E\in (0,E_{*}(\lambda
)).  \label{R2}
\end{equation}
Now, by the estimates of Lemma \ref{lemma6} and (\ref{R2}),
equation (\ref {R1}) follows.
\end{remark}

\begin{lemma}
\label{lemma7}Assume that (\ref{pq1})-(\ref{pq3}) and (\ref{pho})
hold. Then, for all $\lambda >0$, $E\in ( 0,E_{*}^{-}( \lambda )
) $, and for all integer $k\geq 2, $\begin{equation*} k(
p'\lambda ) ^{-1/p}\Theta_+( r_+( \lambda ,E) ) \leq
T_{k}^{\kappa }( \lambda ,E) \leq k( p'\lambda ) ^{-1/p}\Theta_-(
r_-( \lambda ,E) ) .
\end{equation*}
\end{lemma}

\paragraph{Proof.} According to (\ref{a16}) and the definition for
$T_{k}^{\kappa }( \lambda ,E) $, the proof is an immediate
consequence of Lemma \ref{lemma6}.
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\begin{lemma}
\label{lemma75}For all $\lambda >0$ and $E\in ( 0,E_{*}^{-}(
\lambda ) ) $ the following identity holds;
\begin{equation*}
\Theta_+( r_+( \lambda ,E) ) +\Theta_-( r_-( \lambda ,E) ) =S_+(
r_+( \lambda ,E) ) +S_-( r_-( \lambda ,E) ) .
\end{equation*}
\end{lemma}

\paragraph{Proof.} For all $\lambda >0$ and $E\in ( 0,E_{*}^{-}(
\lambda ) ) $, we write $\Theta_-( r_-( \lambda ,E) ) $ as follows
\begin{eqnarray*}
\Theta_-( r_-( \lambda ,E) ) &=&\int_{r_-( \lambda ,E) }^{0}\{F(
r_-( \lambda
,E) ) -F( \xi ) \}^{-1/p}d\xi  \\
&&\ \ -\int_{-r_+( \lambda ,E) }^{0}\{F( r_-(
\lambda ,E) ) -F( \xi ) \}^{-1/p}d\xi  \\
&&\ \ +\int_{-r_+( \lambda ,E) }^{0}\{F( r_-( \lambda ,E) ) -F(
-\xi ) \}^{-1/p}d\xi .
\end{eqnarray*}
The first integral is equal to $S_-( r_-( \lambda ,E) ) $. On the
other hand, the change of variable $\xi =-r_+( \lambda ,E) t$,
and (\ref{dyuk}) imply that the second integral is equal to
$\Theta_+( r_+( \lambda ,E) ) $. The change of variable $\xi
=-t$, and (\ref{dyuk}) imply that the third integral is equal to
$S_+( r_+( \lambda ,E) ) $. Therefore, Lemma \ref{lemma75} is
proved.
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\begin{lemma}
\label{lemma8}Assume that (\ref{pq1})-(\ref{pq4}), $(\ref{pq5})_-$ and (%
\ref{pho}) hold. Then, for all $\lambda >0$, one has

\begin{description}
\item[(i)]  $\lim\limits_{E\rightarrow 0^+}\Theta_{\pm }( r_{\pm
}( \lambda ,E) ) =\left\{
\begin{array}{lcc}
+\infty  & \text{if} & q-p>0 \\
\dfrac{1}{2}( \dfrac{p'\lambda_{1}}{a_{0}}) ^{1/p} &
\text{if} & q-p=0 \\
0 & \text{if} & q-p<0.
\end{array}
\right. $

\item[(ii)]  $\lim\limits_{E\rightarrow E_{*}^{-}( \lambda )
}\Theta_{\pm }( r_{\pm }( \lambda ,E) ) =\ell_{\pm }$ with\newline
$\left\{
\begin{array}{lcl}
\ell_{\pm }=+\infty  & \text{if} & q-p\geq 0\text{ and
}-\alpha_{*}=\alpha
_- \\
\ell_+\in ( 0,+\infty ) \text{ and }\ell_-=+\infty  &
\text{if} & q-p\geq 0\text{ and }-\alpha_{*}>\alpha_- \\
\ell_{\pm }\in ( 0,+\infty )  & \text{if} & q-p<0.
\end{array}
\right. $
\end{description}
\end{lemma}

\paragraph{Proof of Lemma \ref{lemma8}}
\paragraph{Proof of Assertion (i).} For all $\lambda >0$ and $E\in (
0,E_{*}^{-}( \lambda ) ) $ one has $E^p/(p^{\prime }\lambda )=F(
r_+( \lambda ,E) ) $, hence,
\begin{equation*}
\Theta_+( r_+( \lambda ,E) ) =\int_0^{r_+( \lambda ,E) }\left\{
(E^p/p'\lambda )-F(-\xi )\right\} ^{-1/p}d\xi .
\end{equation*}
The change of variable $\xi =r_+( \lambda ,E) t$, yields
\begin{equation*}
\Theta_+( r_+( \lambda ,E) ) =\frac{r_+( \lambda ,E) }E(p'\lambda
)^{1/p}\int_0^1\left\{ 1-p^{\prime }\lambda F(-r_+( \lambda ,E)
t)/E^p\right\} ^{-1/p}dt.
\end{equation*}
By (\ref{bbhu}) one has
\begin{equation}
\lim\limits_{E\rightarrow 0^+}\frac{r_+( \lambda ,E) }%
E=\left\{
\begin{array}{lcl}
+\infty & \text{if} & q-p>0 \\
( \dfrac{p-1}{\lambda a_0}) ^{1/p} & \text{if} & q-p=0 \\
0 & \text{if} & q-p<0.
\end{array}
\right.  \label{Madrid}
\end{equation}
On the other hand, by (\ref{bbhuu}) one has for all $( \kappa
,\eta ) \in \left\{ +,-\right\} ^2$,
\begin{equation}
\lim\limits_{E\rightarrow 0^+}\int_0^1\left\{ 1-p'\lambda
F(\kappa r_\eta ( \lambda ,E) t)/E^p\right\}
^{-1/p}dt=\int_0^1\left\{ 1-t^q\right\} ^{-1/p}  \label{Italy}
\end{equation}
with
\begin{equation}
\int_0^1\left\{ 1-t^q\right\} ^{-1/p}=\left\{
\begin{array}{lcl}
\dfrac 1qB( \dfrac 1q,1-\dfrac 1p) \in {\mathbb R}
& \text{if} & q-p\neq 0 \\
\dfrac 12( \dfrac{\lambda_1}{p-1}) ^{1/p} & \text{if} & q-p=0.
\end{array}
\right.  \label{France}
\end{equation}
Therefore, the limit $\lim_{E\rightarrow 0^+}\Theta_+( r_+(
\lambda ,E) ) $ follows.

