\documentclass[twoside]{article} \usepackage{amssymb} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Exact multiplicity results \hfil EJDE--2000/01} {EJDE--2000/01\hfil Idris Addou \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2000}(2000), No.~01, pp. 1--26. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Exact multiplicity results for quasilinear boundary-value problems with cubic-like nonlinearities \thanks{ {\em 1991 Mathematics Subject Classifications:} 34B15. \hfil\break\indent {\em Key words and phrases:} One dimensional p-Laplacian, multiplicity results, time-maps. \hfil\break\indent \copyright 2000 Southwest Texas State University and University of North Texas. \hfil\break\indent Submitted May 26, 1999. Revised October 1, 1999. Published January 1, 2000.} } \date{} % \author{Idris Addou} \maketitle \begin{abstract} We consider the boundary-value problem $$\displaylines{ -(\varphi_p (u'))' =\lambda f(u) \mbox{ in }(0,1) \cr u(0) = u(1) =0\,, }$$ where $p>1$, $\lambda >0$ and $\varphi_p (x) =| x|^{p-2}x$. The nonlinearity $f$ is cubic-like with three distinct roots $0=a0$. This way we extend a recent result, for $p=2$, by Korman et al. \cite{KormanLiOuyang} to the general case $p>1$. We shall prove that when $12$. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}{Lemma} \section{Introduction}\label{sec1} We consider the question of determining the exact number of solutions of the quasilinear boundary-value problem \begin{eqnarray} &-(\varphi_p (u'))' =g( \lambda ,u) ,\mbox{ in }(0,1)& \label{P1} \\ &u(0) = u(1) =0\,,& \nonumber \end{eqnarray} where $p>1$, $\lambda >0$ and $\varphi_p (u) =| u|^{p-2}u$ for all $u\in{\mathbb R}$ and $g(\lambda ,u) =\lambda f(u)$. Here the nonlinearity $f\in C^{2}({\mathbb R},{\mathbb R})$ is cubic-like satisfying \begin{eqnarray} &f(0) =f(b) =f(c) =0 \mbox{ for some constants } 0 0\mbox{ for } x\in (-\infty ,0) \cup (b,c) & \label{Austine2} \\ &f(x) < 0\mbox{ for } x\in (0,b) \cup (c,+\infty)\,, \nonumber\\ &f''(u) \mbox{ changes sign exactly once when } u\in (0,c)\,,& \label{Austine3}\\ &F(c) >0, \mbox{ where } F(s) =\int_0^{s}f(u) du,\ s\in{\mathbb R}\,. & \label{Austine4} \end{eqnarray} Beside conditions (\ref{Austine1})-(\ref{Austine4}) we shall assume in the case where $p\neq 2$, the following additional conditions: There exists $u_0\in (0,c)$ such that $$(p-2) f'(u) -uf''(u) \leq 0 \mbox{ for } u\in (0,u_0] \label{E1}$$ with strict inequality in an open interval $I \subset (0,u_0)$, and $$(p-2) f'(u) -uf''(u) \geq 0\mbox{ for }u\in [ u_0,c) \,. \label{E2}$$ When $p=2$, we prove in Section \ref{sec3}, that (\ref{E1}) and (\ref{E2}) are consequences of (\ref{Austine1})-(\ref{Austine3}). During this last decade, many articles dealing with boundary-value problems with cubic-like nonlinearities have been published. (See for instance; \cite{Korman97}-\cite{Wei}). However, all the related results have been obtained for the case $p=2$; that is, for the Laplacian operator. The case of cubic-like nonlinearities when the differential operator is the $p$-Laplacian with $p\neq 2$ has yet to be studied. When $p=2$ and $f$ satisfies conditions (\ref{Austine1})-(\ref{Austine4}), the solution set of problem (\ref{P1}) was studied recently by Korman et al. \cite{KormanLiOuyang}. They provide exactness results. They show (among other interesting things) that there exists a critical number $\lambda_0>0$ such that problem (\ref{P1}) has no nontrivial solution for $0<\lambda <\lambda_0$, has a unique nontrivial solution for $\lambda =\lambda_0$ and has exactly two nontrivial solutions for all $\lambda >\lambda_0$. So, a natural question arises; how does the solution set of (\ref{P1}) look like when $p\neq 2$? The purpose of this work is to answer this question. We shall give an exactness result with respect to $p>1$; we prove, in particular, that when $12$. It is known that exactness results are more difficult to derive than a lower bound of the number of solutions to boundary value problems such as (\ref{P1}). The main tool used here is the so-called quadrature method. The delicate part in the process of the proof corresponding to the exactness part of the main results is the study of the exact variations of the time map under consideration over its {\em entire} definition domain (Lemma \ref{Lemma3}). Notice that here, the cubic-like nonlinearity $f$ has three distinct roots $a1$. Also, we have considered in \cite {HalfOdd} a more general case where $a1$, and $f$ is not necessary odd; there we have defined a new kind of functions we called: half-odd. However, the main results of the present paper are directly related to those of Korman et al. \cite{KormanLiOuyang} and not to those of \cite{Addou2} and \cite{HalfOdd}. That is why we do not describe them here. (Also, this would require a large space). The paper is organized as follows. The main results are stated in Section \ref{sec2}. Next, in Section \ref{sec3} we shall state and prove some properties of the nonlinearity $f$. These are of importance in the sequel. Some preliminary lemmas are the aim of Section \ref{sec4}; the first lemma (Lemma \ref{Lemma1}) is technical and in the second one (Lemma \ref{Lemmasup}) we locate all the eventual nontrivial solutions of problem (\ref{P1}). The proof of Lemma \ref{Lemmasup} is postponed to the appendix. After describing the quadrature method used in order to look for the solutions, we devote two lemmas (Lemmas \ref{Lemma2}, \ref{Lemma3}) to study the limits and variations of the time-map. In Section \ref{sec5}, the main results are proved. Finally, in Section \ref{sec6} we ask two questions. \section{Notation and main results}\label{sec2} In order to state the main results, let us first define the subsets of $C^{1}([ 0,1])$ which contain the solutions of the problem (\ref{P1}). \noindent Let $A_1^{+}$ be the subset of $C^1([ 0,1])$ consisting of the functions $u$ satisfying \begin{itemize} \item $u(x) >0$, for all $x\in (0,1)$, $u(0) =u(1) =00$, for all $x\in (0,1)$, $u(0) =u(1) =0 0\mbox{ in }(b_{i},a_{i+1}) ,\mbox{ for all }i\in \{ 0,\cdots ,k-1\} \\ &u \equiv 0\mbox{ in }[ a_{i},b_{i}] ,\mbox{ for all }i\in \{ 0,\cdots ,k-1\}\, .& \end{eqnarray*} \item Every hump of$u$is symmetrical with respect to the center of the interval of its definition. \item The derivative of each hump of$u$vanishes once and only once. \item Each hump is a translated copy of the first one. \end{itemize} \noindent Let$B_k^{+}$be the subset of$B^{+}(k) $consisting of the functions$u$satisfying $a_i(u) =b_i(u) \mbox{ for all }i\in \{0,\cdots ,k\} .$ If there exists$i_0\in \{ 0,\cdots ,k\} $such that$a_{i_0}(u) 1$and$x\in [ r,c]$, define $S_{+}(x) =\int_0^{x}\{ F(x) -F(\xi) \} ^{-1/p}d\xi \,.