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\markboth{\hfil Exact multiplicity results \hfil EJDE--2000/01}
{EJDE--2000/01\hfil Idris Addou \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2000}(2000), No.~01, pp. 1--26. \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} \\
%
 Exact multiplicity results for quasilinear boundary-value problems with
cubic-like nonlinearities 
\thanks{ {\em 1991 Mathematics Subject Classifications:} 34B15.
\hfil\break\indent
{\em Key words and phrases:} One dimensional p-Laplacian, 
multiplicity results, time-maps.
\hfil\break\indent
\copyright 2000 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Submitted May 26, 1999. Revised October 1, 1999. Published January 1, 2000.} }
\date{}
%
\author{Idris Addou}
\maketitle

\begin{abstract} 
We consider the boundary-value problem 
$$\displaylines{
-(\varphi_p (u'))' =\lambda f(u) \mbox{ in }(0,1) \cr
u(0) = u(1) =0\,,
}$$
where $p>1$, $\lambda >0$ and $\varphi_p (x) =| x|^{p-2}x$.
 The nonlinearity $f$ is cubic-like with three distinct roots $0=a<b<c$.
By means of a quadrature method, we provide the exact number of solutions
for all $\lambda >0$. This way we extend a recent result, for $p=2$, by
Korman et al. \cite{KormanLiOuyang} to the general case $p>1$.  We shall
prove that when $1<p\leq 2$ the structure of the solution set is exactly the
same as that studied in the case $p=2$ by Korman et al. \cite{KormanLiOuyang}, 
and strictly different in the case $p>2$.
\end{abstract}

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

\section{Introduction}\label{sec1} 

We consider the question of determining the exact number of
solutions of the quasilinear boundary-value problem 
\begin{eqnarray}
&-(\varphi_p (u'))' =g(
\lambda ,u) ,\mbox{ in }(0,1)&   \label{P1} \\
&u(0)  = u(1) =0\,,&  \nonumber
\end{eqnarray}
where $p>1$, $\lambda >0$ and $\varphi_p (u) =| u|^{p-2}u$ for all 
$u\in{\mathbb R}$ and $g(\lambda ,u) =\lambda f(u) $.  Here the
nonlinearity $f\in C^{2}({\mathbb R},{\mathbb R}) $ is cubic-like satisfying 
\begin{eqnarray}
&f(0) =f(b) =f(c) =0 \mbox{ for some
constants } 0<b<c\,, & \label{Austine1} \\
&f(x) > 0\mbox{ for } x\in (-\infty ,0) \cup (b,c)  & \label{Austine2} \\
&f(x)  < 0\mbox{ for } x\in (0,b) \cup (c,+\infty)\,,   \nonumber\\
&f''(u) \mbox{ changes sign exactly once when } u\in (0,c)\,,&  \label{Austine3}\\
&F(c) >0, \mbox{ where } F(s) =\int_0^{s}f(u) du,\ s\in{\mathbb R}\,. &
 \label{Austine4}
\end{eqnarray}
Beside conditions (\ref{Austine1})-(\ref{Austine4}) we shall assume in the
case where $p\neq 2$, the following additional conditions: There exists 
$u_0\in (0,c) $ such that 
\begin{equation}
(p-2) f'(u) -uf''(u)  \leq 0 \mbox{ for } u\in (0,u_0]  \label{E1} 
\end{equation}
with strict inequality in an open interval $I \subset (0,u_0)$, and 
\begin{equation}
(p-2) f'(u) -uf''(u) \geq 0\mbox{ for }u\in [ u_0,c) \,.  \label{E2}
\end{equation}
When $p=2$, we prove in Section \ref{sec3}, that (\ref{E1}) and (\ref{E2})
are consequences of (\ref{Austine1})-(\ref{Austine3}).

During this last decade, many articles dealing with boundary-value problems
with cubic-like nonlinearities have been published. (See for instance; 
\cite{Korman97}-\cite{Wei}). However, all the related results have been obtained
for the case $p=2$; that is, for the Laplacian operator. The case of
cubic-like nonlinearities when the differential operator is the $p
$-Laplacian with $p\neq 2$ has yet to be studied.

When $p=2$ and $f$ satisfies conditions (\ref{Austine1})-(\ref{Austine4}),
the solution set of problem (\ref{P1}) was studied recently by Korman et al. 
\cite{KormanLiOuyang}. They provide exactness results. They show (among
other interesting things) that there exists a critical number $\lambda_0>0
$ such that problem (\ref{P1}) has no nontrivial solution for $0<\lambda
<\lambda_0$, has a unique nontrivial solution for $\lambda =\lambda_0$
and has exactly two nontrivial solutions for all $\lambda >\lambda_0$.
So, a natural question arises; how does the solution set of (\ref{P1}) look
like when $p\neq 2$? The purpose of this work is to answer this question.
We shall give an exactness result with respect to $p>1$; we prove, in
particular, that when $1<p\leq 2$ the structure of the solution set of (\ref
{P1}) is exactly the same as that studied in the case $p=2$ by Korman et al. 
\cite{KormanLiOuyang}, and is strictly different in the case $p>2$.

It is known that exactness results are more difficult to derive than a lower
bound of the number of solutions to boundary value problems such as (\ref{P1}).

The main tool used here is the so-called quadrature method. The delicate
part in the process of the proof corresponding to the exactness part of the
main results is the study of the exact variations of the time map under
consideration over its {\em entire} definition domain (Lemma \ref{Lemma3}). 

Notice that here, the cubic-like nonlinearity $f$ has three distinct roots 
$a<b<c$ with $a=0$. Recently, together with A. Benmeza\"{i} \cite{Addou2}
(see also, \cite{Addou22}), we considered the case $a<b=0<c=-a$ and $f$ is
odd for the $p$-Laplacian case with $p>1$. Also, we have considered in \cite
{HalfOdd} a more general case where $a<b=0<c$, $p>1$, and $f$ is not
necessary odd; there we have defined a new kind of functions we called:
half-odd. However, the main results of the present paper are directly
related to those of Korman et al. \cite{KormanLiOuyang} and not to those of 
\cite{Addou2} and \cite{HalfOdd}. That is why we do not describe them here.
(Also, this would require a large space).

The paper is organized as follows. The main results are stated in Section 
\ref{sec2}. Next, in Section \ref{sec3} we shall state and prove some
properties of the nonlinearity $f$. These are of importance in the sequel.
Some preliminary lemmas are the aim of Section \ref{sec4}; the first lemma
(Lemma \ref{Lemma1}) is technical and in the second one (Lemma \ref{Lemmasup})
 we locate all the eventual nontrivial solutions of problem (\ref{P1}). The
proof of Lemma \ref{Lemmasup} is postponed to the appendix. After describing
the quadrature method used in order to look for the solutions, we devote two
lemmas (Lemmas \ref{Lemma2}, \ref{Lemma3}) to study the limits and
variations of the time-map. In Section \ref{sec5}, the main results are
proved. Finally, in Section \ref{sec6} we ask two questions.

\section{Notation and main results}\label{sec2}
In order to state the main results, let us first define the
subsets of $C^{1}([ 0,1]) $ which contain the
solutions of the problem (\ref{P1}).

\noindent Let $A_1^{+}$ be the subset of $C^1([ 0,1]) $
consisting of the functions $u$ satisfying

\begin{itemize}
\item  $u(x) >0$, for all $x\in (0,1) $, $u(0) =u(1) =0<u'(0) $.

\item  $u$ is symmetrical with respect to $1/2$.

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

\noindent Let $\tilde A_1^{+}$ be the subset of $C^1([ 0,1]) 
$ consisting of the functions $u$ satisfying

\begin{itemize}
\item  $u(x) >0$, for all $x\in (0,1) $,  $u(0) =u(1) =0<u'(0) $.

\item  $u$ is symmetrical with respect to $1/2$.

\item  There exists a compact interval $K\subset (0,1) $ such
that for all $x\in (0,1)$, 
\[
u'(x) =0\mbox{ if and only if }x\in K\,.
\]
\end{itemize}

\paragraph{Definition} Let $u\in C([ \alpha ,\beta ]) $
be a function with two consecutive zeros $x_1<x_2$.  We call the I-hump
of $u$ the restriction of $u$ to the open interval $I=(
x_1,x_2) $.  When there is no confusion we refer to a hump of $u$.

\noindent Let $B^{+}(k) $, ($k\geq 1$) be the subset of $C^1([
0,1]) $ consisting of the functions $u$ satisfying

\begin{itemize}
\item  For all $i\in \{ 0,\cdots ,k\} $ there exist 
$a_{i}=a_{i}(u) $, $b_{i}=b_{i}(u) $ in $[
0,1] $ such that 
\begin{eqnarray*}
&0 =a_0\leq b_0<\cdots <a_{i}\leq b_{i}<\cdots <a_k\leq b_k=1 &\\
&u > 0\mbox{ in  }(b_{i},a_{i+1}) ,\mbox{ for all }i\in
\{ 0,\cdots ,k-1\}  \\
&u \equiv 0\mbox{ in }[ a_{i},b_{i}] ,\mbox{ for all }i\in
\{ 0,\cdots ,k-1\}\, .&
\end{eqnarray*}

\item  Every hump of $u$ is symmetrical with respect to the center of the
interval of its definition.

\item  The derivative of each hump of $u$ vanishes once and only once.

\item  Each hump is a translated copy of the first one.
\end{itemize}

\noindent Let $B_k^{+}$ be the subset of $B^{+}(k) $ consisting of the
functions $u$ satisfying 
\[
a_i(u) =b_i(u) \mbox{ for all }i\in \{0,\cdots ,k\} . 
\]
If there exists $i_0\in \{ 0,\cdots ,k\} $ such that 
$a_{i_0}(u) <b_{i_0}(u) $ we say that $u\in \tilde B_k^{+}$.  
Therefore, $B^{+}(k) =B_k^{+}\cup \tilde B_k^{+}$ and 
$B_k^{+}\cap \tilde B_k^{+}=\emptyset $.

We call two functions $u_1$,  $u_2$ in $\tilde{B}_k^+$ ($k\geq 1$),
equivalent if for all $i\in \{ 0,\cdots ,k\} $, the $i$-th
hump of $u_1$ is a translated copy of the $i$-th hump of $u_2$, or
equivalently, $u_2$ is obtained from $u_1$ by translating some (maybe
all) of its humps. It is clear that this is an equivalence relation. For all 
$u\in \tilde{B}_k^+$ we denote $\mathop{\rm Cl}(u) $ the equivalence
class of $u$.

Notice that when $f$ satisfies (\ref{Austine1}), (\ref{Austine2}) and (\ref
{Austine4}), there exists a unique $r\in (b,c)$ such that 
\begin{equation}
F(r) =0\,.  \label{Klo}
\end{equation}

For $p>1$ and $x\in [ r,c]$, define
\[
S_{+}(x) =\int_0^{x}\{ F(x) -F(\xi) \} ^{-1/p}d\xi \,.
\]
We shall prove in Lemma \ref{Lemma2} that $S_{+}(r) $ (resp. 
$S_{+}(c) $) is infinite if and only if $1<p\leq 2$.  So, for $p>2$
we can define $\nu =(2S_{+}(c)) ^{p}/p'$,
where $p'=p/(p-1) $, and for all integer $k\geq 0$ we
define $\lambda_k=(2kS_{+}(r)) ^{p}/p'$
and notice that 
\[
0=\lambda_0<\lambda_1<\cdots <\lambda_k=k^{p}\lambda_1\dots 
\mbox{ for all }k\geq 1,\mbox{ and }\lim_{k\to +\infty }\lambda
_k=+\infty \,.
\]
For $\lambda >0$, denote $S_\lambda $ the solution set of problem (\ref
{P1}).

%  fig1.tex	 
\begin{figure}[t]
\setlength{\unitlength}{1mm}
\begin{picture}(90,40)(-15,0)
\linethickness{1pt}
\qbezier(80,30)(-40,20)(80,10)
\qbezier[20](20,0)(20,10)(20,20)
\thinlines
\put(10,0){\vector(0,1){40}}
\put(10,0){\vector(1,0){80}}
\put(18,-5){$\lambda_0$}
\put(90,-5){$\lambda$}
\put(38,28){$A_1^+$}
\end{picture}
\caption{$1< p \leq 2$.}
\end{figure}

The main results are worth being described by means of diagrams.
The first result (Theorem \ref{wmdjkgfh}) concerns the case where $1<p\leq 2$. 
The corresponding diagram (Figure 1) indicates the existence of a unique
branch and it is $\subset $-shaped. So, there is no nontrivial solution for 
$0<\lambda <\lambda_0$, a unique nontrivial solution for $\lambda =\lambda
_0$, and exactly two nontrivial solutions for $\lambda >\lambda_0$. All
these solutions are in $A_1^+$.