Assume that $q-p>0$. By Assertion \textbf{(ii)} of Lemma
\ref{lemma2}, one has \newline $\lim_{E\rightarrow 0^+}T_-(
\lambda ,E) =+\infty $, and by (\ref{a16}) and Assertion
\textbf{(iii)} of Lemma \ref{lemma6}, it follows that
$\lim_{E\rightarrow 0^+}\Theta_-( r_-( \lambda ,E) ) =+\infty $.

Assume that $q-p=0$. In this case, for all $\lambda >0$ and $E\in
( 0,E_{*}^{-}( \lambda ) ) $, we use the identity in Lemma
\ref{lemma75}. That is, we write $\Theta_-( r_-( \lambda ,E) ) $
as follows
\begin{equation*}
\Theta_-( r_-( \lambda ,E) ) =S_+( r_+( \lambda ,E) ) +S_-( r_-(
\lambda ,E) ) -\Theta_+( r_+( \lambda ,E) )
\end{equation*}
and we prove that each term of the right hand side tends to the
same limit; $\dfrac 12( p'\lambda_1/a_0) ^{1/p}$.

The limits of $S_-( r_-( \lambda ,E) ) $ and $S_+( r_+( \lambda
,E) ) $ follows by (\ref{a16}) and Assertion \textbf{(ii)} of
Lemma \ref{lemma2}, and the limit of $\Theta _+( r_+( \lambda ,E)
) $ was computed above. Therefore, $\lim_{E\rightarrow
0^+}\Theta_-( r_-( \lambda ,E) ) =\dfrac 12( p'\lambda_1/a_0)
^{1/p}$ which completes the proof of Assertion \textbf{(i)}.

\paragraph{Proof of Assertion (ii).} By (\ref{dyuk}) it follows that
\begin{equation*}
\Theta_+( r_+( \lambda ,E) ) =\int_0^{r_+( \lambda ,E) }\left\{
F(r_-( \lambda ,E) )-F(-\xi )\right\} ^{-1/p}d\xi .
\end{equation*}
A simple change of variable yields
\begin{equation*}
\Theta_+( r_+( \lambda ,E) ) =\int_{-r_+( \lambda ,E) }^0\left\{
F(r_-( \lambda ,E) )-F(\xi )\right\} ^{-1/p}d\xi
\end{equation*}
and thus,
\begin{equation*}
\lim_{E\rightarrow E_{*}^{-}( \lambda ) }\Theta_+( r_+( \lambda
,E) ) =\int_{-r_+( \lambda ,E_{*}^{-}( \lambda ) ) }^0\left\{
F(r_-( \lambda ,E_{*}^{-}( \lambda ) ) )-F(\xi )\right\}
^{-1/p}d\xi .
\end{equation*}
By (\ref{abhu}), one has $r_-( \lambda ,E_{*}^{-}( \lambda ) )
=\alpha_-$ and by Assertion \textbf{(iii)} of Lemma \ref {lemma2},
it follows that $r_+( \lambda ,E_{*}^{-}( \lambda ) )
=\alpha_{*}$.

Therefore,
\begin{equation*}
\lim_{E\rightarrow E_{*}^{-}( \lambda ) }\Theta_+( r_+( \lambda
,E) ) =\int_{-\alpha_{*}}^0\left\{ F(\alpha_-)-F(\xi )\right\}
^{-1/p}d\xi .
\end{equation*}
According to the definition of $\alpha_{*}$ (see (\ref{12kwdj})),
one has, $\alpha_-\leq -\alpha_{*}$. Thus, one has to distinguish
two cases:

If $-\alpha_{*}=\alpha_-$,
\begin{equation*}
\lim_{E\rightarrow E_{*}^{-}( \lambda ) }\Theta_+( r_+( \lambda
,E) ) =\int_{\alpha_-}^0\left\{ F(\alpha_-)-F(\xi )\right\}
^{-1/p}d\xi =S_-( \alpha_-) .
\end{equation*}
Thus, by Assertion \textbf{(i)} of Lemma \ref{lemma2}, it follows
that
\begin{equation*}
S_-( \alpha_-) =\left\{
\begin{array}{lcc}
+\infty & \text{if} & q-p\geq 0 \\
\ell_+\in ( 0,+\infty ) & \text{if} & q-p<0.
\end{array}
\right.
\end{equation*}

If $-\alpha_{*}>\alpha_-$, it follows that the integral
$\int_{-\alpha _{*}}^0\left\{ F(\alpha_-)-F(\xi )\right\}
^{-1/p}d\xi $ is a positive real number, since the integrand
function is continuous on the compact interval $[ -\alpha_{*},0]
$.

Therefore, the claims related to the limit $\lim_{E\rightarrow
E_{*}^{-}}\Theta_+( r_+( \lambda ,E) ) $ follows.

Assume that $q-p\geq 0$. By Assertion \textbf{(i)} of Lemma
\ref{lemma2}, one has\newline $\lim_{E\rightarrow E_{*}^{-}}T_-(
\lambda ,E) =+\infty , $ and by (\ref{a16}) and Assertion
\textbf{(iii)} of Lemma \ref{lemma6}, it follows that
$\lim_{E\rightarrow E_{*}^{-}}\Theta_-( r_-( \lambda ,E) )
=+\infty $.