$ We shall prove in Lemma \ref{Lemma2} that$S_{+}(r) $(resp.$S_{+}(c) $) is infinite if and only if$12$we can define$\nu =(2S_{+}(c)) ^{p}/p'$, where$p'=p/(p-1) $, and for all integer$k\geq 0$we define$\lambda_k=(2kS_{+}(r)) ^{p}/p'$and notice that $0=\lambda_0<\lambda_1<\cdots <\lambda_k=k^{p}\lambda_1\dots \mbox{ for all }k\geq 1,\mbox{ and }\lim_{k\to +\infty }\lambda _k=+\infty \,.$ For$\lambda >0$, denote$S_\lambda $the solution set of problem (\ref {P1}). % fig1.tex \begin{figure}[t] \setlength{\unitlength}{1mm} \begin{picture}(90,40)(-15,0) \linethickness{1pt} \qbezier(80,30)(-40,20)(80,10) \qbezier[20](20,0)(20,10)(20,20) \thinlines \put(10,0){\vector(0,1){40}} \put(10,0){\vector(1,0){80}} \put(18,-5){$\lambda_0$} \put(90,-5){$\lambda$} \put(38,28){$A_1^+$} \end{picture} \caption{$1< p \leq 2$.} \end{figure} The main results are worth being described by means of diagrams. The first result (Theorem \ref{wmdjkgfh}) concerns the case where$1\lambda_0$. All these solutions are in$A_1^+$. % fig2.tex \begin{figure}[t] \setlength{\unitlength}{1.3mm} \begin{picture}(90,40)(4,-2) \linethickness{1pt} \qbezier(25,0)(-8,12)(57,31) \qbezier[15](13.8,0)(13.8,5)(13.8,9) \qbezier[50](57,0)(57,16)(57,31) \put(38,28){$A_1^+$} \qbezier(57,31)(70,35)(80,37) \put(82,37){$\widetilde{A}_1^+$} \qbezier(25,0)(25,16)(80,28) \put(82,28){$\widetilde{B}_1^+$} \qbezier(38,0)(38,12)(80,21) \put(82,21){$\widetilde{B}_2^+$} \qbezier(51,0)(51,8)(80,14) \put(82,14){$\widetilde{B}_{n-1}^+$} \qbezier(64,0)(64,4)(80,7) \put(82,7){$\widetilde{B}_n^+$} \thinlines \put(10,0){\vector(0,1){40}} \put(10,0){\vector(1,0){76}} \put(13,-3){$\mu$} \put(25,-3){$\lambda_1$} \put(38,-3){$\lambda_2$} \put(49,-3){$\lambda_{n-1}$} \put(57,-3){$\nu$} \put(64,-3){$\lambda_n$} \put(85,-3){$\lambda$} \end{picture} \caption{$p>2$,$\lambda_{n-1}<\nu<\lambda_n$,$1< n$.} \end{figure} % fig3.tex \begin{figure}[t] \setlength{\unitlength}{1.3mm} \begin{picture}(90,42)(8,-3) \linethickness{1pt} \qbezier(25,0)(25,20)(57,31) \qbezier[50](57,0)(57,16)(57,31) \put(38,26){$A_1^+$} \qbezier(57,31)(70,35)(80,37) \put(82,37){$\widetilde{A}_1^+$} \qbezier(25,0)(25,16)(80,28) \put(82,28){$\widetilde{B}_1^+$} \qbezier(38,0)(38,12)(80,21) \put(82,21){$\widetilde{B}_2^+$} \qbezier(51,0)(51,8)(80,14) \put(82,14){$\widetilde{B}_{n-1}^+$} \qbezier(64,0)(64,4)(80,7) \put(82,7){$\widetilde{B}_n^+$} \thinlines \put(15,0){\vector(0,1){40}} \put(15,0){\vector(1,0){75}} \put(25,-3){$\lambda_1$} \put(38,-3){$\lambda_2$} \put(49,-3){$\lambda_{n-1}$} \put(57,-3){$\nu$} \put(64,-3){$\lambda_n$} \put(85,-3){$\lambda$} \end{picture} \caption{$p>2$,$\lambda_{n-1}<\nu<\lambda_n$,$1< n$.} \end{figure} When$p>2$, we have to consider the sequence$(\lambda_k)_{k\geq 0}$and the number$\nu >0$. This number maybe smaller than$\lambda_1$, equal to$\lambda_1$, or greater than$\lambda_1$. In this later case, it may lie between two consecutive points of the sequence:$\lambda_{n-1}<\nu <\lambda_{n}$, with$n>1$(Figures 2 and 3), or it maybe equal to some$\lambda_{n}$with$n>1$. An immediate examination of these bifurcation diagrams, shows that when$\nu$moves from zero to infinity, the upper branch changes but not the others, i.e., beside the upper branch which is different from a diagram to an other, the remaining branches are the same in all these diagrams. Now consider any one of figures 2 or 3 and let us describe each kind of its branches. The$\lambda $-axis designates the trivial solutions, and at each$\lambda_k$,$k\geq 1$, there is a bifurcation point which indicates a pair$(u_k,\lambda_k)$such that$u_k\in B_k^+$. The upper branch contains a point which indicates a pair$(u_1,\nu)$such that$u_1\in A_1^+$. All points lying on this branch which are on the left of$(u_1,\nu)$are in$A_1^+$, and those lying at the right are in$\tilde{A}_1^+$. The remaining branches are in some sense ''singular''. Usually a point$(u,\lambda)$lying on any branch designates a couple where$u$is a solution of some kind and$\lambda $is a real number. This is the case in our diagrams as far as the upper branch or the lower one ($\lambda $-axis) are concerned. However, a point on the remaining branches indicates$(\mathop{\rm Cl}(u),\lambda)$, i.e., the equivalence class of a certain solution$u$lying in some$\tilde{B}_k^+$,$k\geq 1$, and$\lambda $is a real number. So, if$u$is a solution in some$\tilde{B}_k^+$, with$k\geq 1 $, for some$\lambda >0$then any$v\in \mathop{\rm Cl}(u)$is also a solution in the same$\tilde{B}_k^+$. The singularity of these branches maybe removed. In fact, consider the same equivalence relation but defined on$B_k^+$, (for all$k\geq 1$). Then it is clear that $\mathop{\rm Cl}(u)=\{u\}, \mbox{ for all } u\in B_k^+\,.$ So, the bifurcation points on the$\lambda $-axis maybe considered as indicating \\$(\mathop{\rm Cl}(u_k),\lambda_k)=(\{u_k\},\lambda_k)$instead of$(u_k,\lambda_k)$. Also, consider on$A_1^+\cup \tilde{A}_1^+$the same equivalence relation in essence (which maybe formulated differently). It is clear that $\mathop{\rm Cl}(u)=\{u\}, \mbox{ for all } u\in A_1^+\cup \tilde{A}_1^+\,.$ This way, any point on any branch shall designates a couple$(\mathop{\rm Cl}(u),\lambda)$and the elements in$\mathop{\rm Cl}(u)$are solutions of the problem (\ref{P1}) for the same$\lambda $. Therefore, there is coherence in the diagrams and the singularity mentioned above is removed. The statements of the main results below indicate that for$\nu \leq \lambda_1$the upper branch contains a turning point, but when$\nu >\lambda_1$, either it still contains a turning point (Figure 2) or there is no such point (Figure 3). \smallskip The main results read as follows \begin{theorem} \label{wmdjkgfh} Assume that$10$such that \begin{description} \item[(i)] If$0<\lambda <\lambda_0$,$S_\lambda =\{ 0\} $. \item[(ii)] If$\lambda =\lambda_0$, there exists$v_\lambda \in A_1^+$such that$S_\lambda =\{ 0\} \cup \{ v_\lambda\} $. \item[(iii)] If$\lambda >\lambda_0$, there exists$v_\lambda $,$w_\lambda \in A_1^+$such that$v_\lambda \neq w_\lambda $and$S_\lambda =\{ 0\} \cup \{ v_\lambda ,w_{\lambda }\} $. \end{description} \end{theorem} \begin{theorem} \label{wmdjkgfh2}Assume that$p>2$and$f$satisfies conditions (\ref {Austine1})-(\ref{Austine4}), and (\ref{E1}), (\ref{E2}). Moreover, assume that$\nu <\lambda_1$. Then there exists$\mu \in (0,\nu) $such that \begin{description} \item[(i)] If$0<\lambda <\mu $,$S_\lambda =\{ 0\} $. \item[(ii)] If$\lambda =\mu $, there exists$v_\lambda \in A_1^+$such that$S_\lambda =\{ 0\} \cup \{ v_\lambda \} $. \item[(iii)] If$\mu <\lambda \leq \nu $, there exists$v_\lambda $,$w_\lambda \in A_1^+$such that$v_\lambda \neq w_\lambda $and$S_\lambda =\{ 0\} \cup \{ v_\lambda ,w_{\lambda}\} $. \item[(iv)] If$\nu <\lambda <\lambda_1$, there exists$v_\lambda \in A_1^+$and$u_\lambda \in \tilde{A}_1^+$such that$S_{\lambda }=\{ 0\} \cup \{ v_\lambda \} \cup \{u_\lambda \} $. \item[(v)] If$\lambda =\lambda_1$, there exists$u_\lambda \in \tilde{ A}_1^+$and$u_{\lambda ,1}\in B_1^+$such that$S_{\lambda }=\{ 0\} \cup \{ u_\lambda \} \cup \{u_{\lambda ,1}\} $. \item[(vi)] If$\lambda_k<\lambda <\lambda_{k+1}$,$k\geq 1$, there exists$u_\lambda \in \tilde{A}_1^+$and$u_{\lambda ,1},\cdots ,u_{\lambda ,k}$such that$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$for all$i=1,\cdots ,k$and$S_\lambda =\{ 0\} \cup \{ u_\lambda \} \cup \mathop{\rm Cl}(u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k}) $. \item[(vii)] If$\lambda =\lambda_{k+1},k\geq 1$, there exists$u_\lambda \in \tilde{A}_1^+$and$u_{\lambda ,1},\cdots u_{\lambda ,k+1}$such that$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$for all$i=1,\cdots ,k$,$u_{\lambda ,k+1}\in B_{k+1}^{+}$and$S_{\lambda }=\{ 0\} \cup \{ u_\lambda \} \cup \mathop{\rm Cl}( u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k}) \cup \{ u_{\lambda ,k+1}\} $. \end{description} \end{theorem} \begin{theorem} \label{wmdjkgfh3}Assume that$p>2$and$f$satisfies conditions (\ref {Austine1})-(\ref{Austine4}), and (\ref{E1}), (\ref{E2}). Moreover, assume that$\nu =\lambda_1$. Then there exists$\mu \in (0,\lambda_1) $such that \begin{description} \item[(i)] If$0<\lambda <\mu $,$S_\lambda =\{ 0\} $. \item[(ii)] If$\lambda =\mu $, there exists$v_\lambda \in A_1^+$such that$S_\lambda =\{ 0\} \cup \{ v_\lambda \}$. \item[(iii)] If$\mu <\lambda <\lambda_1$, there exists$v_\lambda $,$w_\lambda \in A_1^+$such that$v_\lambda \neq w_\lambda $and$S_\lambda =\{ 0\} \cup \{ v_\lambda ,w_{\lambda }\} $. \item[(iv)] If$\lambda =\lambda_1$, there exists$v_\lambda \in A_1^+$and$u_{\lambda ,1}\in B_1^+$such that$S_{\lambda }=\{ 0\} \cup \{ v_\lambda \} \cup \{ u_{\lambda ,1}\} $. \item[(v)] If$\lambda_k<\lambda <\lambda_{k+1}$,$k\geq 1$, there exists$u_\lambda \in \tilde{A}_1^+$and$u_{\lambda ,1},\cdots ,u_{\lambda ,k}$such that$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$for all$i=1,\cdots ,k$and$S_\lambda =\{ 0\} \cup \{ u_\lambda \} \cup \mathop{\rm Cl}(u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k}) $. \item[(vi)] If$\lambda =\lambda_{k+1},k\geq 1$, there exists$u_\lambda \in \tilde{A}_1^+$and$u_{\lambda ,1},\cdots ,\;u_{\lambda ,k+1}$such that$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$for all$i=1,\cdots ,k$,$u_{\lambda ,k+1}\in B_{k+1}^{+}$and$S_\lambda =\{ 0\} \cup \{ u_\lambda \} \cup \mathop{\rm Cl}(u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k}) \cup \{ u_{\lambda ,k+1}\} $. \end{description} \end{theorem} \begin{theorem} \label{wmdjkgfh4}Assume that$p>2$and$f$satisfies conditions (\ref {Austine1})-(\ref{Austine4}), and (\ref{E1}), (\ref{E2}). Moreover, assume that there exists$n>1$such that$\lambda_{n-1}<\nu <\lambda_{n}$. Then one and only one of the following possibilities occurs: Possibility {\bf A}. There exists$\mu \in (0,\lambda_1) $such that \begin{description} \item[(i)] If$0<\lambda <\mu $,$S_\lambda =\{ 0\} $. \item[(ii)] If$\lambda =\mu $, there exists$v_\lambda \in A_1^+$such that$S_\lambda =\{ 0\} \cup \{ v_\lambda \} $. \item[(iii)] If$\mu <\lambda <\lambda_1$, there exist$v_\lambda $,$w_\lambda \in A_1^+$such that$v_\lambda \neq w_\lambda $and$S_\lambda =\{ 0\} \cup \{ v_\lambda ,w_{\lambda}\} $. \item[(iv)] If$\lambda =\lambda_1$, there exist$v_\lambda \in A_1^+$and$u_{\lambda ,1}\in B_1^+$such that$S_{\lambda }=\{ 0\} \cup \{ v_\lambda \} \cup \{ u_{\lambda ,1}\} $. \item[(v)] If$\lambda_k<\lambda <\min \{ \lambda_{k+1},\nu \} $,$1\leq k\leq n-1$, there exist$v_\lambda \in A_1^+$and$u_{\lambda ,1},\cdots ,u_{\lambda ,k}$such that$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$for all$i=1,\cdots ,k$and$S_{\lambda }=\{ 0\} \cup \{ v_\lambda \} \cup \mathop{\rm Cl}( u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k}) $. \item[(vi)] If$\lambda =\lambda_{k+1},1\leq k\leq n-2$, there exist$v_\lambda \in A_1^+$and$u_{\lambda ,1},\cdots , u_{\lambda ,k+1}$such that$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$for all$i=1,\cdots ,k$,$u_{\lambda ,k+1}\in B_{k+1}^{+}$and$S_{\lambda }=\{ 0\} \cup \{ v_\lambda \} \cup \mathop{\rm Cl}( u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k}) \cup \{ u_{\lambda ,k+1}\} $. \item[(vii)] If$\lambda =\nu $, there exist$v_\lambda \in A_1^+$and$u_{\lambda ,1},\cdots ,u_{\lambda ,n-1}$such that$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$for all$i=1,\cdots ,n-1$, and$S_{\lambda }=\{ 0\} \cup \{ v_\lambda \} \cup \mathop{\rm Cl}( u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,n-1})$. \item[(viii)] If$\max \{ \lambda_k,\nu \} <\lambda <\lambda_{k+1}$,$k\geq n-1$, there exist$u_\lambda \in \tilde{A} _1^+$and$u_{\lambda ,1},\cdots ,u_{\lambda ,k}$such that$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$for all$i=1,\cdots ,k$, and$S_\lambda =\{ 0\} \cup \{ u_\lambda \} \cup \mathop{\rm Cl}(u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}( u_{\lambda,k}) $. \item[(ix)] If$\lambda =\lambda_{k+1}$,$k\geq n-1$, there exist$u_\lambda \in \tilde{A}_1^+$and$u_{\lambda ,1},\cdots ,u_{\lambda ,k+1}$such that$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$for all$i=1,\cdots ,k$,$u_{\lambda ,k+1}\in B_{k+1}^{+}$and$S_{\lambda }=\{ 0\} \cup \{ u_\lambda \} \cup \mathop{\rm Cl}( u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k}) \cup \{ u_{\lambda ,k+1}\} $. \end{description} Possibility {\bf B}. \begin{description} \item[(i)] If$0<\lambda <\lambda_1$,$S_\lambda =\{ 0\} $. \item[(ii)] If$\lambda =\lambda_1$, there exists$u_{\lambda ,1}\in B_1^+$such that$S_\lambda =\{ 0\} \cup \{ u_{\lambda,1}\} $. \item[(iii)] If$\lambda_k<\lambda <\min \{ \lambda_{k+1},\nu \} $,$1\leq k\leq n-1$, there exist$v_\lambda \in A_1^+$and$u_{\lambda ,1},\cdots ,u_{\lambda ,k}$such that$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$for all$i=1,\cdots ,k$and$S_{\lambda }=\{ 0\} \cup \{ v_\lambda \} \cup \mathop{\rm Cl}( u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k}) $. \item[(iv)] If$\lambda =\lambda_{k+1},1\leq k\leq n-2$, there exist$v_\lambda \in A_1^+$and$u_{\lambda ,1},\cdots , u_{\lambda ,k+1}$such that$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$for all$i=1,\cdots ,k$,$u_{\lambda ,k+1}\in B_{k+1}^{+}$and$S_{\lambda }=\{ 0\} \cup \{ v_\lambda \} \cup \mathop{\rm Cl}( u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k}) \cup \{ u_{\lambda ,k+1}\} $. \item[(v)] If$\lambda =\nu $, there exist$v_\lambda \in A_1^+$and$u_{\lambda ,1},\cdots ,u_{\lambda ,n-1}$such