% fig2.tex
\begin{figure}[t]
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\begin{picture}(90,40)(4,-2)
\linethickness{1pt}
\qbezier(25,0)(-8,12)(57,31)
\qbezier[15](13.8,0)(13.8,5)(13.8,9)
\qbezier[50](57,0)(57,16)(57,31)
\put(38,28){$A_1^+$}
\qbezier(57,31)(70,35)(80,37)
\put(82,37){$\widetilde{A}_1^+$}
\qbezier(25,0)(25,16)(80,28)
\put(82,28){$\widetilde{B}_1^+$}
\qbezier(38,0)(38,12)(80,21)
\put(82,21){$\widetilde{B}_2^+$}
\qbezier(51,0)(51,8)(80,14)
\put(82,14){$\widetilde{B}_{n-1}^+$}
\qbezier(64,0)(64,4)(80,7)
\put(82,7){$\widetilde{B}_n^+$}
\thinlines
\put(10,0){\vector(0,1){40}}
\put(10,0){\vector(1,0){76}}
\put(13,-3){$\mu$}
\put(25,-3){$\lambda_1$}
\put(38,-3){$\lambda_2$}
\put(49,-3){$\lambda_{n-1}$}
\put(57,-3){$\nu$}
\put(64,-3){$\lambda_n$}
\put(85,-3){$\lambda$}
\end{picture}
\caption{$p>2$, $\lambda_{n-1}<\nu<\lambda_n$, $1< n$.}
\end{figure}

% fig3.tex

\begin{figure}[t]
\setlength{\unitlength}{1.3mm}
\begin{picture}(90,42)(8,-3)
\linethickness{1pt}
\qbezier(25,0)(25,20)(57,31)
\qbezier[50](57,0)(57,16)(57,31)
\put(38,26){$A_1^+$}
\qbezier(57,31)(70,35)(80,37)
\put(82,37){$\widetilde{A}_1^+$}
\qbezier(25,0)(25,16)(80,28)
\put(82,28){$\widetilde{B}_1^+$}
\qbezier(38,0)(38,12)(80,21)
\put(82,21){$\widetilde{B}_2^+$}
\qbezier(51,0)(51,8)(80,14)
\put(82,14){$\widetilde{B}_{n-1}^+$}
\qbezier(64,0)(64,4)(80,7)
\put(82,7){$\widetilde{B}_n^+$}
\thinlines
\put(15,0){\vector(0,1){40}}
\put(15,0){\vector(1,0){75}}
\put(25,-3){$\lambda_1$}
\put(38,-3){$\lambda_2$}
\put(49,-3){$\lambda_{n-1}$}
\put(57,-3){$\nu$}
\put(64,-3){$\lambda_n$}
\put(85,-3){$\lambda$}
\end{picture}
\caption{$p>2$, $\lambda_{n-1}<\nu<\lambda_n$, $1< n$.}
\end{figure}

When $p>2$, we have to consider the sequence $(\lambda_k)_{k\geq 0}$ 
and the number $\nu >0$.  This number maybe smaller than $\lambda_1$,
 equal to $\lambda_1$, or greater than $\lambda_1$.  
 In this later case, it may lie between two consecutive points of the
sequence: $\lambda_{n-1}<\nu <\lambda_{n}$, with $n>1$ (Figures 2 and 3),
or it maybe equal to some $\lambda_{n}$ with $n>1$.

An immediate examination of these bifurcation diagrams, shows that when 
$\nu$ moves from zero to infinity, the upper branch changes but not the others,
i.e., beside the upper branch which is different from a diagram to an other,
the remaining branches are the same in all these diagrams.

Now consider any one of figures 2 or 3 and let us describe each kind of its
branches. The $\lambda $-axis designates the trivial solutions, and at each 
$\lambda_k$, $k\geq 1$, there is a bifurcation point which indicates a
pair $(u_k,\lambda_k)$ such that $u_k\in B_k^+$.

The upper branch contains a point which indicates a pair 
$(u_1,\nu)$ such that $u_1\in A_1^+$. All points lying on this
branch which are on the left of $(u_1,\nu)$ are in $A_1^+$, and
those lying at the right are in $\tilde{A}_1^+$.

The remaining branches are in some sense ''singular''. Usually a point 
$(u,\lambda)$ lying on any branch designates a couple where $u$ is a
solution of some kind and $\lambda $ is a real number. This is the case in
our diagrams as far as the upper branch or the lower one ($\lambda $-axis)
are concerned. However, a point on the remaining branches indicates 
$(\mathop{\rm Cl}(u),\lambda)$, i.e., the equivalence class of a certain solution $u$
lying in some $\tilde{B}_k^+$, $k\geq 1$, and $\lambda $ is a real
number. So, if $u$ is a solution in some $\tilde{B}_k^+$, with $k\geq 1
$, for some $\lambda >0$ then any $v\in \mathop{\rm Cl}(u)$ is also a solution in the same 
$\tilde{B}_k^+$.

The singularity of these branches maybe removed. In fact, consider the same
equivalence relation but defined on $B_k^+$, (for all $k\geq 1$). Then
it is clear that 
\[
\mathop{\rm Cl}(u)=\{u\}, \mbox{ for all } u\in B_k^+\,. 
\]
So, the bifurcation points on the $\lambda $-axis maybe considered as 
indicating \\ 
$(\mathop{\rm Cl}(u_k),\lambda_k)=(\{u_k\},\lambda_k)$ instead of 
$(u_k,\lambda_k)$.  Also, consider on 
$A_1^+\cup \tilde{A}_1^+$ the same equivalence relation in essence (which maybe formulated
differently). It is clear that 
\[
\mathop{\rm Cl}(u)=\{u\}, \mbox{ for all } u\in A_1^+\cup \tilde{A}_1^+\,. 
\]
This way, any point on any branch shall designates a couple 
$(\mathop{\rm Cl}(u),\lambda)$ and the elements in $\mathop{\rm Cl}(u)$ are solutions of the
problem (\ref{P1}) for the same $\lambda $.  Therefore, there is coherence in
the diagrams and the singularity mentioned above is removed.

The statements of the main results below indicate that for 
$\nu \leq \lambda_1$ the upper branch contains a turning point, but when 
$\nu >\lambda_1$, either it still contains a turning point (Figure 2) or 
there is no such point (Figure 3). \smallskip

 The main results read as follows

\begin{theorem}
\label{wmdjkgfh} Assume that $1<p\leq 2$ and $f$ satisfies conditions (\ref
{Austine1})-(\ref{Austine4}), and (\ref{E1}), (\ref{E2}). Then there exists 
$\lambda_0>0$ such that

\begin{description}
\item[(i)]  If $0<\lambda <\lambda_0$, $S_\lambda =\{ 0\} $.

\item[(ii)]  If $\lambda =\lambda_0$, there exists $v_\lambda \in
A_1^+$ such that $S_\lambda =\{ 0\} \cup \{ v_\lambda\} $.

\item[(iii)]  If $\lambda >\lambda_0$, there exists $v_\lambda $, 
$w_\lambda \in A_1^+$ such that $v_\lambda \neq w_\lambda $ and 
$S_\lambda =\{ 0\} \cup \{ v_\lambda ,w_{\lambda
}\} $.
\end{description}
\end{theorem}

\begin{theorem}
\label{wmdjkgfh2}Assume that $p>2$ and $f$ satisfies conditions (\ref
{Austine1})-(\ref{Austine4}), and (\ref{E1}), (\ref{E2}). Moreover, assume
that $\nu <\lambda_1$.  Then there exists $\mu \in (0,\nu) 
$ such that

\begin{description}
\item[(i)]  If $0<\lambda <\mu $, $S_\lambda =\{ 0\} $.

\item[(ii)]  If $\lambda =\mu $, there exists $v_\lambda \in A_1^+$
such that $S_\lambda =\{ 0\} \cup \{ v_\lambda \} $.

\item[(iii)]  If $\mu <\lambda \leq \nu $, there exists $v_\lambda $,
$w_\lambda \in A_1^+$ such that $v_\lambda \neq w_\lambda $ and 
$S_\lambda =\{ 0\} \cup \{ v_\lambda ,w_{\lambda}\} $.

\item[(iv)]  If $\nu <\lambda <\lambda_1$, there exists $v_\lambda \in
A_1^+$ and $u_\lambda \in \tilde{A}_1^+$ such that $S_{\lambda
}=\{ 0\} \cup \{ v_\lambda \} \cup \{u_\lambda \} $.

\item[(v)]  If $\lambda =\lambda_1$, there exists $u_\lambda \in \tilde{
A}_1^+$ and $u_{\lambda ,1}\in B_1^+$ such that $S_{\lambda
}=\{ 0\} \cup \{ u_\lambda \} \cup \{u_{\lambda ,1}\} $.

\item[(vi)]  If $\lambda_k<\lambda <\lambda_{k+1}$, $k\geq 1$, there
exists $u_\lambda \in \tilde{A}_1^+$ and $u_{\lambda ,1},\cdots
,u_{\lambda ,k}$ such that $u_{\lambda ,i}\in \tilde{B}_{i}^{+}$ for
all $i=1,\cdots ,k$ and $S_\lambda =\{ 0\} \cup \{
u_\lambda \} \cup \mathop{\rm Cl}(u_{\lambda ,1}) \cup \cdots \cup
\mathop{\rm Cl}(u_{\lambda ,k}) $.

\item[(vii)]  If $\lambda =\lambda_{k+1},k\geq 1$, there exists 
$u_\lambda \in \tilde{A}_1^+$and $u_{\lambda ,1},\cdots
u_{\lambda ,k+1}$ such that $u_{\lambda ,i}\in \tilde{B}_{i}^{+}$ for
all $i=1,\cdots ,k$, $u_{\lambda ,k+1}\in B_{k+1}^{+}$ and $S_{\lambda
}=\{ 0\} \cup \{ u_\lambda \} \cup \mathop{\rm Cl}(
u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k})
\cup \{ u_{\lambda ,k+1}\} $.
\end{description}
\end{theorem}

\begin{theorem}
\label{wmdjkgfh3}Assume that $p>2$ and $f$ satisfies conditions (\ref
{Austine1})-(\ref{Austine4}), and (\ref{E1}), (\ref{E2}). Moreover, assume
that $\nu =\lambda_1$.  Then there exists $\mu \in (0,\lambda_1) $ such that

\begin{description}
\item[(i)]  If $0<\lambda <\mu $, $S_\lambda =\{ 0\} $.

\item[(ii)]  If $\lambda =\mu $, there exists $v_\lambda \in A_1^+$
such that $S_\lambda =\{ 0\} \cup \{ v_\lambda \}$.

\item[(iii)]  If $\mu <\lambda <\lambda_1$, there exists $v_\lambda $,  
$w_\lambda \in A_1^+$ such that $v_\lambda \neq w_\lambda $ and 
$S_\lambda =\{ 0\} \cup \{ v_\lambda ,w_{\lambda
}\} $.

\item[(iv)]  If $\lambda =\lambda_1$, there exists $v_\lambda \in
A_1^+$ and $u_{\lambda ,1}\in B_1^+$ such that $S_{\lambda
}=\{ 0\} \cup \{ v_\lambda \} \cup \{
u_{\lambda ,1}\} $.

\item[(v)]  If $\lambda_k<\lambda <\lambda_{k+1}$, $k\geq 1$, there
exists $u_\lambda \in \tilde{A}_1^+$ and $u_{\lambda ,1},\cdots
,u_{\lambda ,k}$ such that $u_{\lambda ,i}\in \tilde{B}_{i}^{+}$ for
all $i=1,\cdots ,k$ and $S_\lambda =\{ 0\} \cup \{
u_\lambda \} \cup \mathop{\rm Cl}(u_{\lambda ,1}) \cup \cdots \cup
\mathop{\rm Cl}(u_{\lambda ,k}) $.

\item[(vi)]  If $\lambda =\lambda_{k+1},k\geq 1$, there exists 
$u_\lambda \in \tilde{A}_1^+$and $u_{\lambda ,1},\cdots
,\;u_{\lambda ,k+1}$ such that $u_{\lambda ,i}\in \tilde{B}_{i}^{+}$
for all $i=1,\cdots ,k$, $u_{\lambda ,k+1}\in B_{k+1}^{+}$ and 
$S_\lambda =\{ 0\} \cup \{ u_\lambda \} \cup
\mathop{\rm Cl}(u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda
,k}) \cup \{ u_{\lambda ,k+1}\} $.
\end{description}
\end{theorem}

\begin{theorem}
\label{wmdjkgfh4}Assume that $p>2$ and $f$ satisfies conditions (\ref
{Austine1})-(\ref{Austine4}), and (\ref{E1}), (\ref{E2}). Moreover, assume
that there exists $n>1$ such that $\lambda_{n-1}<\nu <\lambda_{n}$.  Then
one and only one of the following possibilities occurs:

Possibility {\bf A}. There exists $\mu \in (0,\lambda_1) $
such that

\begin{description}
\item[(i)]  If $0<\lambda <\mu $, $S_\lambda =\{ 0\} $.

\item[(ii)]  If $\lambda =\mu $, there exists $v_\lambda \in A_1^+$
such that $S_\lambda =\{ 0\} \cup \{ v_\lambda \} $.