Assume that $q-p<0$. In this case one has, by Assertion
\textbf{(i)} of Lemma \ref{lemma2}, $\lim_{E\rightarrow
E_{*}^{-}}S_-( r_-( \lambda ,E) ) =S_-( \alpha_-) \in (
0,+\infty ) $, and by Assertion \textbf{(iii)} of Lemma \ref{lemma2}%
,
\begin{equation*}
\lim_{E\rightarrow E_{*}^{-}}S_+( r_+( \lambda ,E) ) =S_+(
\alpha_{*}) =( \lambda p') ^{1/p}T_+( \lambda ,E_{*}^{-}( \lambda
) ) ,
\end{equation*}
and $T_+( \lambda ,E_{*}^{-}( \lambda ) ) \in ( 0,+\infty ) $,
since $E_{*}^{-}( \lambda ) \in int( D_+( \lambda ) ) =int(dom(
T_+( \lambda ,\cdot ) ) )$. Also, $\lim_{E\rightarrow
E_{*}^{-}}\Theta_+( r_+( \lambda ,E) ) =\ell _+\in ( 0,+\infty )
$ (proved above). Thus, by the identity of Lemma \ref{lemma75},
it follows that $\lim_{E\rightarrow E_{*}^{-}}\Theta _-( r_-(
\lambda ,E) ) =\ell_-\in ( 0,+\infty ) $. Therefore, Assertion
\textbf{(ii)} follows, which completes the proof of Lemma
\ref{lemma8}.
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\begin{lemma}
\label{lemma9}Assume that (\ref{pq1})-(\ref{pq3}) and
$(\ref{Miami})_{\pm }$ hold. Then, for any $p>1$ one has,
\begin{equation*}
r^{2}\frac{d^{2}S_{\pm }}{dr^{2}}( r) +2( p-1) r\frac{%
dS_{\pm }}{dr}( r) >0,\text{\ for all }r\in I_{\pm }( \alpha
_{\pm }) ,
\end{equation*}
where $I_+( \alpha_+) =( 0,\alpha_+) $ and $I_-( \alpha_-) =(
\alpha_-,0) $.
\end{lemma}

\paragraph{Proof.} Some easy computations show that for all $\lambda >0$ and
$r\in I_{\pm }( \alpha_{\pm }) $\begin{equation*}
r^{2}\frac{d^{2}S_{\pm }}{dr^{2}}( r) +2( p-1) r\frac{%
dS_{\pm }}{dr}( r)
\end{equation*}
\begin{eqnarray*}
\  &=&\frac{\pm 1}{p}\int_{0}^{r}\frac{\Psi ( r) -\Psi ( s) }{(
F( r) -F( s) ) ^{( p+1) /p}}ds \pm ( \frac{p+1}{p}) \int_{0}^{r}\frac{%
( H( r) -H( s) ) ^{2}}{( F( r) -F( s) ) ^{( 2p+1) /p}}ds
\end{eqnarray*}
where,
\begin{eqnarray*}
H( x)  &=&pF( x) -xf( x) ,\text{for all }%
x\in I_{\pm }( \alpha_{\pm })  \\
\Psi ( x)  &=&p( p-3) F( x) +2xf( x) -x^{2}f'( x) ,\text{for all
}x\in I_{\pm }( \alpha_{\pm }) .
\end{eqnarray*}
A differentiation yields
\begin{equation*}
\Psi '( x) =( p-2) ( p-1) f( x) -x^{2}f''( x) ,\text{ for all
}x\in I_{\pm }( \alpha_{\pm }) .
\end{equation*}
Thus, $(\ref{Miami})_{\pm }$ implies that $\pm \Psi $ is strictly
increasing in $I_{\pm }( \alpha_{\pm }) $. Then, $\Psi ( r) -\Psi
( s) >0$, for all $s\in I_{\pm }( r) \subset I_{\pm }(
\alpha_{\pm }) $. Therefore, Lemma \ref{lemma9} is proved.

\section{{Proof of the main results}}

\label{sec.5}To prove our main results we make use of the
quadrature method; Theorem \ref{quad}. Hence, we have to resolve
equations of the type $T( E) =(1/2)$, where $T$ designates, in
each case, the appropriate time map.

\paragraph{Proof of Theorem \ref{theorem1}}
Let us assume that $1<q<p$.

\paragraph{Proof of Assertion (i)}
For all $\kappa =+,-$, if (\ref{pq1})-(\ref{pq4}) and
$(\ref{pq5})_{\kappa }$ hold, then for each $\lambda >0$, the
function $E\mapsto T_{\kappa }( \lambda ,E) $ is defined in
$D_{\kappa }( \lambda ) =( 0,E_{*}^{\kappa }( \lambda ) ) $ and
\begin{equation*}
\lim_{E\rightarrow 0^+}T_{\kappa }( \lambda ,E) =0,\text{\ \ }%
\lim_{E\mapsto E_{*}^{\kappa }}T_{\kappa }( \lambda ,E) =(
\lambda p') ^{-1/p}S_{\kappa }( \alpha_{\kappa }) <+\infty .
\end{equation*}
(Lemma \ref{lemma2}). Therefore, $\tilde{D}_{\kappa }(\lambda
)=[0,E_{*}^{\kappa }(\lambda )]$ (see Remark \ref{Remark}), and
the equation $T_{\kappa }( \lambda ,E) =(1/2)$ in the variable
$E\in \tilde{D}_{\kappa }( \lambda ) -\left\{ 0\right\} $ admits a
solution in $\tilde{D}_{\kappa }( \lambda ) -\left\{ 0\right\} $
provided that
\begin{equation*}
( \lambda p') ^{-1/p}S_{\kappa }( \alpha_{\kappa }) \geq (1/2),
\end{equation*}
that is, provided that $\lambda \leq ( 2S_{\kappa }( \alpha
_{\kappa }) ) ^{p}/p'$. Furthermore, by Lemma \ref {lemma4}, if
$(\ref{njk})_{\kappa }$ holds, the function $E\mapsto T_{\kappa
}( \lambda ,E) $ is strictly increasing in $\tilde{D}_{\kappa }(
\lambda ) $. Thus, the equation $T_{\kappa }( \lambda ,E) =(1/2)$
in the variable $E\in \tilde{D}_{\kappa }( \lambda ) -\left\{
0\right\} $ admits a solution in $\tilde{D}_{\kappa }( \lambda )
-\left\{ 0\right\} $ if and only if $\lambda \leq ( 2S_{\kappa }(
\alpha_{\kappa }) ) ^{p}/p'$, and in this case the solution is
unique since $T_{\kappa }( \lambda ,\cdot ) $ is strictly
increasing.

