\documentclass[twoside]{article} \usepackage{amssymb, amsmath} \pagestyle{myheadings} \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 $$\gathered -( \varphi_p( u') ) ' =\lambda f( u) ,\text{in }( 0, 1) \label{p1} \\ u( 0) =u( 1) =0, \endgathered$$ 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 $$\gathered -m_+ \geq \frac{f( \alpha_+) -f( \xi ) }{ \varphi_q( \alpha_+-\xi ) }\geq -M_+\text{,\ for all }\xi \in ( \alpha_+-\delta ,\alpha_+) , \\ -m_- \geq \frac{f( \alpha_-) -f( \xi ) }{ \varphi_q( \alpha_--\xi ) }\geq -M_-\text{,\ for all }\xi \in ( \alpha_-,\alpha_-+\delta ) . \endgathered \label{pq5}$$ 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 $$h_{f}( x) \geq 0,\text{ for all }x\in I, \label{Arizona}$$ \item $f$ is \textit{negatively half-odd} (\textit{n.h.o.}, for brevity) on $I\cup ( -I)$, if $$h_{f}( x) \leq 0,\text{ for all }x\in I. \label{Florida}$$ \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 $$f\text{ \ is\ \textit{p.h.o.}\ \ \ on }[ \alpha_-,-\alpha _-] . \label{pho}$$ Note that condition (\ref{pho}) implies that the function $$u\mapsto F( u) :=\int_{0}^{u}f( \xi ) d\xi \text{ \ \ is\ \textit{p.h.e.}\ \ on\ }[ \alpha_-,-\alpha_-] . \label{phe}$$ 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 $$0<-\alpha_-\leq \alpha_+. \label{jzehjf}$$ 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 $10, \label{Miami} for all$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 $$f\in C^2( I_{\pm }( \alpha_{\pm }) ) ,\text{ \ \ }% p\geq 2\text{\ \ and\ \ }\pm f''<0\text{ \ \ in }I_{\pm }( \alpha_{\pm }) . \label{jfghnb}$$ In the case where$10 \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 } $x_{1}0,$\ 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 $1\lambda_{k}/a_{0}$. 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 $10$ 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 $$\gathered -u''( y) =f( u( y) ) , \text{ \ }y\in ( -\lambda ,\lambda ) \label{Pm} \\ u( -\lambda ) =u( \lambda ) =0 \endgathered$$ with the cubic nonlinearity $f( u) =-( u-a) ( u-b) ( u-c)$, and $a0$ 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 $00$ is a real parameter, and $f$ is the cubic nonlinearity $$f( x,u) =a( x) u^2( 1-b( x) u) ,\;\;x\in ( -1,1) ,u\in {\mathbb R} , \label{Lasvegas}$$ 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 $$\gathered a( x) ,b( x) >0\quad \text{ for }-1\leq x\leq 1.\\ xb'( x) >0\quad \text{\and}\quad xa'( x) <0 \quad \text{for } x\in ( -1,1) \backslash \left\{ 0\right\} \\ b''( x) b( x) -2( b') ^2( x) >0\quad\text{for }-1\lambda_{1}. All solutions lie on a single curve of solutions, which is smooth in \lambda . For each \lambda >\lambda_{1} there are finitely many solutions, and different solutions are strictly ordered on ( -1,1) . Moreover, there exists \lambda_{2}\geq \lambda_{1}, so that for \lambda >\lambda_{2} Problem (\ref{p2}) has exactly two solutions denoted by u^{-}( x,\lambda ) 0,a'( x) >0\text{ for } x\in ( 0, 1) \\ a( -x) =a( x) \text{\ for }x\in ( -1,1). \\ a( x) <\frac 12b\text{\ for all }x\in ( -1,1) . \endgathered$$ By the maximum principle every solution satisfies $0\lambda_{1}$. 