\item[(iii)]  If $\mu <\lambda <\lambda_1$, there exist $v_\lambda $,  
$w_\lambda \in A_1^+$ such that $v_\lambda \neq w_\lambda $ and 
$S_\lambda =\{ 0\} \cup \{ v_\lambda ,w_{\lambda}\} $.

\item[(iv)]  If $\lambda =\lambda_1$, there exist $v_\lambda \in
A_1^+$ and $u_{\lambda ,1}\in B_1^+$ such that $S_{\lambda
}=\{ 0\} \cup \{ v_\lambda \} \cup \{
u_{\lambda ,1}\} $.

\item[(v)]  If $\lambda_k<\lambda <\min \{ \lambda_{k+1},\nu
\} $, $1\leq k\leq n-1$, there exist $v_\lambda \in A_1^+$ and 
$u_{\lambda ,1},\cdots ,u_{\lambda ,k}$ such that $u_{\lambda
,i}\in \tilde{B}_{i}^{+}$ for all $i=1,\cdots ,k$ and $S_{\lambda
}=\{ 0\} \cup \{ v_\lambda \} \cup \mathop{\rm Cl}(
u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k}) $.

\item[(vi)]  If $\lambda =\lambda_{k+1},1\leq k\leq n-2$, there exist
$v_\lambda \in A_1^+$ and $u_{\lambda ,1},\cdots ,
u_{\lambda ,k+1}$ such that $u_{\lambda ,i}\in \tilde{B}_{i}^{+}$ for
all $i=1,\cdots ,k$, $u_{\lambda ,k+1}\in B_{k+1}^{+}$ and $S_{\lambda
}=\{ 0\} \cup \{ v_\lambda \} \cup \mathop{\rm Cl}(
u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k})
\cup \{ u_{\lambda ,k+1}\} $.

\item[(vii)]  If $\lambda =\nu $, there exist $v_\lambda \in A_1^+$
and $u_{\lambda ,1},\cdots ,u_{\lambda ,n-1}$ such that $u_{\lambda
,i}\in \tilde{B}_{i}^{+}$ for all $i=1,\cdots ,n-1$, and $S_{\lambda
}=\{ 0\} \cup \{ v_\lambda \} \cup \mathop{\rm Cl}(
u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,n-1})$.

\item[(viii)]  If $\max \{ \lambda_k,\nu \} <\lambda
<\lambda_{k+1}$, $k\geq n-1$, there exist $u_\lambda \in \tilde{A}
_1^+$ and $u_{\lambda ,1},\cdots ,u_{\lambda ,k}$ such that 
$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$ for all $i=1,\cdots ,k$, and 
$S_\lambda =\{ 0\} \cup \{ u_\lambda \} \cup
\mathop{\rm Cl}(u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(
u_{\lambda,k}) $.

\item[(ix)]  If $\lambda =\lambda_{k+1}$, $k\geq n-1$, there exist 
$u_\lambda \in \tilde{A}_1^+$ and $u_{\lambda ,1},\cdots
,u_{\lambda ,k+1}$ such that $u_{\lambda ,i}\in \tilde{B}_{i}^{+}$ for
all $i=1,\cdots ,k$, $u_{\lambda ,k+1}\in B_{k+1}^{+}$and $S_{\lambda
}=\{ 0\} \cup \{ u_\lambda \} \cup \mathop{\rm Cl}(
u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k})
\cup \{ u_{\lambda ,k+1}\} $.
\end{description}

Possibility {\bf B}.

\begin{description}
\item[(i)]  If $0<\lambda <\lambda_1$, $S_\lambda =\{ 0\} $.

\item[(ii)]  If $\lambda =\lambda_1$, there exists $u_{\lambda ,1}\in
B_1^+$ such that $S_\lambda =\{ 0\} \cup \{ u_{\lambda,1}\} $.

\item[(iii)]  If $\lambda_k<\lambda <\min \{ \lambda_{k+1},\nu
\} $, $1\leq k\leq n-1$, there exist $v_\lambda \in A_1^+$ and 
$u_{\lambda ,1},\cdots ,u_{\lambda ,k}$ such that $u_{\lambda
,i}\in \tilde{B}_{i}^{+}$ for all $i=1,\cdots ,k$ and $S_{\lambda
}=\{ 0\} \cup \{ v_\lambda \} \cup \mathop{\rm Cl}(
u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k}) $.

\item[(iv)]  If $\lambda =\lambda_{k+1},1\leq k\leq n-2$, there exist
$v_\lambda \in A_1^+$ and $u_{\lambda ,1},\cdots ,
u_{\lambda ,k+1}$ such that $u_{\lambda ,i}\in \tilde{B}_{i}^{+}$ for
all $i=1,\cdots ,k$, $u_{\lambda ,k+1}\in B_{k+1}^{+}$ and $S_{\lambda
}=\{ 0\} \cup \{ v_\lambda \} \cup \mathop{\rm Cl}(
u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k})
\cup \{ u_{\lambda ,k+1}\} $.

\item[(v)]  If $\lambda =\nu $, there exist $v_\lambda \in A_1^+$ and 
$u_{\lambda ,1},\cdots ,u_{\lambda ,n-1}$ such that $u_{\lambda
,i}\in \tilde{B}_{i}^{+}$ for all $i=1,\cdots ,n-1$, and $S_{\lambda
}=\{ 0\} \cup \{ v_\lambda \} \cup \mathop{\rm Cl}(
u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,n-1})$.

\item[(vi)]  If $\max \{ \lambda_k,\nu \} <\lambda <\lambda
_{k+1}$, $k\geq n-1$, there exist $u_\lambda \in \tilde{A}_1^+$ and 
$u_{\lambda ,1},\cdots ,u_{\lambda ,k}$ such that $u_{\lambda
,i}\in \tilde{B}_{i}^{+}$ for all $i=1,\cdots ,k$, and $S_{\lambda
}=\{ 0\} \cup \{ u_\lambda \} \cup \mathop{\rm Cl}(
u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k}) $.

\item[(vii)]  If $\lambda =\lambda_{k+1}$, $k\geq n-1$, there exist 
$u_\lambda \in \tilde{A}_1^+$ and $u_{\lambda ,1},\cdots
,u_{\lambda ,k+1}$ such that $u_{\lambda ,i}\in \tilde{B}_{i}^{+}$ for
all $i=1,\cdots ,k$, $u_{\lambda ,k+1}\in B_{k+1}^{+}$and $S_{\lambda
}=\{ 0\} \cup \{ u_\lambda \} \cup \mathop{\rm Cl}(
u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k})
\cup \{ u_{\lambda ,k+1}\} $.
\end{description}
\end{theorem}

\begin{theorem}
\label{wmdjkgfh5}Assume that $p>2$ and $f$ satisfies conditions (\ref
{Austine1})-(\ref{Austine4}), and (\ref{E1}), (\ref{E2}). Moreover, assume
that there exists $n>1$ such that $\nu =\lambda_{n}$.  Then one and only
one of the following possibilities occurs:

Possibility {\bf C}. There exists $\mu \in (0,\lambda_1) $
such that

\begin{description}
\item[(i)]  If $0<\lambda <\mu $, $S_\lambda =\{ 0\} $.

\item[(ii)]  If $\lambda =\mu $, there exists $v_\lambda \in A_1^+$
such that $S_\lambda =\{ 0\} \cup \{ v_\lambda \}$.

\item[(iii)]  If $\mu <\lambda <\lambda_1$, there exist $v_\lambda $,  
$w_\lambda \in A_1^+$ such that $v_\lambda \neq w_\lambda $ and 
$S_\lambda =\{ 0\} \cup \{ v_\lambda ,w_{\lambda
}\} $.

\item[(iv)]  If $\lambda =\lambda_1$, there exist $v_\lambda \in
A_1^+$ and $u_{\lambda ,1}\in B_1^+$ such that $S_{\lambda
}=\{ 0\} \cup \{ v_\lambda \} \cup \{u_{\lambda ,1}\} $.

\item[(v)]  If $\lambda_k<\lambda <\lambda_{k+1}$, $1\leq k\leq n-1$,
there exist $v_\lambda \in A_1^+$ and $u_{\lambda ,1},\cdots
,u_{\lambda ,k}$ such that $u_{\lambda ,i}\in \tilde{B}_{i}^{+}$ for
all $i=1,\cdots ,k$ and $S_\lambda =\{ 0\} \cup \{
v_\lambda \} \cup \mathop{\rm Cl}(u_{\lambda ,1}) \cup \cdots \cup
\mathop{\rm Cl}(u_{\lambda ,k}) $.

\item[(vi)]  If $\lambda =\lambda_{k+1},1\leq k\leq n-1$, there exist 
$v_\lambda \in A_1^+$ and $u_{\lambda ,1},\cdots ,
u_{\lambda ,k+1}$ such that $u_{\lambda ,i}\in \tilde{B}_{i}^{+}$ for
all $i=1,\cdots ,k$, $u_{\lambda ,k+1}\in B_{k+1}^{+}$ and $S_{\lambda
}=\{ 0\} \cup \{ v_\lambda \} \cup \mathop{\rm Cl}(
u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k})
\cup \{ u_{\lambda ,k+1}\} $.

\item[(vii)]  If $\lambda_k<\lambda <\lambda_{k+1}$, $k\geq n$, there
exists $u_\lambda \in \tilde{A}_1^+$ and $u_{\lambda ,1},\cdots
,u_{\lambda ,k}$ such that $u_{\lambda ,i}\in \tilde{B}_{i}^{+}$ for
all $i=1,\cdots ,k$, and $S_\lambda =\{ 0\} \cup \{
u_\lambda \} \cup \mathop{\rm Cl}(u_{\lambda ,1}) \cup \cdots \cup
\mathop{\rm Cl}(u_{\lambda ,k}) $.

\item[(viii)]  If $\lambda =\lambda_{k+1}$, $k\geq n$, there exist 
$u_\lambda \in \tilde{A}_1^+$ and $u_{\lambda ,1},\cdots
,u_{\lambda ,k+1}$ such that $u_{\lambda ,i}\in \tilde{B}_{i}^{+}$ for
all $i=1,\cdots ,k$, $u_{\lambda ,k+1}\in B_{k+1}^{+}$ and $S_{\lambda
}=\{ 0\} \cup \{ u_\lambda \} \cup \mathop{\rm Cl}(
u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k})
\cup \{ u_{\lambda ,k+1}\} $.
\end{description}

Possibility {\bf D}.

\begin{description}
\item[(i)]  If $0<\lambda <\lambda_1$, $S_\lambda =\{ 0\} $.

\item[(ii)]  If $\lambda =\lambda_1$, there exists $u_{\lambda ,1}\in
B_1^+$ such that $S_\lambda =\{ 0\} \cup \{ u_{\lambda,1}\} $.

\item[(iii)]  If $\lambda_k<\lambda <\lambda_{k+1}$, $1\leq k\leq n-1$,
there exist $v_\lambda \in A_1^+$ and $u_{\lambda ,1},\cdots
,u_{\lambda ,k}$ such that $u_{\lambda ,i}\in \tilde{B}_{i}^{+}$ for
all $i=1,\cdots ,k$ and $S_\lambda =\{ 0\} \cup \{
v_\lambda \} \cup \mathop{\rm Cl}(u_{\lambda ,1}) \cup \cdots \cup
\mathop{\rm Cl}(u_{\lambda ,k}) $.

\item[(iv)]  If $\lambda =\lambda_{k+1},1\leq k\leq n-1$, there exist 
$v_\lambda \in A_1^+$ and $u_{\lambda ,1},\cdots ,
u_{\lambda ,k+1}$ such that $u_{\lambda ,i}\in \tilde{B}_{i}^{+}$ for
all $i=1,\cdots ,k$, $u_{\lambda ,k+1}\in B_{k+1}^{+}$ and $S_{\lambda
}=\{ 0\} \cup \{ v_\lambda \} \cup \mathop{\rm Cl}(
u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k})
\cup \{ u_{\lambda ,k+1}\} $.

\item[(v)]  If $\lambda_k<\lambda <\lambda_{k+1}$, $k\geq n$, there
exists $u_\lambda \in \tilde{A}_1^+$ and $u_{\lambda ,1},\cdots
,u_{\lambda ,k}$ such that $u_{\lambda ,i}\in \tilde{B}_{i}^{+}$ for
all $i=1,\cdots ,k$, and $S_\lambda =\{ 0\} \cup \{
u_\lambda \} \cup \mathop{\rm Cl}(u_{\lambda ,1}) \cup \cdots \cup
\mathop{\rm Cl}(u_{\lambda ,k}) $.