\paragraph{Proof of Assertion (ii)}
For all $k\geq 2$, if (\ref{pq1})-(\ref{pq4}), $(\ref{pq5})_-$,
and (\ref {pho}) hold, then for each $\lambda >0$, the function
$E\mapsto T_{k}^{\kappa }( \lambda ,E) $ is defined in $D( \lambda
) =( 0,E_{*}^{-}( \lambda ) ) $ and by Lemma \ref{lemma31} its
limit at $0$ is $0$ and by Lemma \ref{lemma32} its limit at
$E_{*}^{-}( \lambda ) $ is

\begin{itemize}
\item  $n( \lambda p') ^{-1/p}( S_-( \alpha
_-) +S_+( \alpha_{*}) ) $ if $k=2n$,

\item  $( \lambda p') ^{-1/p}( nS_-( \alpha
_-) +( n+1) S_+( \alpha_{*}) ) $ if $k=2n+1$, and $\kappa =+$,

\item  $( \lambda p') ^{-1/p}( ( n+1)
S_-( \alpha_-) +nS_+( \alpha_{*}) ) $ if $k=2n+1$, and $\kappa
=-$.
\end{itemize}

Thus, $\tilde{D}( \lambda ) =[0,E_{*}^{-}( \lambda ) ]$ and the
equation $T_{k}^{\kappa }( \lambda ,E) =\frac{1}{2}$ in the
variable $E\in \tilde{D}( \lambda ) -\left\{ 0\right\} $ admits a
solution in $\tilde{D}( \lambda ) -\left\{ 0\right\} $ provided
that,

\begin{itemize}
\item  $\frac{1}{2}\leq (\lambda p')^{-1/p}(nS_-(\alpha
_-)+nS_+(\alpha_{*}))$, if $k=2n$,

\item  $\frac{1}{2}\leq (\lambda p')^{-1/p}(nS_-(\alpha
_-)+(n+1)S_+(\alpha_{*}))$, if $k=2n+1$ and $\kappa =+$,

\item  $\frac{1}{2}\leq (\lambda p')^{-1/p}((n+1)S_-(\alpha
_-)+nS_+(\alpha_{*}))$, if $k=2n+1$ and $\kappa =-$,
\end{itemize}

\noindent that is, provided that,

\begin{itemize}
\item  $0<\lambda \leq (n\frac{2S_-(\alpha_-)}{(p')^{1/p}}+n%
\frac{2S_+(\alpha_{*})}{(p')^{1/p}})^{p}$, if $k=2n$.

\item  $0<\lambda \leq (n\frac{2S_-(\alpha_-)}{(p')^{1/p}}%
+(n+1)\frac{2S_+(\alpha_{*})}{(p')^{1/p}})^{p}$, if $k=2n+1$ and
$\kappa =+$.

\item  $0<\lambda \leq ((n+1)\frac{2S_-(\alpha_-)}{(p')^{1/p}}%
+n\frac{2S_+(\alpha_{*})}{(p')^{1/p}})^{p}$, if $k=2n+1$ and
$\kappa =-$.
\end{itemize}

\noindent Furthermore, by Lemma \ref{lemma4}, if $(\ref{njk})_+$
and $(\ref {njk})_-$ hold, the function $E\mapsto T_{k}^{\kappa
}( \lambda ,E) $ is strictly increasing in $\tilde{D}( \lambda ) $
so that, the equation $T_{k}^{\kappa }( \lambda ,E) =\frac{1}{2}
$ in the variable $E\in \tilde{D}( \lambda ) $ admits a solution
in $\tilde{D}( \lambda ) $ if and only if

\begin{itemize}
\item  $0<\lambda \leq (n\frac{2S_-(\alpha_-)}{(p')^{1/p}}+n%
\frac{2S_+(\alpha_{*})}{(p')^{1/p}})^{p}$, if $k=2n$

\item  $0<\lambda \leq (n\frac{2S_-(\alpha_-)}{(p')^{1/p}}%
+(n+1)\frac{2S_+(\alpha_{*})}{(p')^{1/p}})^{p}$, if $k=2n+1$ and
$\kappa =+$

\item  $0<\lambda \leq ((n+1)\frac{2S_-(\alpha_-)}{(p')^{1/p}}%
+n\frac{2S_+(\alpha_{*})}{(p')^{1/p}})^{p}$, if $k=2n+1$ and
$\kappa =-$,
\end{itemize}

\noindent and in this case the solution is unique since the
function $E\mapsto T_{k}^{\kappa }( \lambda ,E) $ is strictly
increasing in $\tilde{D}( \lambda ) $.

Therefore, if we put $J_{\pm }:=\frac{2S_{\pm }(\alpha_{\pm
})}{(p^{\prime })^{1/p}}$ and
$J_{*}:=\frac{2S_+(\alpha_{*})}{(p')^{1/p}}$, Theorem
\ref{theorem1} is proved.
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\paragraph{Proof of Theorem \ref{theorem2}}
Let us assume that $1<q=p$.