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 $u^{-}( x,\lambda ) u^{-}( x,\lambda )$ 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*} 00$. Notice that here the two positive roots may depend on$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 $$b''( x) <0\text{\ \ \ \ for all }x\in ( -1,1) . \label{Hadjera}$$ 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$has three distinct roots$a0\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 $u^{-}( x,\lambda ) b$ 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 $$\gathered -u'' =\lambda f( u-a) ,\text{\ \ in }(-1,1) \\ u( -1) =u( 1) =0 \endgathered \label{KO}$$ 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 $$f( \beta ) \beta -2[ F( \beta ) -F( -a) ] \geq 0 \label{Austine8}$$ 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 $u^{-}( x,\lambda ) a+b$ 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 $00$ 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 $u^{-}( x,\lambda ) 0$\ for $x\in \left( b,c\right) ,$ where $c>b>0$. Assume that for some $c>\eta >\gamma >b$ we have, $f^{\prime \prime }\left( u\right) >0\text{\ for }00,$ $f(u)-uf^{\prime }(u)>0\text{ for all }u>\eta .$ 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 \gathered \Delta u+\lambda f( u) = 0\;\;\text{ in }\left| x\right| 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 $$\gathered f''( u) >0\text{\ for }u\in ( 0,\alpha) \text{\ \ and} \\ f''( u) <0\text{\ for }u\in ( \alpha ,c) . \endgathered \label{atf25}$$ 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 ( f') ^2u-f'f-ff''u>0\text{\ \ for }b0$for$u<0$. Therefore by the maximum principle, all solutions of (\ref{atf11}) are positive, hence by a well-known result of Gidas, Ni and Nirenberg \cite{GNN} they are radially symmetric. Also, by a result of Lin and Ni \cite{LN} all solutions of the linearized equation \gathered \Delta w+\lambda f'( u) w =0\text{\ \ in\ }\left| x\right| 0$, 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 $0\gamma$ 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 $$\gathered u^{\prime \prime }\left( r\right) +\frac{n-1}{r}u^{\prime }\left( r\right) +\lambda f\left( u\right) =0,\;r\in \left( 0,1\right) ,\text{ }n\geq 1, \label{123} \\ u^{\prime }\left( 0\right)=u\left( 1\right) =0. \endgathered$$ They assume that $f\in C^{2}\left( {\mathbb R}_{+}\right)$ satisfies the following properties: f\left( 0\right) \leq 0,\text{ }f\left( b\right) =f\left( c\right) =0\text{\ for some constants }0&0\text{\ \ for }u\in ( -\infty ,0) \cup ( b,c) , \notag \end{eqnarray} $$\int_0^cf( u) du>0, \label{126}$$ \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) 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$0b$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, g\in C({\mathbb R} ) \text{\ \ and\ \ \ }10\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, $$s\longmapsto G_{\pm }( \lambda ,E,s) :=E^{p}-p^{\prime }\lambda F( s) , \label{Eq.1}$$ 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$00, \label{der} 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, $$s\longmapsto G_{\pm }( \lambda ,E,s) :=E^{p}-p^{\prime }\lambda F( s) , \label{Eq1}$$ 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 $E:01$ 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, $$G_{\pm }( E,s_{\pm }( \lambda ,E) ) =0. \label{Eq.2}$$ 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 $$E^{p}=p'\lambda F( s_{\pm }( \lambda ,E) ) , \label{dklg}$$ 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} & 0E_{*}^{-} \\ ( \alpha_-,0) & \text{if} & E=E_{*}^{-} \\ ( 0,s_-( \lambda ,E) ) & \text{if} & 0E_{*}^+ \\ \alpha_+ & \text{if} & E=E_{*}^+ \\ s_+( \lambda ,E) & \text{if} & 0E_{*}^{-} \\ \alpha_- & \text{if} & E=E_{*}^{-} \\ s_-( \lambda ,E) & \text{if} & 00,\forall \lambda >0,E\in ( 0,E_{*}^{\pm }( p,\lambda ) ) , \label{eq11} $$\lim\limits_{E\rightarrow 0^+}r_{\pm }( \lambda ,E) =0\text{\ \ \ \ \ and \ \ \ }\lim\limits_{E\rightarrow E_{*}^{\pm }}r_{\pm }( \lambda ,E) =\alpha_{\pm }. \label{abhu}$$ $$\lim\limits_{E\rightarrow 0^+}\left| r_{\pm }( \lambda ,E) \right| /E=\left\{ \begin{array}{lcl} +\infty & \text{if} & q-p>0 \\ ( \dfrac{p-1}{\lambda a_0}) ^{1/p} & \text{if} & q-p=0 \\ 0 & \text{if} & q-p<0, \end{array} \right. \label{bbhu}$$ $$\lim\limits_{E\rightarrow 0}F( \kappa r_\eta ( \lambda ,E) \xi ) /E^p=\xi ^q/( \lambda p') ,\;\;\text{ for all }\xi >0\text{ and }( \kappa ,\eta ) \in \left\{ +,-\right\} ^2. \label{bbhuu}$$ On the other hand, \begin{equation*} 0<\left| r_{\pm }( \lambda ,E) \right| <+\infty \text{ \ \ if and only if \ }00\text{ \ \ if and only if \ }00:0<\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*} 00$and$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, $$F( r_+( \lambda ,E) ) -F( r_-( \lambda ,E) ) =0. \label{dyuk}$$ 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$0<-r_-( \lambda _{0},E_{0}) 0$which is a contradiction with (\ref{dyuk}). Therefore, for all$\lambda >0$and$E\in ( 0,E_{*}^{-}( \lambda ) ) $one has, \begin{equation*} 00$ 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, $$T_{\pm }( \lambda ,E) =\left| r_{\pm }( \lambda ,E) \right| \int_{0}^1( E^{p}-p'\lambda F( r_{\pm }( \lambda ,E) \xi ) ) ^{-1/p}d\xi , \label{A}$$ which can be written as, $$T_{\pm }( \lambda ,E) =( \left| r_{\pm }( \lambda ,E) \right| /E) \int_{0}^1( 1-p'\lambda F( r_{\pm }( \lambda ,E) \xi ) /E^{p}) ^{-1/p}d\xi . \label{qze}$$ 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, $$T_{\pm }( \lambda ,E) =\pm ( \lambda p') ^{-1/p}\int_{0}^{r_{\pm }( \lambda ,E) }( F( r_{\pm }( \lambda ,E) ) -F( r_{\pm }( \lambda ,E) \xi ) ) ^{-1/p}d\xi . \label{a15}$$ 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, $$T_{\pm }( \lambda ,E) =( \lambda p') ^{-1/p}S_{\pm }( r_{\pm }( \lambda ,E) ) . \label{a16}$$ On the other hand, we define for any $p>1$, $\lambda >0$ and $E\in D( \lambda ) =( 0,E_{*}^{-})$, the time maps, $$T_{2n}^{\pm }( \lambda ,E) :=n( T_+( \lambda ,E) +T_-( \lambda ,E) ) ,\lambda >0,E\in ( 0,E_{*}^{-}) ,n\geq 0, \label{a18}$$ $$T_{2n+1}^{\pm }( \lambda ,E) :=T_{2n}^{\pm }( \lambda ,E) +T_{\pm }( \lambda ,E) ,\lambda >0,E\in ( 0,E_{*}^{-}) ,n\geq 0. \label{a19}$$ 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 \hfill$\diamondsuit$% %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, $$S_-( r_-( \lambda ,E) ) =\int_0^{-r_-( \lambda ,E) }\{F( r_-( \lambda ,E) ) -F( -\xi ) \}^{-1/p}d\xi . \label{ehjkgr}$$ 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 $0<\xi 0$ 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}. %TCIMACRO{\TeXButton{End