that$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$for all$i=1,\cdots ,n-1$, and$S_{\lambda }=\{ 0\} \cup \{ v_\lambda \} \cup \mathop{\rm Cl}( u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,n-1})$. \item[(vi)] If$\max \{ \lambda_k,\nu \} <\lambda <\lambda _{k+1}$,$k\geq n-1$, there exist$u_\lambda \in \tilde{A}_1^+$and$u_{\lambda ,1},\cdots ,u_{\lambda ,k}$such that$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$for all$i=1,\cdots ,k$, and$S_{\lambda }=\{ 0\} \cup \{ u_\lambda \} \cup \mathop{\rm Cl}( u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k}) $. \item[(vii)] If$\lambda =\lambda_{k+1}$,$k\geq n-1$, there exist$u_\lambda \in \tilde{A}_1^+$and$u_{\lambda ,1},\cdots ,u_{\lambda ,k+1}$such that$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$for all$i=1,\cdots ,k$,$u_{\lambda ,k+1}\in B_{k+1}^{+}$and$S_{\lambda }=\{ 0\} \cup \{ u_\lambda \} \cup \mathop{\rm Cl}( u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k}) \cup \{ u_{\lambda ,k+1}\} $. \end{description} \end{theorem} \begin{theorem} \label{wmdjkgfh5}Assume that$p>2$and$f$satisfies conditions (\ref {Austine1})-(\ref{Austine4}), and (\ref{E1}), (\ref{E2}). Moreover, assume that there exists$n>1$such that$\nu =\lambda_{n}$. Then one and only one of the following possibilities occurs: Possibility {\bf C}. There exists$\mu \in (0,\lambda_1) $such that \begin{description} \item[(i)] If$0<\lambda <\mu $,$S_\lambda =\{ 0\} $. \item[(ii)] If$\lambda =\mu $, there exists$v_\lambda \in A_1^+$such that$S_\lambda =\{ 0\} \cup \{ v_\lambda \}$. \item[(iii)] If$\mu <\lambda <\lambda_1$, there exist$v_\lambda $,$w_\lambda \in A_1^+$such that$v_\lambda \neq w_\lambda $and$S_\lambda =\{ 0\} \cup \{ v_\lambda ,w_{\lambda }\} $. \item[(iv)] If$\lambda =\lambda_1$, there exist$v_\lambda \in A_1^+$and$u_{\lambda ,1}\in B_1^+$such that$S_{\lambda }=\{ 0\} \cup \{ v_\lambda \} \cup \{u_{\lambda ,1}\} $. \item[(v)] If$\lambda_k<\lambda <\lambda_{k+1}$,$1\leq k\leq n-1$, there exist$v_\lambda \in A_1^+$and$u_{\lambda ,1},\cdots ,u_{\lambda ,k}$such that$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$for all$i=1,\cdots ,k$and$S_\lambda =\{ 0\} \cup \{ v_\lambda \} \cup \mathop{\rm Cl}(u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k}) $. \item[(vi)] If$\lambda =\lambda_{k+1},1\leq k\leq n-1$, there exist$v_\lambda \in A_1^+$and$u_{\lambda ,1},\cdots , u_{\lambda ,k+1}$such that$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$for all$i=1,\cdots ,k$,$u_{\lambda ,k+1}\in B_{k+1}^{+}$and$S_{\lambda }=\{ 0\} \cup \{ v_\lambda \} \cup \mathop{\rm Cl}( u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k}) \cup \{ u_{\lambda ,k+1}\} $. \item[(vii)] If$\lambda_k<\lambda <\lambda_{k+1}$,$k\geq n$, there exists$u_\lambda \in \tilde{A}_1^+$and$u_{\lambda ,1},\cdots ,u_{\lambda ,k}$such that$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$for all$i=1,\cdots ,k$, and$S_\lambda =\{ 0\} \cup \{ u_\lambda \} \cup \mathop{\rm Cl}(u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k}) $. \item[(viii)] If$\lambda =\lambda_{k+1}$,$k\geq n$, there exist$u_\lambda \in \tilde{A}_1^+$and$u_{\lambda ,1},\cdots ,u_{\lambda ,k+1}$such that$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$for all$i=1,\cdots ,k$,$u_{\lambda ,k+1}\in B_{k+1}^{+}$and$S_{\lambda }=\{ 0\} \cup \{ u_\lambda \} \cup \mathop{\rm Cl}( u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k}) \cup \{ u_{\lambda ,k+1}\} $. \end{description} Possibility {\bf D}. \begin{description} \item[(i)] If$0<\lambda <\lambda_1$,$S_\lambda =\{ 0\} $. \item[(ii)] If$\lambda =\lambda_1$, there exists$u_{\lambda ,1}\in B_1^+$such that$S_\lambda =\{ 0\} \cup \{ u_{\lambda,1}\} $. \item[(iii)] If$\lambda_k<\lambda <\lambda_{k+1}$,$1\leq k\leq n-1$, there exist$v_\lambda \in A_1^+$and$u_{\lambda ,1},\cdots ,u_{\lambda ,k}$such that$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$for all$i=1,\cdots ,k$and$S_\lambda =\{ 0\} \cup \{ v_\lambda \} \cup \mathop{\rm Cl}(u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k}) $. \item[(iv)] If$\lambda =\lambda_{k+1},1\leq k\leq n-1$, there exist$v_\lambda \in A_1^+$and$u_{\lambda ,1},\cdots , u_{\lambda ,k+1}$such that$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$for all$i=1,\cdots ,k$,$u_{\lambda ,k+1}\in B_{k+1}^{+}$and$S_{\lambda }=\{ 0\} \cup \{ v_\lambda \} \cup \mathop{\rm Cl}( u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k}) \cup \{ u_{\lambda ,k+1}\} $. \item[(v)] If$\lambda_k<\lambda <\lambda_{k+1}$,$k\geq n$, there exists$u_\lambda \in \tilde{A}_1^+$and$u_{\lambda ,1},\cdots ,u_{\lambda ,k}$such that$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$for all$i=1,\cdots ,k$, and$S_\lambda =\{ 0\} \cup \{ u_\lambda \} \cup \mathop{\rm Cl}(u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k}) $. \item[(vi)] If$\lambda =\lambda_{k+1}$,$k\geq n$, there exist$u_\lambda \in \tilde{A}_1^+$and$u_{\lambda ,1},\cdots ,u_{\lambda ,k+1}$such that$u_{\lambda ,i}\in \tilde{B}_{i}^{+}$for all$i=1,\cdots ,k$,$u_{\lambda ,k+1}\in B_{k+1}^{+}$and$S_{\lambda }=\{ 0\} \cup \{ u_\lambda \} \cup \mathop{\rm Cl}( u_{\lambda ,1}) \cup \cdots \cup \mathop{\rm Cl}(u_{\lambda ,k}) \cup \{ u_{\lambda ,k+1}\} $. \end{description} \end{theorem} The novelty in these results concerns the cases$p>1$with$p\neq 2$. The case$p=2$was proved by Korman et al. \cite{KormanLiOuyang}. Of course, the case$p=2$is also studied here. \section{Some properties of the nonlinearity$f$}\label{sec3} In this section we establish some properties of$f$. These are used in the sequel and are of importance in our analysis. We first state the properties and next we give the proofs. \subsection*{Statement of properties} Assume that$f$satisfies (\ref{Austine1})-(\ref{Austine3}), then there exists$u_0\in (0,c) $such that \begin{eqnarray} &f''\geq 0\mbox{ in }(0,u_0]\,, &\label{A1}\\ &f''\leq 0\mbox{ in }[ u_0,c) \,. &\label{A2} \end{eqnarray} Moreover, there exist two open intervals$I$and$J$with$I\subset ( 0,u_0) $and$J\subset (u_0,c) $such that \begin{eqnarray} &f''>0\mbox{ in }I \,,& \label{A3}\\ &f''<0\mbox{ in }J\,.& \label{A4} \end{eqnarray} Hence \begin{eqnarray} &f'\mbox{ is increasing in }(0,u_0], \mbox{ and strictly increasing in } I\,, &\label{A5} \\ &f'\mbox{ is decreasing in }[ u_0,c) ,\mbox{ and strictly decreasing in }J\,. & \label{A6} \end{eqnarray} Furthermore, $$f'(0) <0,f'(u_0)>0,f'(c) <0. \label{A7}$$ Hence, there exist$u_0^{-}\in (0,u_0) $and$u_0^{+}\in (u_0,c) $such that \begin{eqnarray} &f'\leq 0\mbox{ in }[ 0,u_0^{-}) \cup (u_0^{+},c] & \label{A8} \\ &f'(u_0^{-}) =f'(u_0^{+}) =0 &\label{A9}\\ & f'>0\mbox{ in }(u_0^{-},u_0^{+})\, .