\item[(vi)]  If $\lambda =\lambda_{k+1}$, $k\geq n$, there exist 
$u_\lambda \in \tilde{A}_1^+$ and $u_{\lambda ,1},\cdots
,u_{\lambda ,k+1}$ such that $u_{\lambda ,i}\in \tilde{B}_{i}^{+}$ for
all $i=1,\cdots ,k$, $u_{\lambda ,k+1}\in B_{k+1}^{+}$ and $S_{\lambda
}=\{ 0\} \cup \{ u_\lambda \} \cup \mathop{\rm Cl}(
u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k})
\cup \{ u_{\lambda ,k+1}\} $.
\end{description}
\end{theorem}

The novelty in these results concerns the cases $p>1$ with $p\neq 2$.  The
case $p=2$ was proved by Korman et al. \cite{KormanLiOuyang}. Of course, the
case $p=2$ is also studied here.

\section{Some properties of the nonlinearity $f$}\label{sec3} 
In this section we establish some properties of $f$. These are
used in the sequel and are of importance in our analysis. We first state the
properties and next we give the proofs.

\subsection*{Statement of properties}

Assume that $f$ satisfies (\ref{Austine1})-(\ref{Austine3}), then there
exists $u_0\in (0,c) $ such that 
\begin{eqnarray}
&f''\geq 0\mbox{ in }(0,u_0]\,,  &\label{A1}\\
&f''\leq 0\mbox{ in }[ u_0,c) \,. &\label{A2}
\end{eqnarray}
Moreover, there exist two open intervals $I$ and $J$ with $I\subset (
0,u_0) $ and $J\subset (u_0,c) $ such that 
\begin{eqnarray}
&f''>0\mbox{ in }I \,,& \label{A3}\\
&f''<0\mbox{ in }J\,.&  \label{A4}
\end{eqnarray}
Hence 
\begin{eqnarray}
&f'\mbox{ is increasing in }(0,u_0], \mbox{ and strictly increasing in } 
I\,,  &\label{A5} \\
&f'\mbox{ is decreasing in }[ u_0,c) ,\mbox{ and
strictly decreasing in }J\,. & \label{A6}
\end{eqnarray}
Furthermore, 
\begin{equation}
f'(0) <0,f'(u_0)>0,f'(c) <0.  \label{A7}
\end{equation}
Hence, there exist $u_0^{-}\in (0,u_0) $ and 
$u_0^{+}\in (u_0,c) $ such that 
\begin{eqnarray}
&f'\leq 0\mbox{ in }[ 0,u_0^{-}) \cup (u_0^{+},c] & \label{A8} \\
&f'(u_0^{-}) =f'(u_0^{+}) =0 &\label{A9}\\
& f'>0\mbox{ in }(u_0^{-},u_0^{+})\, .&\label{A10}
\end{eqnarray}
Moreover, 
\begin{equation}
f'(b) >0\,.  \label{A11} \end{equation}
So, 
\begin{eqnarray}
&0<u_0^{-}<b<u_0^{+}<c\,,\mbox{ and }&  \label{A12} \\
&f\mbox{ attains its minimum (resp. maximum)  on $[ 0,c]$ 
at $u_0^{-}$ (resp. at $u_0^{+}$)}.  &\label{A13}
\end{eqnarray}

\subsection*{Proof of properties}

By (\ref{Austine1}) and (\ref{Austine2}), it follows that $f''$ must 
change sign at least once in $(0,c) $, say at $u_0$,
and by (\ref{Austine3}), it follows that (\ref{A1}) and (\ref{A2}) hold. If 
$f''\equiv 0$ in $(0,u_0] $ it follows that 
$f'$ is constant in $(0,u_0] $.  But, by (\ref
{Austine1}) and (\ref{Austine2}) it follows that $f'(0)
\leq 0$.  Thus, $f'\leq 0$ in $(0,u_0] $.  On the
other hand, by (\ref{A2}) it follows that $f'$ is decreasing in 
$[ u_0,c) $, thus $f'\leq 0$ in $[
u_0,c) $ and furthermore, $f$ is decreasing in $[
0,c] $.  By $f(0) =f(c) =0$ it follows that 
$f\equiv 0$ in $[ 0,c] $ which contradicts (\ref{Austine2}), and
therefore, since $f\in C^{2}$, the existence of $I$ is proved and that of $J$
is similar. Thus, (\ref{A3}) and (\ref{A4}) are proved. Immediate
consequences are (\ref{A5}) and (\ref{A6}). So, $f'$ attains its
maximum value on $(0,c) $ at $u_0$.  Thus, it can easily be
proved that $f'(u_0) >0$.  In fact, if the contrary
holds, it follows that $f'\leq 0$ in $(0,c) $ and
hence $f$ is monotonic decreasing in $(0,c) $, and by $f(
0) =f(c) =0$ it follows that $f\equiv 0$ in $[
0,c] $ which contradicts (\ref{Austine2}), which proves that 
$f'(u_0) >0$.

Let us prove that $f'(0) <0$ (resp. $f'(
c) <0$). If the contrary holds, that is if $f'(0)
\geq 0$ (resp. $f'(c) \geq 0$), by (\ref{A5}) (resp. by
(\ref{A6})) it follows that $f'(x) \geq 0$ for all 
$x\in (0,u_0) $ (resp. $x\in (u_0,c) $) and
hence, $f$ is increasing in $(0,u_0) $ (resp. in $(
u_0,c) $). Due to the fact that $f$ vanishes at $0$ (resp. at $c
$), it follows that $f\geq 0$ in $(0,u_0) $ (resp. $f\leq 0$
in $(u_0,c) $), which contradicts (\ref{Austine2}).
Therefore, (\ref{A7}) is proved. By making use of continuity arguments it
follows that (\ref{A5}), (\ref{A6}) and (\ref{A7}) imply (\ref{A8}), (\ref
{A9}) and (\ref{A10}).

Let us prove (\ref{A11}). First, by (\ref{Austine2}) it follows that 
$f'(b) \geq 0$.  If $f'(b) =0$, by (%
\ref{A7})-(\ref{A10}) it follows that $b\in (0,u_0^{-}) \cup
(u_0^{+},c) $.  Assume that $b\in (0,u_0^{-}) $
(resp. $b\in (u_0^{+},c) $). By (\ref{A8}), it follows that 
$f'\leq 0$ in $(0,b) $ (resp. in $(b,c) 
$) and therefore $f$ is decreasing in $(0,b) $ (resp. in 
$(b,c) $). By $f(0) =f(b) =0$ (resp. 
$f(b) =f(c) =0$) it follows that $f\equiv 0$ in 
$[ 0,b] $ (resp. in $[ b,c] $) which contradicts (%
\ref{Austine2}). Therefore, (\ref{A11}) is proved, and immediate
consequences are (\ref{A12}) and (\ref{A13}).

\section{Preliminary lemmas}\label{sec4}

Lemma \ref{Lemma1} is a technical one. The aim of the next lemma
is to answer to the question : how does any solution to (\ref{P1}) look like
? We shall prove that if $u$ is a nontrivial solution to (\ref{P1}), then 
\[
u\in A_1^+\cup \tilde{A}_1^+\cup \bigcup_{k\geq 1}B^{+}(k) . 
\]
The proof of Lemma \ref{Lemma1} is the same as that of Lemma 8 in \cite
{Addou6} or Lemmas 6 and 8 in \cite{Addou4} (see also analogous lemmas in 
\cite{Addou1}, \cite{Addou3}). So, it is omitted.
The proof of Lemma \ref{Lemmasup} is not complicated but long and tedious.
So, it is postponed to the appendix.

Next, we define the time map on its interval of definition, compute its
limits at the boundary points of its definition domain in 
Lemma \ref{Lemma2}, and then study its exact variations on its entire 
definition domain in Lemma \ref{Lemma3}.

\begin{lemma}
\label{Lemma1}Assume that $f\in C({\mathbb R}) $ satisfies (\ref{Austine2}) and (\ref{Austine4}). Consider the
function defined in ${\mathbb R}^{\pm }$ by
\begin{equation}
s\longmapsto G_{\pm }(\lambda ,E,s) :=E^{p}-p'\lambda F(s) ,  \label{Eq.1}
\end{equation}
where $E,\lambda >0$ and $p>1$ are real parameters. Then

\begin{description}
\item[(i)]  If $E>E_*(p,\lambda) :=(p'\lambda F(c)) ^{1/p}$ 
(resp. $E>0$), the function  $G_{+}(\lambda ,E,\cdot) $ 
(resp. $G_{-}(\lambda,E,\cdot) $) is strictly positive in
 ${\mathbb R}^{+}$ (resp. in ${\mathbb R}^{-}$).

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

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

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

\item[(b)]  $\lim\limits_{E\to 0^{+}}s_{+}(\lambda ,E)
=r$   and  $\lim\limits_{E\to E_*}s_{+}(\lambda ,E)
=c$, where $r$ is the unique zero of $F$ in $(b,c) $ (see, (%
\ref{Klo})).
\end{description}
\end{description}
\end{lemma}

\noindent The following lemma locate all possible nontrivial solutions.

\begin{lemma}
\label{Lemmasup}Let $u$ be a nontrivial solution of (\ref{P1}). Then
\[
u\in A_1^+\cup \tilde{A}_1^+\cup \bigcup_{k\geq 1}B^{+}(
k) ,\mbox{ and }0\leq u'(0) \leq
E_*(\lambda) =(p'\lambda F(c)
) ^{1/p}. 
\]
\end{lemma}

According to this lemma, for all fixed $\lambda >0$ and $p>1$, we shall
look for the solutions of problem (\ref{P1}) with respect to their
derivative at the origin; $u'(0) =E\in [0,E_*(\lambda) ] $.

For $\lambda >0$, $p>1$ and $E\in [ 0,E_*(\lambda)] $, let 
\[
X_{+}(\lambda ,E) =\{ s>0:E^{p}-p'\lambda F(\xi) >0\,,\ \forall \xi \in (0,s) \}\,. 
\]
By Lemma \ref{Lemma1}, it follows that 
\[
X_{+}(\lambda ,E) =\left\{ 
\begin{array}{lcl}
(0,c) & \mbox{if} & E=E_* \\ 
(0,s_{+}(\lambda ,E)) & \mbox{if} & 0<E<E_*, \\ 
(0,r) & \mbox{if} & E=0
\end{array} \right. 
\]
and therefore, 
\begin{equation}
r_{+}(\lambda ,E) :=\sup X_{+}(\lambda ,E)
=\left\{ 
\begin{array}{lcl}
c & \mbox{if} & E=E_* \\ 
s_{+}(\lambda ,E) & \mbox{if} & 0<E<E_*, \\ 
r & \mbox{if} & E=0\,,
\end{array} \right.  \label{mfrkl1}
\end{equation}
and one deduces from Lemma \ref{Lemma1} the following 
\begin{eqnarray}
&\frac{\partial r_{+}}{\partial E}(\lambda ,E) >0,\forall
\lambda >0,\forall E\in (0,E_*(\lambda)) ,&\label{eq11} \\
&\lim\limits_{E\to 0^{+}}r_{+}(\lambda ,E) =r,\mbox{ and }
\lim\limits_{E\to E_*}r_{+}(\lambda,E) =c\,.  &\label{abhu}
\end{eqnarray}

Define, for any $p>1$, $\lambda >0$ the time map $T_{+}$ \thinspace by 
\begin{equation}
T_{+}(\lambda ,E) :=\int_0^{r_{+}(\lambda ,E)
}(E^p-p'\lambda F(\xi)) ^{-1/p}d\xi
,\;E\in [ 0,E_*(\lambda) ] ,  \label{qdrj}
\end{equation}
with the convention $T_{+}(\lambda ,0) =+\infty $ (resp. 
$T_{+}(\lambda ,E_*(\lambda)) =+\infty $) if
the integral in (\ref{qdrj}) diverges.

Arguing as in Guedda and Veron \cite{GueddaVeron}, it follows that

\begin{itemize}
\item  For each $\lambda >0$ and $E\in (0,E_*(\lambda
)) $, problem (\ref{P1}) admits a solution $u\in A_1^+$
satisfying $u'(0) =E$ if and only if $T_{+}(
\lambda ,E) =1/2$, and in this case the solution is unique and its
sup-norm is equal to $r_{+}(\lambda ,E) $.

\item  For each $\lambda >0$, problem (\ref{P1}) admits a solution $u\in
A_1^+$ satisfying $u'(0) =E_*(\lambda
) $ if and only if $T_{+}(\lambda ,E_*(\lambda)
) =1/2$, and in this case the solution is unique and its sup-norm is
equal to $c$.

\item  For each $\lambda >0$, problem (\ref{P1}) admits a solution $u\in 
\tilde{A}_1^+$ satisfying $u'(0) =E_*(
\lambda) $ if and only if $T_{+}(\lambda ,E_*(\lambda
)) <1/2$, and in this case the solution is unique and its
sup-norm is equal to $c$.

\item  For each $\lambda >0$ and $n\in {\mathbb N}^*$, 
problem (\ref{P1}) admits a solution $u\in B_{n}^{+}$ if and only if 
$nT_{+}(\lambda ,0) =1/2$, and in this case the solution is
unique and its sup-norm is equal to $r$.