Proof}{\hfill$\diamondsuit$}} %BeginExpansion \hfill$\diamondsuit$% %EndExpansion \begin{remark} It is well known that when $f$ is odd then $$T_+(\lambda ,\cdot )=T_-(\lambda ,\cdot )\text{,\ for all }\lambda >0. \label{R1}$$ 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, $$\Theta_-( r_-( \lambda ,E) ) =\Theta_+( r_+( \lambda ,E) ) \text{,\ for all }\lambda >0\text{ and all }E\in (0,E_{*}(\lambda )). \label{R2}$$ 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}. %TCIMACRO{\TeXButton{End Proof}{\hfill$\diamondsuit$}} %BeginExpansion \hfill$\diamondsuit$% %EndExpansion \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. %TCIMACRO{\TeXButton{End Proof}{\hfill$\diamondsuit$}} %BeginExpansion \hfill$\diamondsuit$% %EndExpansion \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 $$\lim\limits_{E\rightarrow 0^+}\frac{r_+( \lambda ,E) }% E=\left\{ \begin{array}{lcl} +\infty & \text{if} & q-p>0 \\ ( \dfrac{p-1}{\lambda a_0}) ^{1/p} & \text{if} & q-p=0 \\ 0 & \text{if} & q-p<0. \end{array} \right. \label{Madrid}$$ On the other hand, by (\ref{bbhuu}) one has for all $( \kappa ,\eta ) \in \left\{ +,-\right\} ^2$, $$\lim\limits_{E\rightarrow 0^+}\int_0^1\left\{ 1-p'\lambda F(\kappa r_\eta ( \lambda ,E) t)/E^p\right\} ^{-1/p}dt=\int_0^1\left\{ 1-t^q\right\} ^{-1/p} \label{Italy}$$ with $$\int_0^1\left\{ 1-t^q\right\} ^{-1/p}=\left\{ \begin{array}{lcl} \dfrac 1qB( \dfrac 1q,1-\dfrac 1p) \in {\mathbb R} & \text{if} & q-p\neq 0 \\ \dfrac 12( \dfrac{\lambda_1}{p-1}) ^{1/p} & \text{if} & q-p=0. \end{array} \right. \label{France}$$ 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}. %TCIMACRO{\TeXButton{End Proof}{\hfill$\diamondsuit$}} %BeginExpansion \hfill$\diamondsuit$% %EndExpansion \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 $10$, 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. %TCIMACRO{\TeXButton{End Proof}{\hfill$\diamondsuit$}} %BeginExpansion \hfill$\diamondsuit$% %EndExpansion \paragraph{Proof of Theorem \ref{theorem2}} Let us assume that $10$, 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 $10$, the function $E\mapsto T_{\pm }( \lambda ,E)$ is defined in $D_{\pm }( \lambda ) =( 0,E_{*}^{\pm }( \lambda ) )$ and by Lemma \ref{lemma2}, $$\lim_{E\rightarrow 0^+}T_{\pm }( \lambda ,E) =\lim_{E\rightarrow E_{*}^{\pm }}T_{\pm }( \lambda ,E) =+\infty . \label{USA}$$ 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} $$\lim_{E\mapsto 0^+}\Theta_-( r_-( \lambda ,E) ) =\lim_{E\mapsto ( E_{*}^{-}) ^{-}}\Theta_-( r_-( \lambda ,E) ) =+\infty . \label{Star}$$ 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 $$k( p'\lambda ) ^{-1/p}(\inf_{E\in ( 0,E_{*}^{-}( \lambda ) ) }\Theta_+( r_+( \lambda ,E) ) )\leq \min_{E\in ( 0,E_{*}^{-}( \lambda ) ) }T_k^\kappa ( \lambda ,E) . \label{jhdyz}$$ 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_{00. \end{equation*} Let us assume momently that $\inf_{00$. It follows that for all $\lambda$ satisfying $k( p'\lambda ) ^{-1/p}\inf_{0(1/2)$, 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 $k( p'\lambda ) ^{-1/p}\inf_{0(1/2)$, that is, for all $0<\lambda <( 2k\inf_{00$. By Lemma \ref{lemma8}, one has $$\lim_{E\mapsto 0^+}\Theta_+( r_+( \lambda ,E) ) =+\infty \text{,\ and }\lim_{E\mapsto ( E_{*}^{-}) ^{-}}\Theta_+( r_+( \lambda ,E) ) =\ell_+% \text{ } \label{mhtsa}$$ \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 $\inf_{00$ 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. \begin{thebibliography}{99} \bibitem{Addou1} \textsc{Addou, I., S. M. Bouguima, M. Derhab and Y. S. Raffed, }\textit{On the number of solutions of a quasilinear elliptic class of B.V.P. with jumping nonlinearities,} Dynamic Syst. Appl. \textbf{7} (4) (1998), pp. 575-599. \bibitem{Addou22} \textsc{Addou, I. and A. Benmeza\"{\i },}\textit{\ On the number of solutions for the one dimensional p-Laplacian with cubic-like nonlinearities, }In:''CIMASI'98, Deuxi\`{e}me Conf\'{e}rence Internationale sur les Math\'{e}matiques Appliqu\'{e}es et les Sciences de l'Ing\'{e}nieur'', held at Casablanca, Morocco, October 27-29, 1998. Actes, \textbf{Vol. 1} (1998), pp. 77-79. \bibitem{Addou3} \textsc{Addou, I. and A. Benmeza\"{\i },}\textit{\ Exact number of positive solutions for a class of quasilinear boundary value problems, }Dynamic Syst. Appl. \textbf{8} (1999), pp. 147-180. \bibitem{Addou4} \textsc{Addou, I. and A. Benmeza\"{\i },}\textit{\ Boundary value problems for the one dimensional p-Laplacian with even superlinearity, }Electron. J. Diff. Eqns., \textbf{1999} (1999), No. 09, pp. 1-29. \bibitem{Addou6} \textsc{Addou, I.,}\textit{\ Multiplicity of solutions for a quasilinear elliptic class of boundary value problems, }Electron. J. Diff. Eqns., \textbf{1999} (1999), No. 21, pp. 1-27. \bibitem{Addou7} \textsc{Addou, I.,}\textit{\ Exact multiplicity results for quasilinear boundary-value problems with cubic-like nonlinearities, }% Electron. J. Diff. Eqns., \textbf{2000} (2000), No. 01, pp. 1-26. (Addendum, pp. 27-29). \bibitem{Thesis} \textsc{Addou, I.,}\textit{\ }Doctoral Thesis, (February 2000), USTHB Institut de Math\'{e}matiques, Algiers, Algeria. \bibitem{Diaz} \textsc{Diaz, J. I.,}\textit{\ Nonlinear partial differential equations and free boundaries vol1: Elliptic equations, }% \textbf{106} Research Notes in Math. (Pitman Advanced Publishing Program, London 1985). \bibitem{GNN} \textsc{Gidas, B., W.-M., Ni, and L. Nirenberg,}\textit{\ Symmetry and related properties via the maximum principle,} Commun. Math. Phys. \textbf{68} (1979), pp. 209-243. \bibitem{Guedda} \textsc{Guedda, M., and L. Veron,}\textit{\ Bifurcation phenomena associated to the p-Laplace operator, }Trans. Amer. Math. Soc. \textbf{310 }(1987), pp. 419-431. \bibitem{Korman97} \textsc{Korman, P.,}\textit{\ Steady states and long time behavior of some convective reaction-diffusion equations,} Funkc. Ekvacioj \textbf{40} (1997), pp. 165-183. \bibitem{KormanPreprint} \textsc{Korman, P.,}\textit{\ The global solution set for a class of semilinear problems,} Preprint. \bibitem{KormanOuyang1} \textsc{Korman, P. and T. Ouyang,}\textit{\ Multiplicity results for two classes of boundary-value problems,} SIAM J. Math. Anal. \textbf{26 }(1995), pp. 180-189. \bibitem{KormanOuyang2} \textsc{Korman, P. and T. Ouyang,}\textit{\ Exact multiplicity results for a class of boundary-value problems with cubic nonlinearities,} J. Math. Anal. Appl. \textbf{194} (1995), pp. 