For all $k\geq 1$ and $\kappa =+,-$, assume that conditions (\ref{pq1})-(%
\ref{pq4}), and $(\mathbf{H})_{k}^{\kappa }$ hold. It follows for
each $\lambda >0$, the function $E\mapsto T_{k}^{\kappa }( \lambda
,E) $ is defined in $D_{k}^{\kappa }( \lambda ) =(
0,E_{k}^{\kappa }( \lambda ) ) $ and by Lemma \ref {lemma31} and
\ref{lemma32}, one has
\begin{equation*}
\lim_{E\rightarrow 0^+}T_{k}^{\kappa }( \lambda ,E) =\frac{1}{%
2}( \frac{\lambda_{k}}{\lambda a_{0}}) ^{1/p}\text{,\ \ \ \ \ }%
\lim_{E\rightarrow E_{k}^{\kappa }}T_{k}^{\kappa }( \lambda ,E)
=+\infty .
\end{equation*}
Therefore, $\tilde{D}_{k}^{\kappa }(\lambda )=[0,E_{k}^{\kappa
}(\lambda )) $ (see Remark \ref{Remark}). So, the equation
$T_{k}^{\kappa }( \lambda ,E) =\frac{1}{2}$ in the variable $E\in
\tilde{D}_{k}^{\kappa
}( \lambda ) -\left\{ 0\right\} $, admits a solution in $\tilde{D}%
_{k}^{\kappa }( \lambda ) -\left\{ 0\right\} $ provided that
\begin{equation*}
\frac{1}{2}( \frac{\lambda_{k}}{\lambda a_{0}}) ^{1/p}<\frac{1}{2%
},
\end{equation*}
that is, provided that, $\lambda >\lambda_{k}/a_{0}$.
Furthermore, by Lemma \ref{lemma4}, if $(\mathbf{K})_{k}^{\kappa
}$ holds, the function $E\mapsto T_{k}^{\kappa }( \lambda ,E) $
is strictly increasing in $\tilde{D}_{k}^{\kappa }( \lambda )
-\left\{ 0\right\} $ and then, the equation $T_{k}^{\kappa }(
\lambda ,E) =\frac{1}{2}$ in the variable $E\in
\tilde{D}_{k}^{\kappa }( \lambda ) -\left\{ 0\right\} $, admits a
solution in $\tilde{D}_{k}^{\kappa }( \lambda ) -\left\{
0\right\} $ if and only if $\lambda >\lambda_{k}/a_{0}$, and in
this case the solution is unique since $T_{k}^{\kappa }( \lambda
,\cdot ) $ is strictly increasing.

\paragraph{Proof of Theorem \ref{theorem3}}
Let us assume that $1<p<q$.

\paragraph{Proof of Assertion (i)}
If (\ref{pq1})-(\ref{pq4}), and $(\ref{pq5})_{\pm }$ hold, then
for each $\lambda >0$, the function $E\mapsto T_{\pm }( \lambda
,E) $ is defined in $D_{\pm }( \lambda ) =( 0,E_{*}^{\pm }(
\lambda ) ) $ and by Lemma \ref{lemma2},
\begin{equation}
\lim_{E\rightarrow 0^+}T_{\pm }( \lambda ,E) =\lim_{E\rightarrow
E_{*}^{\pm }}T_{\pm }( \lambda ,E) =+\infty .  \label{USA}
\end{equation}
Therefore, $\tilde{D}_{\pm }( \lambda ) =D_{\pm }( \lambda ) =(
0,E_{*}^{\pm }( \lambda ) ) $, and the function $E\mapsto T_{\pm
}( \lambda ,E) $ admits at least a minimum value. Recall that for
all $\lambda >0$, the function $E\mapsto \pm r_{\pm }( \lambda
,E) $ is an increasing $C^1$-diffeomorphism from $( 0,E_{*}^{\pm
}( \lambda ) ) $ onto $( 0,\pm \alpha_{\pm }) $. So, regarding
(\ref{a16}) it follows that the local maximum and minimum values
of $T_{\pm }( \lambda ,\cdot ) $ are in a one to one
correspondence with those of $S_{\pm }(\cdot ) $ respectively.
That is, $S_{\pm }( \cdot ) $ attains a local maximum (resp.
minimum) value at $r_{\pm }^{*}\in I_{\pm }( \alpha_{\pm }) $ if
and only if $T_{\pm }( \lambda ,\cdot ) $ does so at $r_{\pm
}^{-1}( \lambda ,r_{\pm }^{*}) $, where $r_{\pm }^{-1}( \lambda
,\cdot ) $ is the function inverse to $r_{\pm }( \lambda ,\cdot )
$. Let $r_{\pm }^{*}\in I_{\pm }( \alpha_{\pm }) $ be a point
where $S_{\pm }$ attains its global minimum value in $I_{\pm }(
\alpha_{\pm }) $. (The existence of $r_{\pm }^{*}$ is guaranteed
by the limits $\lim_{r\rightarrow 0^+}S_{\pm }( r)
=\lim_{r\rightarrow \pm \alpha_{\pm }}S_{\pm }( r) =+\infty $
which follow from (\ref {a16}) and (\ref{USA})). Thus, for each
fixed $\lambda >0$, there exists a unique $\tilde{E}^{\pm
}=\tilde{E}^{\pm }( \lambda ) \in ( 0,E_{*}^{\pm }( \lambda ) ) $
such that $r_{\pm }^{*}=r_{\pm }( \lambda ,\tilde{E}^{\pm }) $
and then for all $\lambda >0$, and all $E\in ( 0,E_{*}^{\pm }(
\lambda ) ) , $\begin{equation*}
\begin{array}{ccl}
T_{\pm }( \lambda ,E) & = & ( \lambda p')
^{-1/p}S_{\pm }( r_{\pm }( \lambda ,E) ) \\
& \geq & ( \lambda p') ^{-1/p}S_{\pm }( r_{\pm
}^{*}) \\
& = & T_{\pm }( \lambda ,\tilde{E}^{\pm }) ,
\end{array}
\end{equation*}
that is, $T_{\pm }( \lambda ,\cdot ) $ attains its global minimum
value at $\tilde{E}^{\pm }( \lambda ) \in ( 0,E_{*}^{\pm }(
\lambda ) ) $. It follows that

\begin{itemize}
\item  If $( \lambda p') ^{-1/p}S_{\pm }( r_{\pm
}^{*}) >(1/2)$, the equation $T_{\pm }( \lambda ,E) =(1/2) $ in
the variable $E\in ( 0,E_{*}^{\pm }( \lambda ) ) $ admits no
solution.

\item  If $( \lambda p') ^{-1/p}S_{\pm }( r_{\pm
}^{*}) =(1/2)$, the equation $T_{\pm }( \lambda ,E) =(1/2) $ in
the variable $E\in ( 0,E_{*}^{\pm }( \lambda ) ) $ admits at
least a solution (Notice that $S_{\pm }$ may attains its global
minimum at two-or more-distinct points; $r_{\pm }^{*}$ and other
point(s)!).