\item  For each $\lambda >0$ and $n\in {\mathbb N}^*$,
problem (\ref{P1}) admits a solution $u\in \tilde{B}_{n}^{+}$ if and
only if $nT_{+}(\lambda ,0) <1/2$, and in this case $v$ is an
other solution in $\tilde{B}_{n}^{+}$ if and only if $v\in \mathop{\rm Cl}(u) 
$, and the sup-norm of each solution is equal to $r$.
\end{itemize}

A simple change of variables shows that, 
\begin{equation}
T_{+}(\lambda ,E) =r_{+}(\lambda ,E)
\int_0^{1}(E^{p}-p'\lambda F(r_{+}(\lambda
,E) \xi)) ^{-1/p}d\xi ,  \label{A}
\end{equation}
which can be written as, 
\begin{equation}
T_{+}(\lambda ,E) =(r_{+}(\lambda ,E)
/E) \int_0^{1}(1-p'\lambda F(r_{+}(
\lambda ,E) \xi) /E^{p}) ^{-1/p}d\xi .  \label{qze}
\end{equation}
Also, observe that one has from the definition of $s_{+}(\lambda
,E) $, (Lemma \ref{Lemma1}, Assertion {\bf (iii)}), $E^{p}=\lambda
p'F(r_{+}(\lambda ,E)) $, so, (\ref{A})
may be written as, 
\begin{equation}
T_{+}(\lambda ,E) =(\lambda p')
^{-1/p}\int_0^{r_{+}(\lambda ,E) }(F(r_{+}(
\lambda ,E)) -F(\xi)) ^{-1/p}d\xi .
\label{a15}
\end{equation}
For any $p>1$ and $x\in [ r,c] $ let us define $S_{+}(
x) $ by 
\[
S_{+}(x) :=\int_0^{x}(F(x) -F(\xi
)) ^{-1/p}d\xi \in [ 0,+\infty ] . 
\]
Thus, (\ref{a15}) may be written as, 
\begin{equation}
T_{+}(\lambda ,E) =(\lambda p')^{-1/p}S_{+}(r_{+}(\lambda ,E)) .  \label{a16}
\end{equation}

The limits of the time map $T_{+}(\lambda ,\cdot) $ are the
aim of the following.

\begin{lemma}
\label{Lemma2}
\begin{description}
\item[(i)]  $S_{+}(r) =+\infty $ if and only if $1<p\leq 2$, and 
$S_{+}(c) =+\infty $ if and only if $1<p\leq 2$.

\item[(ii)]  $\lim\limits_{E\to 0^{+}}T_{+}(\lambda
,E) =(\lambda p') ^{-1/p}S_{+}(r) $
and $\lim\limits_{E\to E_*}T_{+}(\lambda ,E) =(
\lambda p') ^{-1/p}S_{+}(c) $.
\end{description}
\end{lemma}


\paragraph{Proof of (i). } By (\ref{Austine2}) and (\ref{Klo}), one has 
\[
\lim_{x\to r}\frac{F(r) -F(x) }{r-x}=f(r) >0\,. 
\]
Thus, there exist $\delta >0$ and $M>0$ such that 
\[
F(r) -F(x) >M(r-x) ,\mbox{ for all } x\in (r-\delta ,r) \,. 
\]
Therefore, 
\[
\int_{r-\delta }^{r}\frac{dx}{(F(r) -F(x)
) ^{1/p}}<M^{-1/p}\int_{r-\delta }^{r}\frac{dx}{(r-x)
^{1/p}}<+\infty \mbox{ for all }p>1\,. 
\]
On the other hand, using L'Hopital's rule twice and (\ref{A7}) it follows
that 
\[
\lim_{x\to 0^{+}}\frac{F(0) -F(x) }{-x^{2}}=\frac{f'(0) }{2}<0\,. 
\]
Thus, there exist $\varepsilon >0$, $m_{-}<0$ and $M_{-}<0$ such that 
\[
m_{-}\leq \frac{F(0) -F(x) }{-x^{2}}\leq M_{-},
\mbox{ for all }x\in (0,\varepsilon)\, . 
\]
Therefore, 
\[
(-m_{-}) ^{-1/p}\int_0^{\varepsilon }\frac{dx}{x^{2/p}}\leq
\int_0^{\varepsilon }\frac{dx}{(F(0) -F(x)
) ^{1/p}}\leq (-M_{-}) ^{-1/p}\int_0^{\varepsilon }\frac{dx}{x^{2/p}}\,. 
\]
The first part of Assertion {\bf (i)} follows from $F(r)
=F(0) =0$ and the well-known fact 
\[
\int_0^{\varepsilon }\frac{dx}{x^{2/p}}<+\infty \mbox{   if and only if } p>2. 
\]
The second part may be proved similarly. In fact, using L'Hopital's rule
twice and (\ref{A7}) it follows that 
\[
\lim_{x\to c^{-}}\frac{F(c) -F(x) }{(
c-x) ^{2}}=-\frac{f'(c) }{2}>0. 
\]
Thus 
\[
M_{+}^{-1/p}\int_{c-\varepsilon }^{c}\frac{dx}{(c-x) ^{2/p}}\leq
\int_{c-\varepsilon }^{c}\frac{dx}{(F(c) -F(x)
) ^{1/p}}\leq m_{+}^{-1/p}\int_{c-\varepsilon }^{c}\frac{dx}{(
c-x) ^{2/p}} 
\]
for some strictly positive constants $M_{+},m_{+}$ and $\varepsilon $.
Therefore, the second assertion of {\bf (i)} follows from the well-known
fact 
\[
\int_{c-\varepsilon }^{c}\frac{dx}{(c-x) ^{2/p}}<+\infty \mbox{
  if and only if }p>2. 
\]

\paragraph{Proof of (ii).} The value of the limits follows by passing to the limit
in (\ref{a16}) as $E$ tends to $0$ and $E_*$ respectively.
Then Lemma \ref{Lemma2} is proved.  \hfill$\diamondsuit$\medskip

To study the exact number of solutions of (\ref{P1}) we need to know the
exact variations of the time map $T_{+}(\lambda ,\cdot) $
over all its definition domain $(0,E_*(\lambda)
) $.  These variations are the aim of the following,

\begin{lemma}
\label{Lemma3} If $1<p\leq 2$, for all $\lambda >0$ the time map 
$T_{+}(\lambda ,\cdot) $ admits a unique critical point; a
minimum.
If $p>2$, for all $\lambda >0$ {\em either} the time map $T_{+}(
\lambda ,\cdot) $ is strictly increasing {\em or} admits a unique
critical point; a minimum in $(0,E_*(\lambda))$.
\end{lemma}

\paragraph{Proof}
By (\ref{a16}), recall that for all $\lambda >0$ and $E\in (
0,E_*(\lambda))$. 
\[
T_{+}(\lambda ,E) =(\lambda p')^{-1/p}S_{+}(r_{+}(\lambda ,E)) . 
\]
On the other hand, by (\ref{eq11}) and (\ref{abhu}), for each fixed $\lambda
>0$, the function $E\mapsto r_{+}(\lambda ,E) $ is an
increasing $C^{1}-$diffeomorphism from $(0,E_*(\lambda
)) $ onto $(r,c) $, where $r$ is the unique zero
of $F$ in $(b,c) $.  A differentiation yields 
\[
\frac{\partial T_{+}}{\partial E}(\lambda ,E) =(\lambda
p') ^{-1/p}\times \frac{\partial r_{+}}{\partial E}(
\lambda ,E) \times S_{+}'(r_{+}(\lambda,E)) . 
\]
Thus, to study the variations of $T_{+}(\lambda ,\cdot) $ in 
$(0,E_*(\lambda)) $ it suffices to study those
of $S_{+}(\cdot) $ in $(r,c) $.
One has 
\[
S_{+}(\rho) =\int_0^\rho \{ F(\rho) -F(u) \} ^{-1/p}du,\;\rho \in (r,c) 
\]
and 
\begin{equation}
S_{+}'(\rho) =\frac 1{p\rho }\int_0^\rho \frac{%
H_p(\rho) -H_p(u) }{\{ F(\rho)
-F(u) \} ^{(p+1)/p}}du,\;\rho \in (r,c)\,,
\label{Dop}
\end{equation}
where 
\begin{equation}
H_p(u) =pF(u) -uf(u) ,\mbox{ for all }
u\in [ 0,c] \mbox{ and }p>1\,.  \label{B1}
\end{equation}

To study the sign of the derivative $S_{+}'(\cdot) $ we need to study that 
of expression 
\[
H_p(\rho) -H_p(u) \;\mbox{ for all }0<u<\rho \mbox{ and }r<\rho <c\,. 
\]
So, a careful analysis of the variations of $H_p(\cdot) $ is
required. One has 
\begin{equation}
H_p'(u) =(p-1) f(u) -uf'(u), \mbox{ for all }u\in [ 0,c] 
\mbox{ and }p>1 \label{B2}
\end{equation}
and 
\begin{equation}
H_p''(u) =(p-2) f'(
u) -uf''(u) ,\mbox{ for all }u\in [0,c] \mbox{ and }p>1\,.  \label{B3}
\end{equation}
By (\ref{Austine1}) it follows that 
\begin{equation}
H_p(0) =H_p'(0) =0, \mbox{ for all }p>1\,,
\label{B4}
\end{equation}
and by (\ref{Austine1}) and (\ref{Austine3}), it follows that 
\begin{equation}
H_p(c) >0, \mbox{ for all }p>1,  \label{B5}
\end{equation}
and by (\ref{A7}), 
\begin{equation}
H_p'(c) >0, \mbox{ for all }p>1.  \label{B6}
\end{equation}

Now, let us look closely to the special case where $p=2$.  By (\ref{A1}), it
follows that 
\[
H_2''(u) \leq 0,\mbox{ for all }u\in (0,u_0] \,, 
\]
and by (\ref{B4}) it follows that 
\begin{equation}
H_2'(u) \leq 0,\mbox{ for all }u\in [0,u_0]\, .  \label{krs}
\end{equation}
By (\ref{A3}) it follows that there exists a unique $\alpha \in [0,u_0) $ 
such that 
\begin{eqnarray}
&H_2(u) =H_2'(u) =0\mbox{ for all } u\in [ 0,\alpha ] & \label{B7} \\
&H_2(u) < 0\mbox{ and }H_2'(u) <0 \mbox{ for all } 
u\in (\alpha ,u_0]\,.&  \nonumber
\end{eqnarray}
On the other hand, by (\ref{A2}) it follows that 
$H_2''(u) \geq 0$,  for all $u\in [u_0,c)$, 
and by (\ref{B6}) and (\ref{B7}) there exist $\beta $ and $\gamma $ in 
$(u_0,c) $ such that 
$$\displaylines{
u_0<\beta \leq \gamma <c \cr
H_2'(u) <0,\mbox{ for all }u\in [u_0,\beta) \cr
H_2'(u) =0, \mbox{ for all }u\in [ \beta,\gamma ]  \cr
H_2'(u) >0,\mbox{ for all }u\in (\gamma ,c]\, . \cr
}$$
Therefore, regarding (\ref{B4}) and (\ref{krs}), there exists a unique 
$\delta \in (\gamma ,c) $ such that 
$$\displaylines{
H_2(u) <0,\mbox{ for all }u\in [ u_0,\beta ]  \cr
H_2(u) =H_2(\beta) <0,\mbox{ for all }u\in [ \beta ,\gamma ]  \cr
H_2(u) <0,\mbox{ for all }u\in [ \gamma ,\delta) \mbox{ and }H_2(\delta) =0 \cr
H_2(u) >0,\mbox{ for all }u\in (\delta ,c]\,. \cr
}$$
Thus, for all fixed $\rho \in (0,\alpha ]$,
\begin{equation}
H_2(\rho) -H_2(u) =0,\mbox{ for all }u\in(0,\rho) ,  \label{S0}
\end{equation}
and for all fixed $\rho \in (\alpha ,\gamma ] $
\begin{equation}
H_2(\rho) -H_2(u) <0\mbox{ for all }u\in (0,\min (\rho ,\beta)), 
 \label{S1}
\end{equation}
and for all fixed $\rho \in [ \delta ,c) $,
\begin{equation}
H_2(\rho) -H_2(u) >0\mbox{ for all }u\in
(0,\rho) .  \label{S2}
\end{equation}
Notice that to obtain (\ref{S0})-(\ref{S2}) the starting conditions were
conditions (\ref{A1}), (\ref{A2}) and (\ref{A3}). In contrast, if $p\neq 2$, 
$H_p''$ changes sign in $(0,u_0) $ since by
(\ref{A1}), $f''$ is of constant sign in $(0,u_0) $ and by (\ref{A8}),
 (\ref{A9}) and (\ref{A10}), $f'$ changes sign in $(0,u_0) $.  
This leads us to consider the additional conditions (\ref{E1}) and (\ref{E2}) 
for all $p>1$, and therefore 
\begin{equation}
H_p''(u) \leq 0,\mbox{ for all }u\in (
0,u_0] ,\mbox{ and  }p>1  \label{B88}
\end{equation}
with strict inequality in an open interval $I_p\subset (0,u_0)$,
and 
\begin{equation}
H_p''(u) \geq 0,\mbox{ for all }u\in [u_0,c) \mbox{ and }p>1\,. \label{B9}
\end{equation}
Let us emphasize that (\ref{E1}) and (\ref{E2}) are automatically satisfied
if $p=2$.  In fact they are reduced to (\ref{A1})-(\ref{A3}). So, (\ref{E1})
and (\ref{E2}) do not consist as additional conditions for the special case
where $p=2$.