&\label{A10} \end{eqnarray} Moreover, $$f'(b) >0\,. \label{A11}$$ So, \begin{eqnarray} &00$. In fact, if the contrary holds, it follows that $f'\leq 0$ in $(0,c)$ and hence $f$ is monotonic decreasing in $(0,c)$, and by $f( 0) =f(c) =0$ it follows that $f\equiv 0$ in $[ 0,c]$ which contradicts (\ref{Austine2}), which proves that $f'(u_0) >0$. Let us prove that $f'(0) <0$ (resp. $f'( c) <0$). If the contrary holds, that is if $f'(0) \geq 0$ (resp. $f'(c) \geq 0$), by (\ref{A5}) (resp. by (\ref{A6})) it follows that $f'(x) \geq 0$ for all $x\in (0,u_0)$ (resp. $x\in (u_0,c)$) and hence, $f$ is increasing in $(0,u_0)$ (resp. in $( u_0,c)$). Due to the fact that $f$ vanishes at $0$ (resp. at $c$), it follows that $f\geq 0$ in $(0,u_0)$ (resp. $f\leq 0$ in $(u_0,c)$), which contradicts (\ref{Austine2}). Therefore, (\ref{A7}) is proved. By making use of continuity arguments it follows that (\ref{A5}), (\ref{A6}) and (\ref{A7}) imply (\ref{A8}), (\ref {A9}) and (\ref{A10}). Let us prove (\ref{A11}). First, by (\ref{Austine2}) it follows that $f'(b) \geq 0$. If $f'(b) =0$, by (% \ref{A7})-(\ref{A10}) it follows that $b\in (0,u_0^{-}) \cup (u_0^{+},c)$. Assume that $b\in (0,u_0^{-})$ (resp. $b\in (u_0^{+},c)$). By (\ref{A8}), it follows that $f'\leq 0$ in $(0,b)$ (resp. in $(b,c)$) and therefore $f$ is decreasing in $(0,b)$ (resp. in $(b,c)$). By $f(0) =f(b) =0$ (resp. $f(b) =f(c) =0$) it follows that $f\equiv 0$ in $[ 0,b]$ (resp. in $[ b,c]$) which contradicts (% \ref{Austine2}). Therefore, (\ref{A11}) is proved, and immediate consequences are (\ref{A12}) and (\ref{A13}). \section{Preliminary lemmas}\label{sec4} Lemma \ref{Lemma1} is a technical one. The aim of the next lemma is to answer to the question : how does any solution to (\ref{P1}) look like ? We shall prove that if $u$ is a nontrivial solution to (\ref{P1}), then $u\in A_1^+\cup \tilde{A}_1^+\cup \bigcup_{k\geq 1}B^{+}(k) .$ The proof of Lemma \ref{Lemma1} is the same as that of Lemma 8 in \cite {Addou6} or Lemmas 6 and 8 in \cite{Addou4} (see also analogous lemmas in \cite{Addou1}, \cite{Addou3}). So, it is omitted. The proof of Lemma \ref{Lemmasup} is not complicated but long and tedious. So, it is postponed to the appendix. Next, we define the time map on its interval of definition, compute its limits at the boundary points of its definition domain in Lemma \ref{Lemma2}, and then study its exact variations on its entire definition domain in Lemma \ref{Lemma3}. \begin{lemma} \label{Lemma1}Assume that $f\in C({\mathbb R})$ satisfies (\ref{Austine2}) and (\ref{Austine4}). Consider the function defined in ${\mathbb R}^{\pm }$ by $$s\longmapsto G_{\pm }(\lambda ,E,s) :=E^{p}-p'\lambda F(s) , \label{Eq.1}$$ where $E,\lambda >0$ and $p>1$ are real parameters. Then \begin{description} \item[(i)] If $E>E_*(p,\lambda) :=(p'\lambda F(c)) ^{1/p}$ (resp. $E>0$), the function $G_{+}(\lambda ,E,\cdot)$ (resp. $G_{-}(\lambda,E,\cdot)$) is strictly positive in ${\mathbb R}^{+}$ (resp. in ${\mathbb R}^{-}$). \item[(ii)] If $E=E_*(\lambda)$, the function $G_{+}( \lambda ,E,\cdot)$ is strictly positive in $(0,c)$ and vanishes at $c$. \item[(iii)] If $00, \label{der} for all$E\in (0,E_*(\lambda)) $. \item[(b)]$\lim\limits_{E\to 0^{+}}s_{+}(\lambda ,E) =r$and$\lim\limits_{E\to E_*}s_{+}(\lambda ,E) =c$, where$r$is the unique zero of$F$in$(b,c) $(see, (% \ref{Klo})). \end{description} \end{description} \end{lemma} \noindent The following lemma locate all possible nontrivial solutions. \begin{lemma} \label{Lemmasup}Let$u$be a nontrivial solution of (\ref{P1}). Then $u\in A_1^+\cup \tilde{A}_1^+\cup \bigcup_{k\geq 1}B^{+}( k) ,\mbox{ and }0\leq u'(0) \leq E_*(\lambda) =(p'\lambda F(c) ) ^{1/p}.$ \end{lemma} According to this lemma, for all fixed$\lambda >0$and$p>1$, we shall look for the solutions of problem (\ref{P1}) with respect to their derivative at the origin;$u'(0) =E\in [0,E_*(\lambda) ] $. For$\lambda >0$,$p>1$and$E\in [ 0,E_*(\lambda)] $, let $X_{+}(\lambda ,E) =\{ s>0:E^{p}-p'\lambda F(\xi) >0\,,\ \forall \xi \in (0,s) \}\,.$ By Lemma \ref{Lemma1}, it follows that $X_{+}(\lambda ,E) =\left\{ \begin{array}{lcl} (0,c) & \mbox{if} & E=E_* \\ (0,s_{+}(\lambda ,E)) & \mbox{if} & 00,\forall \lambda >0,\forall E\in (0,E_*(\lambda)) ,&\label{eq11} \\ &\lim\limits_{E\to 0^{+}}r_{+}(\lambda ,E) =r,\mbox{ and } \lim\limits_{E\to E_*}r_{+}(\lambda,E) =c\,. &\label{abhu} \end{eqnarray} Define, for any p>1, \lambda >0 the time map T_{+} \thinspace by $$T_{+}(\lambda ,E) :=\int_0^{r_{+}(\lambda ,E) }(E^p-p'\lambda F(\xi)) ^{-1/p}d\xi ,\;E\in [ 0,E_*(\lambda) ] , \label{qdrj}$$ with the convention T_{+}(\lambda ,0) =+\infty (resp. T_{+}(\lambda ,E_*(\lambda)) =+\infty ) if the integral in (\ref{qdrj}) diverges. Arguing as in Guedda and Veron \cite{GueddaVeron}, it follows that \begin{itemize} \item For each \lambda >0 and E\in (0,E_*(\lambda )) , problem (\ref{P1}) admits a solution u\in A_1^+ satisfying u'(0) =E if and only if T_{+}( \lambda ,E) =1/2, and in this case the solution is unique and its sup-norm is equal to r_{+}(\lambda ,E) . \item For each \lambda >0, problem (\ref{P1}) admits a solution u\in A_1^+ satisfying u'(0) =E_*(\lambda ) if and only if T_{+}(\lambda ,E_*(\lambda) ) =1/2, and in this case the solution is unique and its sup-norm is equal to c. \item For each \lambda >0, problem (\ref{P1}) admits a solution u\in \tilde{A}_1^+ satisfying u'(0) =E_*( \lambda) if and only if T_{+}(\lambda ,E_*(\lambda )) <1/2, and in this case the solution is unique and its sup-norm is equal to c. \item For each \lambda >0 and n\in {\mathbb N}^*, problem (\ref{P1}) admits a solution u\in B_{n}^{+} if and only if nT_{+}(\lambda ,0) =1/2, and in this case the solution is unique and its sup-norm is equal to r. \item For each \lambda >0 and n\in {\mathbb N}^*, problem (\ref{P1}) admits a solution u\in \tilde{B}_{n}^{+} if and only if nT_{+}(\lambda ,0) <1/2, and in this case v is an other solution in \tilde{B}_{n}^{+} if and only if v\in \mathop{\rm Cl}(u) , and the sup-norm of each solution is equal to r. \end{itemize} A simple change of variables shows that, $$T_{+}(\lambda ,E) =r_{+}(\lambda ,E) \int_0^{1}(E^{p}-p'\lambda F(r_{+}(\lambda ,E) \xi)) ^{-1/p}d\xi , \label{A}$$ which can be written as, $$T_{+}(\lambda ,E) =(r_{+}(\lambda ,E) /E) \int_0^{1}(1-p'\lambda F(r_{+}( \lambda ,E) \xi) /E^{p}) ^{-1/p}d\xi . \label{qze}$$ Also, observe that one has from the definition of s_{+}(\lambda ,E) , (Lemma \ref{Lemma1}, Assertion {\bf (iii)}), E^{p}=\lambda p'F(r_{+}(\lambda ,E)) , so, (\ref{A}) may be written as, $$T_{+}(\lambda ,E) =(\lambda p') ^{-1/p}\int_0^{r_{+}(\lambda ,E) }(F(r_{+}( \lambda ,E)) -F(\xi)) ^{-1/p}d\xi . \label{a15}$$ For any p>1 and x\in [ r,c] let us define S_{+}( x) by \[ S_{+}(x) :=\int_0^{x}(F(x) -F(\xi )) ^{-1/p}d\xi \in [ 0,+\infty ] .