\item  If $( \lambda p') ^{-1/p}S_{\pm }( r_{\pm
}^{*}) <(1/2)$, the equation $T_{\pm }( \lambda ,E) =(1/2) $ in
the variable $E\in ( 0,E_{*}^{\pm }( \lambda ) ) $ admits at
least two solutions.
\end{itemize}

Hence, the first part of Assertion \textbf{(i)} of Theorem
\ref{theorem3} follows if we put $\mu_1^{\pm }=( 2S_{\pm }( r_{\pm
}^{*}) ) ^p/p'$. Notice that $S_{\pm }( r_{\pm }^{*}) $ is the
(unique) global minimum value of the function $S_{\pm }( \cdot )
$, and do not depends on $r_{\pm }^{*}$ which may be not unique.

\bigskip \ Now, if $(\ref{Miami})_{\pm }$ holds, let us show that the
function $S_{\pm }( \cdot ) $ admits at most a minimum value (and
no maximum one) in $I_{\pm }( \alpha_{\pm }) $. To this end,
since the set $I_{\pm }( \alpha_{\pm }) $ is an interval, that
is, a connected set, it suffices to show that $S_{\pm }( \cdot ) $
admits a minimum value at each of its critical points. This
follows if $S_{\pm }( \cdot ) $ is convex in a neighborhood of
each of its critical points, that is
\begin{equation*}
\frac{dS_{\pm }}{dr}( r) =0\Longrightarrow \frac{d^{2}S_{\pm }}{%
dr^{2}}( r) >0,\text{\ \ for all }r\in I_{\pm }( \alpha _{\pm }) .
\end{equation*}
But this holds as an immediate consequence of Lemma \ref{lemma9}.
Therefore, $S_{\pm }( \cdot ) $ admits at most a minimum value
(and no maximum one) in $I_{\pm }( \alpha_{\pm }) $, and for all
$\lambda >0$, $T_{\pm }( \lambda ,\cdot ) $ admits a unique
minimum value (and no maximum one) in $\tilde{D}_{\pm }( \lambda
) $. So, if we denote (as above) $r_{\pm }^{*}$ the point of
$I_{\pm }( \alpha_{\pm }) $ at which the function $S_{\pm }( \cdot
) $ attains its global minimum, it follows that,

\begin{itemize}
\item  If $( \lambda p') ^{-1/p}S_{\pm }( r_{\pm
}^{*}) >(1/2)$, the equation $T_{\pm }( \lambda ,E) =(1/2) $ in
the variable $E\in \tilde{D}_{\pm }( \lambda ) $ admits no
solution.

\item  If $( \lambda p') ^{-1/p}S_{\pm }( r_{\pm
}^{*}) =(1/2)$, the equation $T_{\pm }( \lambda ,E) =(1/2) $ in
the variable $E\in \tilde{D}_{\pm }( \lambda ) $ admits a unique
solution.

\item  If $( \lambda p') ^{-1/p}S_{\pm }( r_{\pm
}^{*}) <(1/2)$, the equation $T_{\pm }( \lambda ,E) =(1/2) $ in
the variable $E\in \tilde{D}_{\pm }( \lambda ) $ admits exactly
two solutions.
\end{itemize}

Hence, the second part of  \textbf{(i)} of Theorem \ref{theorem3}
follows with $\mu_1^{\pm }=( 2S_{\pm }( r_{\pm }^{*}) ) ^p/p'$.
Notice that this $\mu_1^{\pm }$ is the same as that of the first
part of Assertion \textbf{(i)} above. Therefore, Assertion
(\textbf{i}) of Theorem \ref{theorem3} is proved.

\paragraph{Proof of Assertion (ii)}

Assume that (\ref{pq1})-(\ref{pq4}), $(\ref{pq5})_-$ and
(\ref{pho}) hold. For each $\lambda >0$, the function $E\mapsto
\Theta_-( r_-( \lambda ,E) ) $ is defined in $D( \lambda ) =(
0,E_{*}^{-}( \lambda ) ) $ and by Lemma \ref{lemma8}
\begin{equation}
\lim_{E\mapsto 0^+}\Theta_-( r_-( \lambda ,E) ) =\lim_{E\mapsto (
E_{*}^{-}) ^{-}}\Theta_-( r_-( \lambda ,E) ) =+\infty .
\label{Star}
\end{equation}