328-341. \bibitem{KormanOuyang3} \textsc{Korman, P. and T. Ouyang,}\textit{\ Solution curves for two classes of boundary-value problems,} Nonlinear Analysis T. M. A. \textbf{27} (9) (1996), pp. 1031-1047. \bibitem{KormanShi} \textsc{Korman, P. and J. Shi,}\textit{\ Instability and multiplicity of solutions of semilinear equations,} Proc. of Conference honoring Alan Lazer for his 60th birthday, Miami, Florida, USA, 1999. \bibitem{KormanEtAl} \textsc{Korman, P., Y. Li and T. Ouyang,}\textit{\ Exact multiplicity results for boundary problems with nonlinearities generalizing cubic,} Proc. Royal Soc. Edinb. \textbf{126A} (1996), pp. 599-616. \bibitem{KormanEtAl2} \textsc{Korman, P., Y. Li and T. Ouyang,}\textit{\ An exact multiplicity result for a class of semilinear equations,} Commun. in Partial Diff. Equations, \textbf{22} (3 \& 4), (1997), pp. 661-684. \bibitem{LN} \textsc{Lin, C. S., and W.-M., Ni,}\textit{\ A counterexample to the nodal domain conjecture and a related semilinear equation,} Proc. Amer. Math. Soc. \textbf{102} (1988), pp. 271-277. \bibitem{Otani} \textsc{\^{O}tani, M.,}\textit{\ A remark on certain nonlinear elliptic equations, }Proc. Faculty of Science, Tokai Univ. \textbf{% 19} (1984), 23-28. \bibitem{OuyangShi} \textsc{Ouyang, T., and J. Shi,}\textit{\ Exact multiplicity of positive solutions for a class of semilinear problems,} J. Differential Equations, II, \textbf{158}, No. 1, (1999), pp. 94-151. \bibitem{Pimbley} \textsc{G. H. Pimbley, Jr.,}\textit{\ Eigenfunction branches of nonlinear operators, and their bifurcations}, Lecture Notes in Math. \textbf{104} (1969), Springer-Verlag. (Sect. 9, p. 92). \bibitem{Schaaf} \textsc{Schaaf, R.,}\textit{\ Global solution branches of two point boundary value problems,} Lecture Notes in Mathematics \textbf{% 1458,} Berlin, Springer 1990. \bibitem{ShiShivaji} \textsc{Shi, J., and R. Shivaji,}\textit{\ Exact multiplicity of solutions for classes of semipositone problems with concave-convex nonlinearity,} Preprint, May 2000. \bibitem{SmollerWasserman} \textsc{Smoller, J. and A. Wasserman,}\textit{\ Global bifurcation of steady-state solutions,} J. Diff. Equations, \textbf{39% } (1981), pp. 269-290. \bibitem{Urabe} \textsc{Urabe, M.,}\textit{\ Relations between periods and amplitudes of periodic solutions of }$\ddot{x}+g( x) =0$, Funkc. Ekvacioj \textbf{6} (1964), pp. 63-88. \bibitem{Wang} \textsc{Wang, S.-H.,}\textit{\ A correction for a paper by J. Smoller and A. Wasserman,} J. Diff. Equations, \textbf{77} (1989), pp. 199-202. \bibitem{WangKaza1} \textsc{Wang, S.-H., and Kazarinoff}\textit{, Bifurcation and stability of positive solutions of two-point boundary value problem,} J. Austral. Math. Soc. Ser. A \textbf{52} (1992), pp. 334-342. \bibitem{WangKaza2} \textsc{Wang, S.-H., and Kazarinoff}\textit{, Bifurcation and steady-state solutions of a scalar reaction-diffusion equation in one space variable,} J. Austral. Math. Soc. Ser. A \textbf{52} (1992), pp. 343-355.. \bibitem{Wei} \textsc{Wei, J.,}\textit{\ Exact multiplicity for some nonlinear elliptic equations in balls, Proc. Amer. Math. Soc. }\textbf{125} (1997), pp. 3235-3242. \end{thebibliography} \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} \end{document}