$ Thus, (\ref{a15}) may be written as, $$T_{+}(\lambda ,E) =(\lambda p')^{-1/p}S_{+}(r_{+}(\lambda ,E)) . \label{a16}$$ The limits of the time map$T_{+}(\lambda ,\cdot) $are the aim of the following. \begin{lemma} \label{Lemma2} \begin{description} \item[(i)]$S_{+}(r) =+\infty $if and only if$10\,. \] Thus, there exist $\delta >0$ and $M>0$ such that $F(r) -F(x) >M(r-x) ,\mbox{ for all } x\in (r-\delta ,r) \,.$ Therefore, $\int_{r-\delta }^{r}\frac{dx}{(F(r) -F(x) ) ^{1/p}}1\,.$ On the other hand, using L'Hopital's rule twice and (\ref{A7}) it follows that $\lim_{x\to 0^{+}}\frac{F(0) -F(x) }{-x^{2}}=\frac{f'(0) }{2}<0\,.$ Thus, there exist $\varepsilon >0$, $m_{-}<0$ and $M_{-}<0$ such that $m_{-}\leq \frac{F(0) -F(x) }{-x^{2}}\leq M_{-}, \mbox{ for all }x\in (0,\varepsilon)\, .$ Therefore, $(-m_{-}) ^{-1/p}\int_0^{\varepsilon }\frac{dx}{x^{2/p}}\leq \int_0^{\varepsilon }\frac{dx}{(F(0) -F(x) ) ^{1/p}}\leq (-M_{-}) ^{-1/p}\int_0^{\varepsilon }\frac{dx}{x^{2/p}}\,.$ The first part of Assertion {\bf (i)} follows from $F(r) =F(0) =0$ and the well-known fact $\int_0^{\varepsilon }\frac{dx}{x^{2/p}}<+\infty \mbox{ if and only if } p>2.$ The second part may be proved similarly. In fact, using L'Hopital's rule twice and (\ref{A7}) it follows that $\lim_{x\to c^{-}}\frac{F(c) -F(x) }{( c-x) ^{2}}=-\frac{f'(c) }{2}>0.$ Thus $M_{+}^{-1/p}\int_{c-\varepsilon }^{c}\frac{dx}{(c-x) ^{2/p}}\leq \int_{c-\varepsilon }^{c}\frac{dx}{(F(c) -F(x) ) ^{1/p}}\leq m_{+}^{-1/p}\int_{c-\varepsilon }^{c}\frac{dx}{( c-x) ^{2/p}}$ for some strictly positive constants $M_{+},m_{+}$ and $\varepsilon$. Therefore, the second assertion of {\bf (i)} follows from the well-known fact $\int_{c-\varepsilon }^{c}\frac{dx}{(c-x) ^{2/p}}<+\infty \mbox{ if and only if }p>2.$ \paragraph{Proof of (ii).} The value of the limits follows by passing to the limit in (\ref{a16}) as $E$ tends to $0$ and $E_*$ respectively. Then Lemma \ref{Lemma2} is proved. \hfill$\diamondsuit$\medskip To study the exact number of solutions of (\ref{P1}) we need to know the exact variations of the time map $T_{+}(\lambda ,\cdot)$ over all its definition domain $(0,E_*(\lambda) )$. These variations are the aim of the following, \begin{lemma} \label{Lemma3} If $10$ the time map $T_{+}(\lambda ,\cdot)$ admits a unique critical point; a minimum. If $p>2$, for all $\lambda >0$ {\em either} the time map $T_{+}( \lambda ,\cdot)$ is strictly increasing {\em or} admits a unique critical point; a minimum in $(0,E_*(\lambda))$. \end{lemma} \paragraph{Proof} By (\ref{a16}), recall that for all $\lambda >0$ and $E\in ( 0,E_*(\lambda))$. $T_{+}(\lambda ,E) =(\lambda p')^{-1/p}S_{+}(r_{+}(\lambda ,E)) .$ On the other hand, by (\ref{eq11}) and (\ref{abhu}), for each fixed $\lambda >0$, the function $E\mapsto r_{+}(\lambda ,E)$ is an increasing $C^{1}-$diffeomorphism from $(0,E_*(\lambda ))$ onto $(r,c)$, where $r$ is the unique zero of $F$ in $(b,c)$. A differentiation yields $\frac{\partial T_{+}}{\partial E}(\lambda ,E) =(\lambda p') ^{-1/p}\times \frac{\partial r_{+}}{\partial E}( \lambda ,E) \times S_{+}'(r_{+}(\lambda,E)) .$ Thus, to study the variations of $T_{+}(\lambda ,\cdot)$ in $(0,E_*(\lambda))$ it suffices to study those of $S_{+}(\cdot)$ in $(r,c)$. One has $S_{+}(\rho) =\int_0^\rho \{ F(\rho) -F(u) \} ^{-1/p}du,\;\rho \in (r,c)$ and $$S_{+}'(\rho) =\frac 1{p\rho }\int_0^\rho \frac{% H_p(\rho) -H_p(u) }{\{ F(\rho) -F(u) \} ^{(p+1)/p}}du,\;\rho \in (r,c)\,, \label{Dop}$$ where $$H_p(u) =pF(u) -uf(u) ,\mbox{ for all } u\in [ 0,c] \mbox{ and }p>1\,. \label{B1}$$ To study the sign of the derivative $S_{+}'(\cdot)$ we need to study that of expression $H_p(\rho) -H_p(u) \;\mbox{ for all }01 \label{B2} and $$H_p''(u) =(p-2) f'( u) -uf''(u) ,\mbox{ for all }u\in [0,c] \mbox{ and }p>1\,. \label{B3}$$ By (\ref{Austine1}) it follows that $$H_p(0) =H_p'(0) =0, \mbox{ for all }p>1\,, \label{B4}$$ and by (\ref{Austine1}) and (\ref{Austine3}), it follows that $$H_p(c) >0, \mbox{ for all }p>1, \label{B5}$$ and by (\ref{A7}), $$H_p'(c) >0, \mbox{ for all }p>1. \label{B6}$$ Now, let us look closely to the special case where p=2. By (\ref{A1}), it follows that \[ H_2''(u) \leq 0,\mbox{ for all }u\in (0,u_0] \,,$ and by (\ref{B4}) it follows that $$H_2'(u) \leq 0,\mbox{ for all }u\in [0,u_0]\, . \label{krs}$$ By (\ref{A3}) it follows that there exists a unique $\alpha \in [0,u_0)$ such that \begin{eqnarray} &H_2(u) =H_2'(u) =0\mbox{ for all } u\in [ 0,\alpha ] & \label{B7} \\ &H_2(u) < 0\mbox{ and }H_2'(u) <0 \mbox{ for all } u\in (\alpha ,u_0]\,.& \nonumber \end{eqnarray} On the other hand, by (\ref{A2}) it follows that $H_2''(u) \geq 0$, for all $u\in [u_0,c)$, and by (\ref{B6}) and (\ref{B7}) there exist $\beta$ and $\gamma$ in $(u_0,c)$ such that $$\displaylines{ u_0<\beta \leq \gamma 0,\mbox{ for all }u\in (\gamma ,c]\, . \cr }$$ Therefore, regarding (\ref{B4}) and (\ref{krs}), there exists a unique $\delta \in (\gamma ,c)$ such that $$\displaylines{ H_2(u) <0,\mbox{ for all }u\in [ u_0,\beta ] \cr H_2(u) =H_2(\beta) <0,\mbox{ for all }u\in [ \beta ,\gamma ] \cr H_2(u) <0,\mbox{ for all }u\in [ \gamma ,\delta) \mbox{ and }H_2(\delta) =0 \cr H_2(u) >0,\mbox{ for all }u\in (\delta ,c]\,. \cr }$$ Thus, for all fixed $\rho \in (0,\alpha ]$, $$H_2(\rho) -H_2(u) =0,\mbox{ for all }u\in(0,\rho) , \label{S0}$$ and for all fixed $\rho \in (\alpha ,\gamma ]$ $$H_2(\rho) -H_2(u) <0\mbox{ for all }u\in (0,\min (\rho ,\beta)), \label{S1}$$ and for all fixed $\rho \in [ \delta ,c)$, $$H_2(\rho) -H_2(u) >0\mbox{ for all }u\in (0,\rho) . \label{S2}$$ Notice that to obtain (\ref{S0})-(\ref{S2}) the starting conditions were conditions (\ref{A1}), (\ref{A2}) and (\ref{A3}). In contrast, if $p\neq 2$, $H_p''$ changes sign in $(0,u_0)$ since by (\ref{A1}), $f''$ is of constant sign in $(0,u_0)$ and by (\ref{A8}), (\ref{A9}) and (\ref{A10}), $f'$ changes sign in $(0,u_0)$. This leads us to consider the additional conditions (\ref{E1}) and (\ref{E2}) for all $p>1$, and therefore $$H_p''(u) \leq 0,\mbox{ for all }u\in ( 0,u_0] ,\mbox{ and }p>1 \label{B88}$$ with strict inequality in an open interval $I_p\subset (0,u_0)$, and $$H_p''(u) \geq 0,\mbox{ for all }u\in [u_0,c) \mbox{ and }p>1\,. \label{B9}$$ Let us emphasize that (\ref{E1}) and (\ref{E2}) are automatically satisfied if $p=2$. In fact they are reduced to (\ref{A1})-(\ref{A3}). So, (\ref{E1}) and (\ref{E2}) do not consist as additional conditions for the special case where $p=2$. By (\ref{B4}) and (\ref{B88}) it follows that there exists a unique $\alpha _p\in [ 0,u_0)$ such that \begin{eqnarray} &H_p(u) =H_p'(u) =0,\mbox{ for all }u\in [ 0,\alpha_p ]\,, &\label{M0}\\ &H_p(u) <0,\ H_p'(u) <0,\mbox{ for all } u\in (\alpha_p ,u_0]\, .