Recall that for all $\lambda >0$, the function $E\mapsto r_-(
\lambda ,E) $ is a decreasing $C^1$-diffeomorphism from $(
0,E_{*}^{-}( \lambda ) ) $ onto $( \alpha _-,0) $. Thus,
(\ref{Star}) implies that
\begin{equation*}
\lim_{x\mapsto \alpha_-}\Theta_-( x) =\lim_{x\mapsto
0^{-}}\Theta_-( x) =+\infty .
\end{equation*}
Therefore, there exists at least a $x_{*}\in ( \alpha_-,0) $ at
which the function $x\mapsto \Theta_-( x) $ attains its global
minimum on $( \alpha_-,0) $, say $\Theta_-( x_{*}) $. Note that
$x_{*}$ is independent of $\lambda $. On the other hand, for each
$\lambda >0$, the function $E\mapsto T_{k}^{\kappa }( \lambda ,E)
$ is defined on $D( \lambda ) $ and by Lemmas \ref{lemma31} and
\ref{lemma32}
\begin{equation*}
\lim_{E\mapsto 0^+}T_{k}^{\kappa }( \lambda ,E) =\lim_{E\mapsto (
E_{*}^{-}) ^{-}}T_{k}^{\kappa }( \lambda ,E) =+\infty .
\end{equation*}
Therefore, $\tilde{D}(\lambda )=D(\lambda )$\ (see Remark
\ref{Remark}). Moreover, it attains its global minimum value at a
certain point in $( 0,E_{*}^{-}( \lambda ) ) $ which may depends
on $\lambda ,k$ and $\kappa $. Using Lemma \ref{lemma7} it
follows that for $\lambda
>0,
$\begin{equation*} \min_{E\in ( 0,E_{*}^{-}( \lambda ) )
}T_{k}^{\kappa }( \lambda ,E) \leq k( p'\lambda )
^{-1/p}\Theta_-( r_-( \lambda ,E) ) ,\text{%
for all }E\in ( 0,E_{*}^{-}( \lambda ) ) .
\end{equation*}
In particular,
\begin{equation*}
\min_{E\in ( 0,E_{*}^{-}( \lambda ) ) }T_{k}^{\kappa }( \lambda
,E) \leq k( p'\lambda ) ^{-1/p}\Theta_-( r_-( \lambda
,\tilde{E}_-( \lambda ) ) ) ,
\end{equation*}
where $\tilde{E}_-( \lambda ) \in ( 0,E_{*}^{-}(
\lambda ) ) $ is such that $r_-( \lambda ,\tilde{E}%
_-( \lambda ) ) =x_{*}$. The existence of $\tilde{E}%
_-( \lambda ) $ is guaranteed from the fact that $E\mapsto r_-(
\lambda ,E) $ is a $C^1$-diffeomorphism from $( 0,E_{*}^{-}(
\lambda ) ) $ onto $( \alpha _-,0) $. Thus
\begin{equation*}
\min_{E\in ( 0,E_{*}^{-}( \lambda ) ) }T_{k}^{\kappa }( \lambda
,E) \leq k( p'\lambda ) ^{-1/p}\Theta_-( x_{*}) ,\text{for all
}\lambda >0.
\end{equation*}
Therefore, if $k( p'\lambda ) ^{-1/p}\Theta_-( x_{*}) <(1/2)$, it
follows that $ \min_{E\in ( 0,E_{*}^{-}( \lambda ) )
}T_{k}^{\kappa }( \lambda ,E) <(1/2).$ Thus, the equation
$T_{k}^{\kappa }( \lambda ,E) =\frac{1}{2}$ in the variable $E\in
\tilde{D}( \lambda ) $ admits at least two solutions in
$\tilde{D}( \lambda ) $, for all $\lambda $ satisfying: $k(
p'\lambda ) ^{-1/p}\Theta_-( x_{*}) <(1/2)$, that is, for all
$\lambda >( 2k\Theta_-( x_{*}) ) ^{p}/p'$. Therefore, the
existence part of Assertion \textbf{(ii)} of Theorem
\ref{theorem3} follows by taking $\mu _{k}=( 2k\Theta_-( x_{*}) )
^{p}/p'$. (Notice that $\mu_{k}>0$ since $\Theta_-( x) >0$ for all
$x\in ( \alpha_-,0) $, in particular, $\Theta_-( x_{*}) >0$).

We have find a range of $\lambda $ where there is existence of at
least two solutions for the equation $T_k^\kappa ( \lambda ,E)
=(1/2)$. Let us, now, look for an other range where there is no
solution.

Using Lemma \ref{lemma7} it follows that for $\lambda >0,
$\begin{equation*} k( p'\lambda ) ^{-1/p}(\inf_{E\in (
0,E_{*}^{-}( \lambda ) ) }\Theta_+( r_+( \lambda ,E) ) )\leq
T_k^\kappa ( \lambda ,E) \text{,}
\end{equation*}
for all $E\in ( 0,E_{*}^{-}( \lambda ) ) $. In particular, for
all $\lambda >0$, one has
\begin{equation}
k( p'\lambda ) ^{-1/p}(\inf_{E\in ( 0,E_{*}^{-}( \lambda ) )
}\Theta_+( r_+( \lambda ,E) ) )\leq \min_{E\in ( 0,E_{*}^{-}(
\lambda ) ) }T_k^\kappa ( \lambda ,E) . \label{jhdyz}
\end{equation}

Recall that for all $\lambda >0$, the function $E\mapsto r_+(
\lambda ,E) $ is strictly increasing on the interval $D( \lambda
) =( 0,E_{*}^{-}( \lambda ) ) \subset ( 0,E_{*}^+( \lambda ) ) $.
Hence, for all $\lambda >0$, and $E\in D( \lambda ) $, $r_+(
\lambda ,E) \in ( 0,r_+( \lambda ,E_{*}^{-}( \lambda ) ) ) $. On
the other hand, by Assertion \textbf{(iii)} of Lemma \ref
{lemma2}, the quantity $r_+( \lambda ,E_{*}^{-}( \lambda ) ) $ is
independent of $\lambda >0$ and $r_+( \lambda ,E_{*}^{-}( \lambda
) ) =y( \alpha_-) $. Thus, for all $\lambda >0$ the function
$E\mapsto r_+( \lambda ,E) $ is an increasing diffeomorphism from
$( 0,E_{*}^{-}( \lambda ) ) $ onto $( 0,y( \alpha_-) ) $.
Therefore, the quantity $\inf_{E\in ( 0,E_{*}^{-}( \lambda ) )
}\Theta_+( r_+( \lambda ,E) ) $ is independent of $\lambda >0$ and
\begin{equation*}
\inf_{E\in ( 0,E_{*}^{-}( \lambda ) ) }\Theta _+( r_+( \lambda
,E) ) =\inf_{0<x<y( \alpha _-) }\Theta_+( x) .
\end{equation*}
Thus, (\ref{jhdyz}) becomes,
\begin{equation*}
k( p'\lambda ) ^{-1/p}\inf_{0<x<y( \alpha _-) }\Theta_+( x) \leq
\min_{E\in ( 0,E_{*}^{-}( \lambda ) ) }T_k^\kappa ( \lambda ,E)
,\text{ \ for all }\lambda >0.
\end{equation*}

Let us assume momently that $\inf_{0<x<y( \alpha_-) }\Theta _+(
x) >0$. It follows that for all $\lambda $ satisfying $k(
p'\lambda ) ^{-1/p}\inf_{0<x<y( \alpha _-) }\Theta_+( x) >(1/2)$,
one has
\begin{equation*}
\min_{E\in ( 0,E_{*}^{-}( \lambda ) ) }T_{k}^{\kappa }( \lambda
,E) >(1/2).
\end{equation*}
Hence, the equation $T_{k}^{\kappa }( \lambda ,E) =\frac{1}{2}$
in the variable $E\in \tilde{D}( \lambda ) $ admits no solution
in $\tilde{D}( \lambda ) $ for all $\lambda $ satisfying $k(
p'\lambda ) ^{-1/p}\inf_{0<x<y( \alpha_-) }\Theta_+( x) >(1/2)$,
that is, for all $0<\lambda <( 2k\inf_{0<x<y( \alpha_-)
}\Theta_+( x) ) ^{p}/p'$, provided that\newline $\inf_{0<x<y(
\alpha_-) }\Theta_+( x) $ is strictly positive. In this case, we
put, for all $k\geq 2, $\begin{equation*} \nu_{k}=(
2k\inf_{0<x<y( \alpha_-) }\Theta_+( x) ) ^{p}/p',
\end{equation*}
and thus, Problem (\ref{p1}) admits no solution in $A_{k}^{\kappa
}$ for $\lambda \in ( 0,\nu_{k}) $, $k\geq 2$.