By (\ref{B4}) and (\ref{B88}) it follows that there exists a unique $\alpha
_p\in [ 0,u_0) $ such that 
\begin{eqnarray}
&H_p(u) =H_p'(u) =0,\mbox{ for all }u\in [ 0,\alpha_p ]\,, &\label{M0}\\
&H_p(u) <0,\ H_p'(u) <0,\mbox{ for all } u\in (\alpha_p ,u_0]\, .&\label{M05}
\end{eqnarray}
By (\ref{B6}) and (\ref{B9}) it follows that for all $p>1$, there exist 
$\beta_p $ and $\gamma_p $ in $(u_0,c) $ such that 
\begin{eqnarray}
&u_0<\beta_p \leq \gamma_p <c &\nonumber\\
&H_p'(u) <0,\mbox{ for all }u\in [u_0,\beta_p) & \label{M1}\\
&H_p'(u) =0,\mbox{ for all }u\in [ \beta_p,\gamma_p ] & \label{M2}\\
&H_p'(u) >0,\mbox{ for all }u\in (\gamma_p,c] \,. & \label{M3}
\end{eqnarray}
Therefore, there exists a unique $\delta_p \in (\gamma_p,c) $ such that 
\begin{eqnarray}
&H_p(u) <0,\mbox{ for all }u\in [ u_0,\beta_p] & \label{M4}\\
&H_p(u) =H_p(\beta_p) <0,\mbox{ for all } u\in [ \beta_p ,\gamma_p ]  
   &\label{M5} \\
&H_p(u) <0,\mbox{ for all }u\in [ \gamma_p ,\delta
_p) \mbox{ and    }H_p(\delta_p) =0  &\label{M6} \\
&H_p(u) >0,\mbox{ for all }u\in (\delta_p,c] .  &\label{M7}\end{eqnarray}
This implies that: for all $p>1$ and all fixed $\rho \in (0,\alpha_p]$,
\begin{equation}
H_p(\rho) -H_p(u) =0,\mbox{ for all }u\in (0,\rho) ,  \label{C0}
\end{equation}
and for all fixed $\rho \in (\alpha_p,\gamma_p] $,
\begin{equation}
H_p(\rho) -H_p(u) <0,\mbox{ for all }u\in (0,\min (\rho ,\beta_p)) \,, 
\label{C1}
\end{equation}
and for all fixed $\rho \in [ \delta_p,c) $,
\begin{equation}
H_p(\rho) -H_p(u) >0,\mbox{ for all }u\in(0,\rho) \,.  \label{C2}
\end{equation}
Notice that 
\[
H_p(r) =pF(r) -rf(r) =-rf(r)<0\,. 
\]
Thus, by (\ref{M0}), (\ref{M05}), (\ref{M6}) and (\ref{M7}) it follows that 
\begin{equation}
\alpha_p<r<\delta_p,\mbox{ for all }p>1.  \label{S3}
\end{equation}
By (\ref{C2}) it follows that 
\begin{equation}
S_{+}'(\rho) >0\mbox{ for all }\rho \in [
\delta_p,c) \subset (r,c) .  \label{S4}
\end{equation}
It remains to study the variations of $S_{+}(\cdot) $ on the
interval $(r,\delta_p) $.  Notice that one has to distinguish
two cases 
\begin{eqnarray}
&\alpha_p<r<\gamma_p<\delta_p<c \,,& \label{Case1}\\
&\gamma_p\leq r<\delta_p<c\,.  &\label{Case2}
\end{eqnarray}
It is easy to see that if (\ref{Case1}) holds, then  
\begin{equation}
S_{+}'(\rho) <0\mbox{ for all }\rho \in (r,\gamma_p] .  \label{I}
\end{equation}
In fact, this follows by (\ref{C0}), (\ref{C1}), (\ref{M5}) and (\ref{Dop}).
Now, we will show in the case (\ref{Case1}) (resp. case (\ref{Case2})) that 
$S_{+}'$ admits at most one zero in $(\gamma_p,\delta
_p) $ (resp. in $(r,\delta_p) $).

Easy computations show that for all $\rho \in (r,c) $,
\[
S_{+}''(\rho) =\frac{p+1}{(p\rho
) ^{2}} \int_0^{\rho }\frac{(H_p(\rho)
-H_p(u)) ^{2}}{\{ F(\rho) -F(
u) \} ^{(2p+1)/p}}\,du+\frac{1}{p\rho ^{2}}\int_0^{\rho }\frac{%
\Phi_p (\rho) -\Phi_p (u) }{\{ F(
\rho) -F(u) \} ^{(p+1)/p}}\,du 
\]
where $\Phi_p (u) :=-p(p+1) F(u) +2puf(u) -u^{2}f'(u) $,
 for all $u\in (0,c)$.

Let $K$ be a real number. Thus, for all $\rho \in (r,c)$,
\begin{eqnarray}
\lefteqn{p\rho^{2}S_{+}''(\rho) +p\rho KS_{+}'(\rho) }\label{Kol}\\
&=& \int_0^{\rho }\frac{\Psi_p (\rho) -\Psi_p (u) 
}{\{ F(\rho) -F(u) \} ^{(p+1)/p}}\,du
+\frac{p+1}{p}\int_0^{\rho }\frac{(H_p(\rho) -H_p(u)) ^{2}}{\{ F(\rho)
-F(u) \} ^{(2p+1)/p}}\,du   \nonumber
\end{eqnarray}
where $\Psi_p (u) =\Phi_p (u) +KH_p(u)$, for all $u\in (0,c)$. 
Choose $K=p+1$.  Thus $\Psi_p (u) =uH_p'(u)$,  for all $u\in (0,c)$. 

Next, in the case (\ref{Case1}) (resp. case (\ref{Case2})), we split the
first integral in (\ref{Kol}) as follows 
\begin{eqnarray}
\lefteqn{ \int_0^{\rho }\frac{\Psi_p (\rho) -\Psi_p (u) 
}{\{ F(\rho) -F(u) \} ^{(p+1)/p}} \,du  }\label{Jil2}\\
&=&\int_0^{\beta_p }\frac{\rho H_p'(\rho)
-uH_p'(u) }{\{ F(\rho) -F(u) \} ^{(p+1)/p}}\,du+  
\int_{\beta_p }^{\gamma_p }\frac{\rho H_p'(\rho)
-uH_p'(u) }{\{ F(\rho) -F(u) \} ^{(p+1)/p}}\,du \nonumber\\
&&+\int_{\gamma_p }^{\rho }\frac{\rho H_p'(\rho) -uH_p'(u) }{
\{ F(\rho) -F(u) \} ^{(p+1)/p}}\,du  \nonumber
\end{eqnarray}
for all $\rho \in (\gamma_p ,\delta_p) $ (resp. for all 
$\rho \in (r,\delta_p) \subset (\gamma_p ,\delta_p) $).

By (\ref{M3}), (\ref{M0}), (\ref{M05}) and (\ref{M1}) the first integral in 
(\ref{Jil2}) is strictly positive, and by (\ref{M2}) and (\ref{M3}) the
second one is also strictly positive.

By (\ref{B9}) it follows that $H_p'$ is increasing in $(
\gamma_p ,\delta_p) $.  Using (\ref{M3}) it follows that for all 
$\rho \in (\gamma_p ,\delta_p) $ (resp. $\rho \in (r,\delta_p) $), 
\[
0<H_p'(u) \leq H_p'(\rho) \mbox{ for all }u\in (\gamma_p ,\rho) \subset (
\gamma_p ,\delta_p) . 
\]
Therefore, $\rho H_p'(\rho) -uH_p'(u)\geq 0$,  for all 
$u\in (\gamma_p ,\rho)$. 
Thus, the third integral in (\ref{Jil2}) is positive. It follows that 
\[
\rho S_{+}''(\rho) +(p+1)
S_{+}'(\rho) >0,\mbox{ for all }\rho \in (
\gamma_p ,\delta_p) ,\mbox{ (resp. }\rho \in (r,\delta_p))\,,
\]
which implies that $S_{+}$ is convex in a neighborhood of each of its
critical points lying in $(\gamma_p ,\delta_p) $ (resp.
in $(r,\delta_p) $). Thus, $S_{+}'$ vanishes at
most once in $(\gamma_p ,\delta_p) $ (resp. in $(
r,\delta_p) $) for all $p>1$.  Therefore; regarding (\ref{S4}), it
follows that $S_{+}$ is either strictly increasing in $(r,c) $
or strictly decreasing in $(r,s_p) $ for some $s_p\in
(r,\delta_p) $ and then strictly increasing in $(
s_p,c) $. For the special case $1<p\leq 2$, by Lemma \ref{Lemma2}
one has 
\[
\lim_{\rho \to r^{+}}S_{+}(\rho) =\lim_{\rho
\to c^{-}}S_{+}(\rho) =+\infty . 
\]
Thus, the first possibility above cannot occur, and in this case $S_{+}$
admits a unique critical point which is a minimum. Therefore, Lemma \ref
{Lemma3} is proved. \hfill$\diamondsuit$

\section{Proofs of main results}\label{sec5}
Assume that $1<p\leq 2$.  By Lemma \ref{Lemma2} and \ref{Lemma3},
for all fixed $\lambda >0$, the time map $T_{+}(\lambda ,\cdot
) $ admits a unique critical point which is a minimum in $(
0,E_*(\lambda)) $ and satisfies 
\[
\lim_{E\to 0^{+}}T_{+}(\lambda ,E) =\lim_{E\to
E_*}T_{+}(\lambda ,E) =+\infty . 
\]
Also, by Lemma \ref{Lemma2} 
\[
\lim_{\rho \to r^{+}}S_{+}(\rho) =\lim_{\rho
\to c^{-}}S_{+}(\rho) =+\infty , 
\]
and by the proof of Lemma \ref{Lemma3}, $S_{+}$ admits a unique critical
point, a minimum in $(r,c) $ at $r_*$, say. Therefore,
based upon the fact that for all $\lambda >0$, $r_{+}(\lambda ,\cdot
) $ is strictly increasing from $(0,E_*(\lambda
)) $ onto $(r,c) $, it follows that there exists
a unique $\tilde E=\tilde E(\lambda) \in (0,E_*(
\lambda)) $ such that $r_*=r_{+}(\lambda ,\tilde
E(\lambda)) $.  Thus, by (\ref{a16}), for all $E\in
(0,E_*(\lambda)) $ 
\begin{eqnarray*}
T_{+}(\lambda ,\tilde E(\lambda)) &=&(
p'\lambda) ^{-1/p}S_{+}(r_*) \\
  &\leq &(p'\lambda) ^{-1/p}S_{+}(r_{+}(
\lambda ,E)) =T_{+}(\lambda ,E) ,
\end{eqnarray*}
hence, $T_{+}(\lambda ,\cdot) $ attains its unique global
minimum value at $\tilde E(\lambda) \in (0,E_*(
\lambda)) $.  It follows that

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

\item  If $(p'\lambda) ^{-1/p}S_{+}(r_*)
=(1/2) $, the equation $T_{+}(\lambda ,E) =(1/2)$
in the variable $E\in (0,E_*(\lambda)) $
admits a unique solution; $\tilde{E}(\lambda) $.

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

Hence, Theorem \ref{wmdjkgfh} is proved if we let $\lambda_0=(
2S_{+}(r_*)) ^p/p'$. \hfill$\diamondsuit$\medskip  

Now, assume that $p>2$ and let us prove Theorem \ref{wmdjkgfh2}. By the
assumption 
\[
\nu =(2S_{+}(c)) ^p/p'<(2S_{+}(
r)) ^p/p'=\lambda_1, 
\]
it follows that, for all fixed $\lambda >0,
$\[
\lim_{E\to E_*}T_{+}(\lambda ,E) <\lim_{E\to 0}T_{+}(\lambda ,E) . 
\]
According to Lemma \ref{Lemma3}, it follows that $T_{+}(\lambda
,\cdot) $ admits a unique critical point; a minimum. Thus, as in
the case where $1<p\leq 2$, there exists a unique $r_*\in (
r,c) $ and a unique $\tilde E=\tilde E(\lambda) \in
(0,E_*(\lambda)) $ such that 
\begin{eqnarray*}
\min_{r\leq \rho \leq c}S_{+}(\rho) &=&S_{+}(r_*),\mbox{ and} \\
\min_{0\leq E\leq E_*}T_{+}(\lambda ,E) &=&T_{+}(
\lambda ,\tilde E(\lambda)) =(p'\lambda) ^{-1/p}S_{+}(r_*) .
\end{eqnarray*}
Define 
\begin{eqnarray*}
J_0 &=&\{ u\in C^1([ 0,1]) :u\neq 0\mbox{ and }u'(0) =0\} \\
J_1(\lambda) &=&\{ u\in C^1([ 0,1]
) :0<u'(0) <E_*(\lambda) \}
\\
J_2(\lambda) &=&\{ u\in C^1([ 0,1]) :u'(0) =E_*(\lambda) \} .
\end{eqnarray*}
According to Lemma \ref{Lemma45}, each nontrivial solution to (\ref{P1})
belongs to $J_0\cup J_1(\lambda) \cup J_2(\lambda)
$.  So, let us look for the nontrivial solutions in $J_1(\lambda
) $.

\begin{itemize}
\item  If $(p'\lambda) ^{-1/p}S_{+}(r_*)
>1/2$, the equation $T_{+}(\lambda ,E) =1/2$ in the variable 
$E\in (0,E_*(\lambda)) $ admits no solution.
Thus, if $0<\lambda <\mu :=(2S_{+}(r_*))
^{p}/p'$, problem (\ref{P1}) admits no solution in $J_1(
\lambda) $.