&\label{M05} \end{eqnarray} By (\ref{B6}) and (\ref{B9}) it follows that for all $p>1$, there exist $\beta_p$ and $\gamma_p$ in $(u_0,c)$ such that \begin{eqnarray} &u_0<\beta_p \leq \gamma_p 0,\mbox{ for all }u\in (\gamma_p,c] \,. & \label{M3} \end{eqnarray} Therefore, there exists a unique $\delta_p \in (\gamma_p,c)$ such that \begin{eqnarray} &H_p(u) <0,\mbox{ for all }u\in [ u_0,\beta_p] & \label{M4}\\ &H_p(u) =H_p(\beta_p) <0,\mbox{ for all } u\in [ \beta_p ,\gamma_p ] &\label{M5} \\ &H_p(u) <0,\mbox{ for all }u\in [ \gamma_p ,\delta _p) \mbox{ and }H_p(\delta_p) =0 &\label{M6} \\ &H_p(u) >0,\mbox{ for all }u\in (\delta_p,c] . &\label{M7}\end{eqnarray} This implies that: for all $p>1$ and all fixed $\rho \in (0,\alpha_p]$, $$H_p(\rho) -H_p(u) =0,\mbox{ for all }u\in (0,\rho) , \label{C0}$$ and for all fixed $\rho \in (\alpha_p,\gamma_p]$, $$H_p(\rho) -H_p(u) <0,\mbox{ for all }u\in (0,\min (\rho ,\beta_p)) \,, \label{C1}$$ and for all fixed $\rho \in [ \delta_p,c)$, $$H_p(\rho) -H_p(u) >0,\mbox{ for all }u\in(0,\rho) \,. \label{C2}$$ Notice that $H_p(r) =pF(r) -rf(r) =-rf(r)<0\,.$ Thus, by (\ref{M0}), (\ref{M05}), (\ref{M6}) and (\ref{M7}) it follows that \alpha_p1. \label{S3} By (\ref{C2}) it follows that $$S_{+}'(\rho) >0\mbox{ for all }\rho \in [ \delta_p,c) \subset (r,c) . \label{S4}$$ It remains to study the variations of $S_{+}(\cdot)$ on the interval $(r,\delta_p)$. Notice that one has to distinguish two cases \begin{eqnarray} &\alpha_p0,\mbox{ for all }\rho \in ( \gamma_p ,\delta_p) ,\mbox{ (resp. }\rho \in (r,\delta_p))\,, \] which implies that $S_{+}$ is convex in a neighborhood of each of its critical points lying in $(\gamma_p ,\delta_p)$ (resp. in $(r,\delta_p)$). Thus, $S_{+}'$ vanishes at most once in $(\gamma_p ,\delta_p)$ (resp. in $( r,\delta_p)$) for all $p>1$. Therefore; regarding (\ref{S4}), it follows that $S_{+}$ is either strictly increasing in $(r,c)$ or strictly decreasing in $(r,s_p)$ for some $s_p\in (r,\delta_p)$ and then strictly increasing in $( s_p,c)$. For the special case $10$, the time map $T_{+}(\lambda ,\cdot )$ admits a unique critical point which is a minimum in $( 0,E_*(\lambda))$ and satisfies $\lim_{E\to 0^{+}}T_{+}(\lambda ,E) =\lim_{E\to E_*}T_{+}(\lambda ,E) =+\infty .$ Also, by Lemma \ref{Lemma2} $\lim_{\rho \to r^{+}}S_{+}(\rho) =\lim_{\rho \to c^{-}}S_{+}(\rho) =+\infty ,$ and by the proof of Lemma \ref{Lemma3}, $S_{+}$ admits a unique critical point, a minimum in $(r,c)$ at $r_*$, say. Therefore, based upon the fact that for all $\lambda >0$, $r_{+}(\lambda ,\cdot )$ is strictly increasing from $(0,E_*(\lambda ))$ onto $(r,c)$, it follows that there exists a unique $\tilde E=\tilde E(\lambda) \in (0,E_*( \lambda))$ such that $r_*=r_{+}(\lambda ,\tilde E(\lambda))$. Thus, by (\ref{a16}), for all $E\in (0,E_*(\lambda))$ \begin{eqnarray*} T_{+}(\lambda ,\tilde E(\lambda)) &=&( p'\lambda) ^{-1/p}S_{+}(r_*) \\ &\leq &(p'\lambda) ^{-1/p}S_{+}(r_{+}( \lambda ,E)) =T_{+}(\lambda ,E) , \end{eqnarray*} hence, $T_{+}(\lambda ,\cdot)$ attains its unique global minimum value at $\tilde E(\lambda) \in (0,E_*( \lambda))$. It follows that \begin{itemize} \item If $(p'\lambda) ^{-1/p}S_{+}(r_*) >(1/2)$, the equation $T_{+}(\lambda ,E) =(1/2)$ in the variable $E\in (0,E_*(\lambda))$ admits no solution. \item If $(p'\lambda) ^{-1/p}S_{+}(r_*) =(1/2)$, the equation $T_{+}(\lambda ,E) =(1/2)$ in the variable $E\in (0,E_*(\lambda))$ admits a unique solution; $\tilde{E}(\lambda)$. \item If $(p'\lambda) ^{-1/p}S_{+}(r_*) <(1/2)$, the equation $T_{+}(\lambda ,E) =(1/2)$ in the variable $E\in (0,E_*(\lambda))$ admits exactly two solutions. \end{itemize} Hence, Theorem \ref{wmdjkgfh} is proved if we let $\lambda_0=( 2S_{+}(r_*)) ^p/p'$. \hfill$\diamondsuit$\medskip Now, assume that $p>2$ and let us prove Theorem \ref{wmdjkgfh2}. By the assumption $\nu =(2S_{+}(c)) ^p/p'<(2S_{+}( r)) ^p/p'=\lambda_1,$ it follows that, for all fixed $\lambda >0,$$\lim_{E\to E_*}T_{+}(\lambda ,E) <\lim_{E\to 0}T_{+}(\lambda ,E) .$ According to Lemma \ref{Lemma3}, it follows that $T_{+}(\lambda ,\cdot)$ admits a unique critical point; a minimum. Thus, as in the case where $11/2$, the equation $T_{+}(\lambda ,E) =1/2$ in the variable $E\in (0,E_*(\lambda))$ admits no solution. Thus, if $0<\lambda <\mu :=(2S_{+}(r_*)) ^{p}/p'$, problem (\ref{P1}) admits no solution in $J_1( \lambda)$. \item If $(p'\lambda) ^{-1/p}S_{+}(r_*) =1/2$, the equation $T_{+}(\lambda ,E) =1/2$ in the variable $E\in (0,E_*(\lambda))$ admits a unique solution; $\tilde{E}(\lambda)$. Thus, if $\lambda =\mu$, problem (\ref{P1}) admits a unique solution $v_\lambda$ in $J_1( \lambda)$, and this solution belongs to $A_1^+$. \item If $(p'\lambda) ^{-1/p}S_{+}(r_*) <1/2<(p'\lambda) ^{-1/p}S_{+}(c)$, the equation $T_{+}(\lambda ,E) =1/2$ in the variable $E\in ( 0,E_*(\lambda))$ admits exactly two solutions. Thus, if $\mu <\lambda <\nu$, problem (\ref{P1}) admits exactly two solutions $v_\lambda$, $w_\lambda$ in $J_1(\lambda) ,$ and they belong to $A_1^+$. \item If $(p'\lambda) ^{-1/p}S_{+}(c) =1/2$, the equation $T_{+}(\lambda ,E) =1/2$ in the variable $E\in (0,E_*(\lambda))$ admits a unique solution; $E_1<\tilde{E}(\lambda)$. Thus, if $\lambda =\nu$, problem (\ref{P1}) admits a unique solution $v_\lambda$ in $J_1( \lambda)$, and it belongs to $A_1^+$. \item If $(p'\lambda) ^{-1/p}S_{+}(c) <1/2<(p'\lambda) ^{-1/p}S_{+}(r)$, the equation $T_{+}(\lambda ,E) =1/2$ in the variable $E\in ( 0,E_*(\lambda))$ admits a unique solution; $E_1<% \tilde{E}(\lambda)$. Thus, if $\nu <\lambda <\lambda_1$, problem (\ref{P1}) admits a unique solution $v_\lambda$ in $J_1( \lambda)$, and it belongs to $A_1^+$. \item If $(p'\lambda) ^{-1/p}S_{+}(r) \geq 1/2$, the equation $T_{+}(\lambda ,E) =1/2$ in the variable $E\in (0,E_*(\lambda))$ admits no solution. Thus, if $\lambda \geq \lambda_1$, problem (\ref{P1}) admits no solution in $J_1(\lambda)$. \end{itemize} Now, let us look for the nontrivial solutions in $J_2(\lambda)$. For all $\lambda >0$, $T_{+}(\lambda ,E_*(\lambda)) =1/2\mbox{ if and only if }(p'\lambda) ^{-1/p}S_{+}(c)=1/2\,.$ Thus, problem (\ref{P1}) admits a solution in $J_2(\lambda) \cap A_1^{+}$ if and only if $\lambda =\nu$, and in this case the solution is unique. For all $\lambda >0$, $T_{+}(\lambda ,E_*(\lambda)) <1/2\mbox{ if and only if }(p'\lambda) ^{-1/p}S_{+}(c)<1/2\,.$ Thus, problem (\ref{P1}) admits a solution in $J_2(\lambda) \cap \tilde A_1^{+}$ if and only if $\lambda >\nu$, and in this case the solution is unique. Now let us look for the nontrivial solutions in $J_0$. Let $n\in {\mathbb N}^*$. For all $\lambda >0$, $nT_{+}(\lambda ,0) =1/2\mbox{ if and only if }n( p'\lambda) ^{-1/p}S_{+}(r) =1/2\,.