Now, let us prove that $\inf_{0<x<y( \alpha_-) }\Theta _+( x)
>0$. By Lemma \ref{lemma8}, one has
\begin{equation}
\lim_{E\mapsto 0^+}\Theta_+( r_+( \lambda ,E) ) =+\infty \text{,\
and }\lim_{E\mapsto ( E_{*}^{-})
^{-}}\Theta_+( r_+( \lambda ,E) ) =\ell_+%
\text{ }  \label{mhtsa}
\end{equation}
\begin{equation*}
\text{with }\left\{
\begin{array}{lcc}
\ell_+\in ( 0,+\infty ) & \text{if} & y( \alpha
_-) <-\alpha_- \\
\ell_+=+\infty & \text{if} & y( \alpha_-) =-\alpha_-.
\end{array}
\right.
\end{equation*}
However, since for all $\lambda >0$, the function $E\mapsto r_+(
\lambda ,E) $ is an increasing diffeomorphism from $(
0,E_{*}^{-}( \lambda ) ) $ onto $( 0,y( \alpha_-) ) $, it follows
from (\ref{mhtsa}) that
\begin{eqnarray*}
\lim_{x\mapsto 0^+}\Theta_+( x) &=&+\infty \text{,\ and} \\
\lim_{x\rightarrow y( \alpha_-) }\Theta_+( x) &=&\ell_+\text{
with }\left\{
\begin{array}{lcc}
\ell_+\in ( 0,+\infty ) & \text{if} & y( \alpha
_-) <-\alpha_- \\
\ell_+=+\infty & \text{if} & y( \alpha_-) =-\alpha_-.
\end{array}
\right.
\end{eqnarray*}
Therefore, if $y( \alpha_-) <-\alpha_-$ (resp. $y( \alpha_-)
=-\alpha_-$) there exists at least a $x^{*}\in ( 0,y( \alpha_-) ]
$ (resp. $x^{*}\in ( 0,y( \alpha_-) ) $) at which the function
$x\mapsto \Theta _+( x) $ attains its global minimum on $( 0,y(
\alpha_-) ] $ (resp. on $( 0,y( \alpha _-) ) $).

Thus $\inf_{0<x<y( \alpha_-) }\Theta_+( x) =\Theta_+( x^{*}) $.
But, it is clear from its definition that $\Theta_+( x) >0$ for
all $x\in ( 0,-\alpha_-) $. In particular $\Theta_+( x^{*}) >0$.
Therefore, the non existence part of Assertion \textbf{(ii)} of
Theorem \ref{theorem3} is proved, which completes the proof of
Theorem \ref{theorem3}.
%TCIMACRO{\TeXButton{End Proof}{\hfill$\diamondsuit$}}
%BeginExpansion
\hfill$\diamondsuit$%
%EndExpansion

\section{{Open questions}}
\label{sec.6}
\begin{enumerate}
\item  Regarding the identity of Lemma \ref{lemma75} one may ask if there
exists a nonlinearity $\tilde{f}$ such that the corresponding
time-maps would be
\begin{equation*}
\tilde{T}_{\pm }( \lambda ,E) =( \lambda p^{\prime })
^{-1/p}\Theta_{\pm }( r_{\pm }( \lambda ,E)) .
\end{equation*}
In the affirmative, the identity of Lemma \ref{lemma75} implies
that
\begin{equation*}
\tilde{T}_{2n}( \lambda ,E) =T_{2n}( \lambda ,E) .
\end{equation*}
So, does $\tilde{T}_{2n+1}^{\pm }( \lambda ,E) =T_{2n+1}^{\pm }(
\lambda ,E) $, and if not can one compare them? On the other hand
what kind of symmetry does $\tilde{f}$ have: odd, \textit{p.h.o}.,
\textit{n.h.o.,} or something else ? A comparison of the two
solution sets corresponding to $f$ and $\tilde{f}$ would be
interesting.

Notice that this is an inverse problem. Related results are
available in the literature, see for instance Urabe \cite{Urabe},
Schaaf \cite[Chap. 4] {Schaaf}.

\item  A description of the entire solution set should be interesting.
Indeed the main results of this paper describe only the solutions
which are
inside the set $\cup_{k}A_{k}$. So, how does the solution set of Problem (%
\ref{p1}) look like outside $\cup_{k}A_{k}$? (Such kinds of
descriptions can be found in Guedda and Veron \cite{Guedda} or
Addou \cite{Addou7}).

\item  Open questions are numerous. In fact, in view of the known results
for $p=2$ described in Section \ref{sec.2}, one can ask to extend
each one of them to the general case where $p>1$, either in one
or higher dimensions. But a question very close to our main
results is to consider Problem (\ref
{p1}) with $f$ satisfying our conditions but with $q_{0}$ instead of $q$ in (%
\ref{pq4}) and $q_{\pm }$ instead of $q$ in $( \ref{pq5})_{\pm }$
and to give a description of the solution set with respect to all
these parameters: $p$, $q_{0}$, $q_+$, $q_->1$ when $f$ is either
\textit{p.h.o.} or \textit{ n.h.o. }We have considered only the
cases $p>1$, and $q_{0}=q_+=q_->1$, when $f$ is \textit{p.h.o.}
We believe that the same method (Theorem \ref{quad}) works, but
much more patience is required!
\end{enumerate}

\paragraph{ Acknowledgment.} It is a pleasure to thank Professor P.
Korman for sending me many of his recent papers.

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\noindent{\sc  Idris Addou  }\\ 
 U.S.T.H.B., Institut de Math\'ematiques \\
 El-Alia, B.P. no. 32, Bab-Ezzouar \\
 16111, Alger, Alg\'erie. \\
 e-mail: {\tt idrisaddou@hotmail.com}

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