\item  If $(p'\lambda) ^{-1/p}S_{+}(r_*)
=1/2$, the equation $T_{+}(\lambda ,E) =1/2$ in the variable 
$E\in (0,E_*(\lambda)) $ admits a unique
solution; $\tilde{E}(\lambda) $. Thus, if $\lambda =\mu $,
problem (\ref{P1}) admits a unique solution $v_\lambda $ in $J_1(
\lambda) $, and this solution belongs to $A_1^+$.

\item  If $(p'\lambda) ^{-1/p}S_{+}(r_*)
<1/2<(p'\lambda) ^{-1/p}S_{+}(c) $, the
equation $T_{+}(\lambda ,E) =1/2$ in the variable $E\in (
0,E_*(\lambda)) $ admits exactly two solutions.
Thus, if $\mu <\lambda <\nu $, problem (\ref{P1}) admits exactly two
solutions $v_\lambda $, $w_\lambda $ in $J_1(\lambda) ,$
and they belong to $A_1^+$.

\item  If $(p'\lambda) ^{-1/p}S_{+}(c)
=1/2 $, the equation $T_{+}(\lambda ,E) =1/2$ in the variable 
$E\in (0,E_*(\lambda)) $ admits a unique
solution; $E_1<\tilde{E}(\lambda) $. Thus, if $\lambda =\nu 
$, problem (\ref{P1}) admits a unique solution $v_\lambda $ in $J_1(
\lambda) $, and it belongs to $A_1^+$.

\item  If $(p'\lambda) ^{-1/p}S_{+}(c)
<1/2<(p'\lambda) ^{-1/p}S_{+}(r) $, the
equation $T_{+}(\lambda ,E) =1/2$ in the variable $E\in (
0,E_*(\lambda)) $ admits a unique solution; $E_1<%
\tilde{E}(\lambda) $. Thus, if $\nu <\lambda <\lambda_1$,
problem (\ref{P1}) admits a unique solution $v_\lambda $ in $J_1(
\lambda) $, and it belongs to $A_1^+$.

\item  If $(p'\lambda) ^{-1/p}S_{+}(r)
\geq 1/2$, the equation $T_{+}(\lambda ,E) =1/2$ in the
variable $E\in (0,E_*(\lambda)) $ admits no
solution. Thus, if $\lambda \geq \lambda_1$, problem (\ref{P1}) admits no
solution in $J_1(\lambda) $.
\end{itemize}

Now, let us look for the nontrivial solutions in $J_2(\lambda)$.
 For all $\lambda >0$,
 \[
T_{+}(\lambda ,E_*(\lambda)) =1/2\mbox{ if
and only if }(p'\lambda) ^{-1/p}S_{+}(c)=1/2\,. 
\]
Thus, problem (\ref{P1}) admits a solution in $J_2(\lambda)
\cap A_1^{+}$ if and only if $\lambda =\nu $, and in this case the solution
is unique. For all $\lambda >0$,
\[
T_{+}(\lambda ,E_*(\lambda)) <1/2\mbox{ if
and only if   }(p'\lambda) ^{-1/p}S_{+}(c)<1/2\,. 
\]
Thus, problem (\ref{P1}) admits a solution in $J_2(\lambda)
\cap \tilde A_1^{+}$ if and only if $\lambda >\nu $, and in this case the
solution is unique.

Now let us look for the nontrivial solutions in $J_0$.  Let $n\in {\mathbb N}^*$.
For all $\lambda >0$,
\[
nT_{+}(\lambda ,0) =1/2\mbox{ if and only if   }n(
p'\lambda) ^{-1/p}S_{+}(r) =1/2\,. 
\]
Thus, problem (\ref{P1}) admits a solution in $J_0\cap B_n^{+}$ if and only
if $\lambda =\lambda_n$, and in this case the solution is unique.
For all $\lambda >0$,
\[
nT_{+}(\lambda ,0) <1/2\mbox{ if and only if   }n(
p'\lambda) ^{-1/p}S_{+}(r) <1/2\,. 
\]
Thus, problem (\ref{P1}) admits a solution $u_{\lambda ,n}$ in $J_0\cap
\tilde B_n^{+}$ if and only if $\lambda >\lambda_n$, and in this case each
function $u$ in $\mathop{\rm Cl}(u_{\lambda ,n}) $ is a solution to (\ref
{P1}).
Therefore, Theorem \ref{wmdjkgfh2} is proved. \hfill$\diamondsuit$\smallskip 

To prove Theorems \ref{wmdjkgfh3}, \ref{wmdjkgfh4} and \ref{wmdjkgfh5}, the
same reasoning works. However, for Theorems \ref{wmdjkgfh4} and \ref
{wmdjkgfh5}, one has 
\[
\lim_{E\to 0}T_{+}(\lambda ,E) <\lim_{E\to
E_*}T_{+}(\lambda ,E) . 
\]
Thus, according to Lemma \ref{Lemma3}, $T_{+}(\lambda ,\cdot) 
$ may have a unique critical point; a minimum, or may be strictly
increasing. These two alternatives lead for Theorem \ref{wmdjkgfh4} to the
possibilities {\bf A} and {\bf B}, and for Theorem \ref{wmdjkgfh5} to the
possibilities {\bf C} and {\bf D}.  \hfill$\diamondsuit$

\section{Open questions}\label{sec6} 
\begin{enumerate}
\item  For $p>2$, Theorems \ref{wmdjkgfh4} and \ref{wmdjkgfh5} provide
alternative results. Do there exist some sufficient conditions ensuring
that possibility {\bf A} (resp. {\bf B}, {\bf C}, {\bf D}) holds? Can one
find an example of $f$ such that possibility {\bf A} (resp. {\bf B}, {\bf C},
{\bf D}) holds? Or maybe among the two alternatives Theorem \ref{wmdjkgfh4}
(resp. Theorem \ref{wmdjkgfh5}) provides, the same one holds always?

\item  In the literature, there are some examples of nonlinearities 
$g(\lambda ,u)$ such that the structure of the solution set of (\ref{P1})
does change when $p$ varies (as that studied in this paper) but in others it
does not change; for example as that studied by Addou and Benmeza\"{\i }
\cite{Addou3} for $g(\lambda ,u)=\lambda \exp (u) $.

Thus, we ask the question of providing sufficient or necessary conditions on 
$g$ insuring that the structure of (at least) the set of (positive)
solutions of problem (\ref{P1}) does not change when $p$ varies.
\end{enumerate}

\section{\bf Appendix}

In this section, we prove Lemma \ref{Lemmasup} which is a
consequence of the following two lemmas.

\begin{lemma} \label{Lemma45}
Let $u$ be a nontrivial solution of (\ref{P1}). Then 
\[
u\geq 0\mbox{ in }[ 0,1] \mbox{ and }0\leq
u'(0) \leq E_*(\lambda) =(p'\lambda F(c)) ^{1/p}.
\]
Moreover,

\begin{itemize}
\item  If $0\leq u'(0) <E_*(\lambda) ,$\
then $\max_{0\leq x\leq 1}u(x) <c$.

\item  If $u'(0) =E_*(\lambda)$ , then 
$\max_{0\leq x\leq 1}u(x) =c$.
\end{itemize}
\end{lemma}

\begin{lemma}
\label{Lemma451}Let $u$ be a nontrivial solution of (\ref{P1}). Then

\begin{description}
\item[(a)]  $u'(0) \in (0,E_*(\lambda
)) $ implies $u\in A_1^+$,

\item[(b)]  $u'(0) =E_*(\lambda) $
implies $u\in A_1^+\cup \tilde{A}_1^+,$

\item[(c)]  $u'(0) =0$ implies $u\in \bigcup_{k\geq 1}B^{+}(k) $.
\end{description}
\end{lemma}

\paragraph{Proof of Lemma \ref{Lemma45}.} Assume that there exists $x_0\in
(0,1) $ such that 
\[
u(x_0) =\min_{0\leq x\leq 1}u(x) <0\mbox{ and }u'(x_0) =0\,. 
\]
The variations of $F$ imply that $F(u(x_0)) <0$.
On the other hand, by the energy relation (see \cite[Lemma 7]{Addou6}) 
\[
| u'(x) | ^{p}=| u'(
0) | ^{p}-p'\lambda F(u(x)),\mbox{ for all }x\in [ 0,1] , 
\]
it follows that 
\[
0=| u'(x_0) | ^{p}=| u'(0) | ^{p}-p'\lambda F(u(x_0)) . 
\]
Thus, $F(u(x_0)) =(1/p'\lambda) | u'(0) | ^{p}\geq 0$.  A
contradiction. Therefore $u\geq 0$ in $[ 0,1] $.  Let $x_*\in
(0,1) $ be such that $u(x_*) =\max_{0\leq x\leq
1}u(x) $.  By the energy relation, it follows that 
\[
0=| u'(x_*) | ^{p}=| u'(0) | ^{p}-p'\lambda F(u(x_*)) . 
\]
Thus, $u(x_*) $ is a positive zero of the function $s\mapsto
| u'(0) | ^{p}-p'\lambda F(
s) $.  By Lemma \ref{Lemma1} this function vanishes at least once if
and only if $0\leq | u'(0) | \leq
E_*(\lambda) $.  Since, $u\geq 0$ in $[ 0,1] $ it
follows that $u'(0) \geq 0$.  Thus $| u'(0) | =u'(0) $ and therefore $0\leq
u'(0) \leq E_*(\lambda) $.

Assume that $u'(0) =0$ and there exists $x_*\in
(0,1) $ such that $u(x_*) =\max_{0\leq x\leq
1}u(x) \geq c$.  Thus, there exists $x_0\in (0,1) $
such that $u(x_0) =c$.  By the energy relation, it follows that 
\[
0\leq | u'(x_0) | ^p=| u'(0) | ^p-p'\lambda F(u(x_0)
) =-p'\lambda F(c) . 
\]
Thus, $F(c) \leq 0$, which contradicts hypothesis (\ref{Austine4}). 
Therefore, $u'(0) =0$ implies that $\max_{0\leq x\leq1}u(x) <c$.

Assume that $0<u'(0) <E_*(\lambda)
=(p'\lambda F(c)) ^{1/p}$, and there
exists $x_*\in (0,1) $ such that $u(x_*)
=\max_{0\leq x\leq 1}u(x) \geq c$.  Thus, there exists $x_0\in
(0,1) $ such that $u(x_0) =c$.  By the energy
relation, it follows that 
\[
0\leq | u(x_0) | ^p=| u'(0)
| ^p-p'\lambda F(u(x_0)) =|u'(0) | ^p-p'\lambda F(c) . 
\]
Thus, $F(c) \leq (1/p'\lambda) |
u'(0) | ^p$ which is impossible since $u'(0) <E_*(\lambda) $ 
implies that $(1/p'\lambda) | u'(0) |^p<F(c) $.  Therefore, 
$0<u'(0) <E_*(\lambda) $ implies that $\max_{0\leq x\leq 1}u(x) <c$.

Assume that $u'(0) =E_*(\lambda) =(p'\lambda F(c)) ^{1/p}$. 
 Let $x_*\in (0,1) $ be such that $\max_{0\leq x\leq 1}u(x) =u(x_*) $ and 
$u'(x_*) =0$.  By the energy relation it follows that 
\[
0=| u'(x_*) | ^{p}=| u'(0) | ^{p}-p'\lambda F(u(
x_*)) =p'\lambda F(c) -p'\lambda F(u(x_*))\,. 
\]
Thus, $F(c) =F(u(x_*)) $ and
therefore $u(x_*) =c$ since $F(c) >F(x) $ for all $x\geq 0$ and $x\neq c$.
 Lemma \ref{Lemma45} is proved.  \hfill$\diamondsuit$

\paragraph{Proof of Lemma \ref{Lemma451}.} Each assertion is a consequence of
several steps.
If $u$ is a nontrivial solution of (\ref{P1}) and satisfying $u'(0) \in (0,E_*(\lambda)) $, then
Assertion {\bf (a)} is an immediate consequence of the following steps:

\begin{description}
\item[(a1)]  For all $x_*\in (0,1) $, $u'(
x_*) =0$ implies $u(x_*) =s_{+}(u'(0)) \in (r,c) $.

\item[(a2)]  For all $x_1,x_2\in (0,1)$, $x_1<x_2$, $u'(x_1) =u'(x_2) =0$
implies $u\equiv s_{+}(u'(0)) $ in $[
x_1,x_2] $.

\item[(a3)]  The derivative $u'$ vanishes exactly once in $(0,1) $.

\item[(a4)]  The solution $u$ is symmetric with respect to $1/2$.
\end{description}

\noindent If $u$ is a nontrivial solution of (\ref{P1}) and satisfying $u'(0) =E_*(\lambda) $, then Assertion {\bf (b)} is
an immediate consequence of the following steps:

\begin{description}
\item[(b1)]  For all $x_*\in (0,1) $, $u'(x_*) =0$ implies $u(x_*) =c$.

\item[(b2)]  For all $x_1,x_2\in (0,1)$, $x_1<x_2$, 
$u'(x_1) =u'(x_2) =0$
implies $u\equiv c$ in $[ x_1,x_2] $.

\item[(b3)]  There exist $x_1,x_2\in (0,1) $, such
that $x_1\leq x_2$, and for all $x\in (0,1)$,
\[
u'(x) =0,\mbox{ if and only if }x\in [x_1,x_2] .
\]

\item[(b4)]  There exist $x_1,x_2\in (0,1)$,  such
that $0<x_1\leq x_2<1,$\ and $u'>0$ on $(0,x_1) $, $u'\equiv 0$ 
on $[ x_1,x_2] $, and $u'<0$ on $(x_2,1) $.

\item[(b5)]  The solution $u$ is symmetric with respect to $1/2$.
\end{description}

If $u$ is a nontrivial solution of (\ref{P1}) and satisfying 
$u'(0) =0$, then Assertion {\bf (c)} is an immediate consequence
of the following steps:

\begin{description}
\item[(c1)]  For all $x_*\in (0,1) $, $u'(x_*) =0$ implies $u(x_*) =0$
 or $u(x_*) =r$.

\item[(c2)]  Each local maxima of $u$ is a strict one.

\item[(c3)]  There are finitely many critical points at which $u$ attains
its maximum value; $r$.

\item[(c4)]  If $u$ attains its maximum value at the $n$ points of the
strictly increasing sequence $(x_{i})_{1\leq i\leq n}$ then
for all $i\in \{ 1,\cdots ,n\} $ there exists $a_{i}\leq b_{i}
$ in $[ 0,1] $ such that
\end{description} 
\begin{eqnarray}
&x_{i}<a_{i}\leq b_{i}<x_{i+1},,\mbox{ for all }i\in \{ 1,\cdots,n-1\},&
  \label{f1} \\
&0=a_0\leq b_0<x_1\mbox{ and }x_{n}<a_{n}\leq b_{n}=1\,, &\label{f2} \\
&u\equiv 0\mbox{ on }[ a_{i},b_{i}] \mbox{ for all }i\in
\{ 0,\cdots ,n\}, &  \label{f3} \\
&u'>0\mbox{ on  }(b_{i},x_{i+1}) \mbox{ for all }i\in \{ 0,\cdots ,n-1\},&
  \label{f4}\\
&u'<0\mbox{ on  }(x_{i},a_{i}) \mbox{ for all }i\in \{ 1,\cdots ,n\},&
\label{f5} \\
&b_{i}+a_{i+1}=2x_{i+1},\mbox{ for all }i\in \{ 0,\cdots,n-1\}, &\label{f6} \\
&u|_{[ b_{i},a_{i+1}] },\mbox{ is symmetric with respect to 
   $x_{i+1}$ for $i\in \{ 0,\cdots ,n-1\}$}, & \label{f77} \\
&u|_{[ b_{i},a_{i+1}] },\mbox{ is a translation of 
$u|_{[ b_0,a_1]}$, for all $i\in \{ 0,\cdots ,n-1\}$}\,.&   \label{f88}
\end{eqnarray}


\noindent The proofs of all these steps are simple and  therefore omitted. 
Full details can be found in the author's doctoral thesis \cite{Thesis}.

\paragraph{Acknowledgments.} Many thanks to Professors P. Korman, S.-H. Wang and J. Wei 
for sending me some of their publications.

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

\section*{ Addendum:  May 3, 2000.}

In this addendum we answer the  question about alternative results for 
Theorems 2.4 and 2.5.  We shall prove that for $p>2$, 
Possibility B of Theorem 2.4  and  Possibility D of Theorem 2.5 never happen. 
Therefore, the diagram in Fig. 3 does not occur. That is, for $p>2$ the
upper branch has always a turning point (which is unique).

The alternatives in Theorems 2.4  and 2.5 come from the
alternative situation on the time map  $T_{+}(\lambda, \cdot )$. 
Indeed,  Lemma 4 states that for all $p>2$ and $\lambda >0$, 
\emph{either} the time map $T_{+}(\lambda , \cdot )$ is
strictly increasing \emph{or} it admits a unique critical point; a minimum in 
$(0, E_{*}(\lambda ))$.

We shall prove that for all $p>2$ and all $\lambda >0$, 
$T_{+}(\lambda, \cdot )$ admits at least one minimum in 
$(0, E_{*}(\lambda)) $. Therefore, it admits a unique critical point for all 
$p>1$ (according to the first part of Lemma 4) and it is never strictly 
increasing on $(0, E_{*}(\lambda ))$. As a consequence, for all $p>2$,
Possibility B of Theorem 2.4  and Possibility D of Theorem 2.5 do not occur.
To prove this statement, it suffices to show that: 

\paragraph{Lemma.}
 $S_{+}'(r)=-\infty $ and  $S_{+}'(c)=+\infty$ for all $p>2$.

\paragraph{Proof.}
Since  $F(r)=F(0)=0$, the integral in the expression 
\[
S_{+}'(r)=\frac{1}{pr}\int_0^{r}\frac{H_p(r)-H_p(u)}
{(F(r)-F(u))^{1+\frac{1}{p}}}\,du
\]
has two singularities: one at $0$ and one at $r$. So, we shall write
\[  S_{+}'(r)=\frac{1}{pr}(I_0+I_{r}),
\]
where
\[
I_0=\int_0^{r/2}\frac{H_p(r)-H_p(u)}{(F(r)-F(u))^{1+\frac{1}{p}}}\,du\,,
\quad
I_{r}=\int_{r/2}^{r}\frac{H_p(r)-H_p(u)}{(F(r)-F(u))^{1+\frac{1}{p}}}\,du\,.
\]

Next we prove that $I_0=-\infty $ and
$I_{r}\in [-\infty , +\infty )$, so that $S_{+}'(r)=-\infty $.

\paragraph{Proof of $I_0=-\infty$.}
Using l'Hopital's rule twice it follows that
\[
\lim_{u\rightarrow 0}\frac{F(u)}{u^{2}}=\frac{f'(0)}{2}<0\,.
\]
This last inequality follows from (15). Therefore,
\[
\frac{H_p(r)-H_p(u)}{(F(r)-F(u))^{1+\frac{1}{p}}} 
=\frac{H_p(r)-H_p(u)}{(-\frac{F(u)}{u^{2}})^{1+\frac{1}{p}}}\cdot 
\frac{1}{u^{2(1+\frac{1}{p})}}  
\approx \frac{H_p(r)}{(-\frac{f'(0)}{2})^{1+\frac{1}{p}}}\cdot 
\frac{1}{u^{2(1+\frac{1}{p})}}
\]
for all $u \in (0,\varepsilon )$ and for some $\varepsilon>0$.
 Since $2(1+\frac{1}{p})>1$ and
\[
\frac{H_p(r)}{(-\frac{f'(0)}{2})^{1+\frac{1}{p}}}
=\frac{-rf(r)}{(-\frac{f'(0)}{2})^{1+\frac{1}{p}}}<0
\]
it follows that
\[
\int_0^{r/2}\frac{H_p(r)}{(-\frac{f'(0)}{2})^{1+\frac{1}{p}}}
\cdot \frac{1}{u^{2(1+\frac{1}{p})}}du=-\infty
\]
which proves that $I_0=-\infty $.

\paragraph{Proof of $I_{r}\in [-\infty , +\infty )$.}
We distinguish two cases.

\noindent Case $H_p'(r)\neq 0$. \quad Since 
\[
\lim_{u \rightarrow r}\frac{H_p(r)-H_p(u)}{r-u}=H_p' (r)\neq 0
\]
and
\[
\lim_{u \rightarrow r}\frac{F(r)-F(u)}{r-u}=f(r)>0\,,
\]
it follows that
\[
\frac{H_p(r)-H_p(u)}{(F(r)-F(u))^{1+\frac{1}{p}}} 
=\frac{(\frac{H_p(r)-H_p(u)}{r-u})}{(\frac{F(r)-F(u)}{r-u})^{1+\frac{1}{p}}}
\cdot\frac{1}{(r-u)^{\frac{1}{p}}} 
\approx \frac{H_p'(r)}{(f(r))^{1+\frac{1}{p}}}\cdot 
\frac{1}{(r-u)^{\frac{1}{p}}} 
\] 
for all $u \in (r-\varepsilon, r)$ and for some $\varepsilon>0$. 
Since $\frac{1}{p}<1$ and $\frac{H_p'(r)}{(f(r))^{1+(1/p)}}\neq 0$, 
\[
\int_{r/2}^{r}\frac{H_p'(r)}{(f(r))^{1+\frac{1}{p}}}\cdot \frac{1%
}{(r-u)^{\frac{1}{p}}}du\in (-\infty , +\infty )
\]
which proves that $I_{r}\in [-\infty , +\infty )$.
\medskip

\noindent Case $H_p'(r)=0$. \quad From equations (46)-(50), it follows that
$\beta _p\leq r\leq \gamma _p$.


First assume that $\beta _p \neq r$. Then in a left
neighborhood of $ r $ the integrand function is identically zero.
That is, there exists $ \varepsilon >0 $ such that
\[
\frac{H_p(r)-H_p(u)}{(F(r)-F(u))^{1+\frac{1}{p}}}=0
\]
 for all $u \in (r-\varepsilon , r)$. Therefore,
\[
\int_{r-\varepsilon}^{r}\frac{H_p(r)-H_p(u)}{(F(r)-F(u))^{1
 +\frac{1}{p}} }\,du=0\,.
\]
So, the integral $I_{r}$ presents no singularity at $r$ and
$I_{r}\in (-\infty , +\infty )$.
\medskip

Now assume that $r=\beta _p$. Then, by (46)-(48)
$H_p(r)-H_p(u)\leq 0$ \  for all $u \in (0, r)$.
Therefore,
\[
\int_{r/2}^{r}\frac{H_p(r)-H_p(u)}{(F(r)-F(u))^{1+\frac{1}{p}}}\,du\in
[-\infty , 0],
\]
which proves that $I_{r}\in [-\infty , +\infty )$.
Therefore, $S'(r)=-\infty $. \medskip

 Now, we shall prove that $S'(c)=+\infty $. Since
\[
\lim_{u \rightarrow c} \frac{H_p(c)-H_p(u)}{c-u} 
= H_p'(c) =-cf'(c)>0\,,
\]
because of (15), and 
\[
\lim_{u \rightarrow c}\frac{F(c)-F(u)}{(c-u)^{2}}=-\frac{f'(c)}{2}>0\,,
\]
it follows that
\[
\frac{H_p(c)-H_p(u)}{(F(c)-F(u))^{1+\frac{1}{p}}} 
=\frac{(\frac{H_p(c)-H_p(u)}{c-u})}
{(\frac{F(c)-F(u)}{(c-u)^{2}})^{1+\frac{1}{p}}}
\cdot \frac{1}{(c-u)^{1+\frac{2}{p}}} 
\approx \frac{(-cf'(c))}{(-\frac{f'(c)}{2})^{1+\frac{1}{p}}}
\cdot \frac{1}{(c-u)^{1+\frac{2}{p}}} 
\]
for all $u \in (c-\varepsilon , c)$ and for some $\varepsilon >0$.
Since $1+\frac{2}{p}>1$ and 
$\frac{(-cf'(c))}{(-\frac{f'(c)}{2})^{1+\frac{1}{p}}}>0$,
 it follows that
\[
\int_0^{c}\frac{(-cf'(c))}{(-\frac{f'(c)}{2})^{1+\frac{1}{p}}}
\cdot \frac{1}{(c-u)^{1+\frac{2}{p}}}du=+\infty,
\]
which proves that $S'(c)=+\infty $. Therefore, the present proof is
complete, and the claim of the addendum is proved.  \hfill$\diamondsuit$

\end{document}