$ Thus, problem (\ref{P1}) admits a solution in $J_0\cap B_n^{+}$ if and only if $\lambda =\lambda_n$, and in this case the solution is unique. For all $\lambda >0$, $nT_{+}(\lambda ,0) <1/2\mbox{ if and only if }n( p'\lambda) ^{-1/p}S_{+}(r) <1/2\,.$ Thus, problem (\ref{P1}) admits a solution $u_{\lambda ,n}$ in $J_0\cap \tilde B_n^{+}$ if and only if $\lambda >\lambda_n$, and in this case each function $u$ in $\mathop{\rm Cl}(u_{\lambda ,n})$ is a solution to (\ref {P1}). Therefore, Theorem \ref{wmdjkgfh2} is proved. \hfill$\diamondsuit$\smallskip To prove Theorems \ref{wmdjkgfh3}, \ref{wmdjkgfh4} and \ref{wmdjkgfh5}, the same reasoning works. However, for Theorems \ref{wmdjkgfh4} and \ref {wmdjkgfh5}, one has $\lim_{E\to 0}T_{+}(\lambda ,E) <\lim_{E\to E_*}T_{+}(\lambda ,E) .$ Thus, according to Lemma \ref{Lemma3}, $T_{+}(\lambda ,\cdot)$ may have a unique critical point; a minimum, or may be strictly increasing. These two alternatives lead for Theorem \ref{wmdjkgfh4} to the possibilities {\bf A} and {\bf B}, and for Theorem \ref{wmdjkgfh5} to the possibilities {\bf C} and {\bf D}. \hfill$\diamondsuit$ \section{Open questions}\label{sec6} \begin{enumerate} \item For $p>2$, Theorems \ref{wmdjkgfh4} and \ref{wmdjkgfh5} provide alternative results. Do there exist some sufficient conditions ensuring that possibility {\bf A} (resp. {\bf B}, {\bf C}, {\bf D}) holds? Can one find an example of $f$ such that possibility {\bf A} (resp. {\bf B}, {\bf C}, {\bf D}) holds? Or maybe among the two alternatives Theorem \ref{wmdjkgfh4} (resp. Theorem \ref{wmdjkgfh5}) provides, the same one holds always? \item In the literature, there are some examples of nonlinearities $g(\lambda ,u)$ such that the structure of the solution set of (\ref{P1}) does change when $p$ varies (as that studied in this paper) but in others it does not change; for example as that studied by Addou and Benmeza\"{\i } \cite{Addou3} for $g(\lambda ,u)=\lambda \exp (u)$. Thus, we ask the question of providing sufficient or necessary conditions on $g$ insuring that the structure of (at least) the set of (positive) solutions of problem (\ref{P1}) does not change when $p$ varies. \end{enumerate} \section{\bf Appendix} In this section, we prove Lemma \ref{Lemmasup} which is a consequence of the following two lemmas. \begin{lemma} \label{Lemma45} Let $u$ be a nontrivial solution of (\ref{P1}). Then $u\geq 0\mbox{ in }[ 0,1] \mbox{ and }0\leq u'(0) \leq E_*(\lambda) =(p'\lambda F(c)) ^{1/p}.$ Moreover, \begin{itemize} \item If $0\leq u'(0) F(x)$ for all $x\geq 0$ and $x\neq c$. Lemma \ref{Lemma45} is proved. \hfill$\diamondsuit$ \paragraph{Proof of Lemma \ref{Lemma451}.} Each assertion is a consequence of several steps. If $u$ is a nontrivial solution of (\ref{P1}) and satisfying $u'(0) \in (0,E_*(\lambda))$, then Assertion {\bf (a)} is an immediate consequence of the following steps: \begin{description} \item[(a1)] For all $x_*\in (0,1)$, $u'( x_*) =0$ implies $u(x_*) =s_{+}(u'(0)) \in (r,c)$. \item[(a2)] For all $x_1,x_2\in (0,1)$, $x_10$ on $(0,x_1)$, $u'\equiv 0$ on $[ x_1,x_2]$, and $u'<0$ on $(x_2,1)$. \item[(b5)] The solution $u$ is symmetric with respect to $1/2$. \end{description} If $u$ is a nontrivial solution of (\ref{P1}) and satisfying $u'(0) =0$, then Assertion {\bf (c)} is an immediate consequence of the following steps: \begin{description} \item[(c1)] For all $x_*\in (0,1)$, $u'(x_*) =0$ implies $u(x_*) =0$ or $u(x_*) =r$. \item[(c2)] Each local maxima of $u$ is a strict one. \item[(c3)] There are finitely many critical points at which $u$ attains its maximum value; $r$. \item[(c4)] If $u$ attains its maximum value at the $n$ points of the strictly increasing sequence $(x_{i})_{1\leq i\leq n}$ then for all $i\in \{ 1,\cdots ,n\}$ there exists $a_{i}\leq b_{i}$ in $[ 0,1]$ such that \end{description} \begin{eqnarray} &x_{i}0\mbox{ on }(b_{i},x_{i+1}) \mbox{ for all }i\in \{ 0,\cdots ,n-1\},& \label{f4}\\ &u'<0\mbox{ on }(x_{i},a_{i}) \mbox{ for all }i\in \{ 1,\cdots ,n\},& \label{f5} \\ &b_{i}+a_{i+1}=2x_{i+1},\mbox{ for all }i\in \{ 0,\cdots,n-1\}, &\label{f6} \\ &u|_{[ b_{i},a_{i+1}] },\mbox{ is symmetric with respect to $x_{i+1}$ for $i\in \{ 0,\cdots ,n-1\}$}, & \label{f77} \\ &u|_{[ b_{i},a_{i+1}] },\mbox{ is a translation of $u|_{[ b_0,a_1]}$, for all $i\in \{ 0,\cdots ,n-1\}$}\,.& \label{f88} \end{eqnarray} \noindent The proofs of all these steps are simple and therefore omitted. 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Therefore, the diagram in Fig. 3 does not occur. That is, for $p>2$ the upper branch has always a turning point (which is unique). The alternatives in Theorems 2.4 and 2.5 come from the alternative situation on the time map $T_{+}(\lambda, \cdot )$. Indeed, Lemma 4 states that for all $p>2$ and $\lambda >0$, \emph{either} the time map $T_{+}(\lambda , \cdot )$ is strictly increasing \emph{or} it admits a unique critical point; a minimum in $(0, E_{*}(\lambda ))$. We shall prove that for all $p>2$ and all $\lambda >0$, $T_{+}(\lambda, \cdot )$ admits at least one minimum in $(0, E_{*}(\lambda))$. Therefore, it admits a unique critical point for all $p>1$ (according to the first part of Lemma 4) and it is never strictly increasing on $(0, E_{*}(\lambda ))$. As a consequence, for all $p>2$, Possibility B of Theorem 2.4 and Possibility D of Theorem 2.5 do not occur. To prove this statement, it suffices to show that: \paragraph{Lemma.} $S_{+}'(r)=-\infty$ and $S_{+}'(c)=+\infty$ for all $p>2$. \paragraph{Proof.} Since $F(r)=F(0)=0$, the integral in the expression $S_{+}'(r)=\frac{1}{pr}\int_0^{r}\frac{H_p(r)-H_p(u)} {(F(r)-F(u))^{1+\frac{1}{p}}}\,du$ has two singularities: one at $0$ and one at $r$. So, we shall write $S_{+}'(r)=\frac{1}{pr}(I_0+I_{r}),$ where $I_0=\int_0^{r/2}\frac{H_p(r)-H_p(u)}{(F(r)-F(u))^{1+\frac{1}{p}}}\,du\,, \quad I_{r}=\int_{r/2}^{r}\frac{H_p(r)-H_p(u)}{(F(r)-F(u))^{1+\frac{1}{p}}}\,du\,.$ Next we prove that $I_0=-\infty$ and $I_{r}\in [-\infty , +\infty )$, so that $S_{+}'(r)=-\infty$. \paragraph{Proof of $I_0=-\infty$.} Using l'Hopital's rule twice it follows that $\lim_{u\rightarrow 0}\frac{F(u)}{u^{2}}=\frac{f'(0)}{2}<0\,.$ This last inequality follows from (15). Therefore, $\frac{H_p(r)-H_p(u)}{(F(r)-F(u))^{1+\frac{1}{p}}} =\frac{H_p(r)-H_p(u)}{(-\frac{F(u)}{u^{2}})^{1+\frac{1}{p}}}\cdot \frac{1}{u^{2(1+\frac{1}{p})}} \approx \frac{H_p(r)}{(-\frac{f'(0)}{2})^{1+\frac{1}{p}}}\cdot \frac{1}{u^{2(1+\frac{1}{p})}}$ for all $u \in (0,\varepsilon )$ and for some $\varepsilon>0$. Since $2(1+\frac{1}{p})>1$ and $\frac{H_p(r)}{(-\frac{f'(0)}{2})^{1+\frac{1}{p}}} =\frac{-rf(r)}{(-\frac{f'(0)}{2})^{1+\frac{1}{p}}}<0$ it follows that $\int_0^{r/2}\frac{H_p(r)}{(-\frac{f'(0)}{2})^{1+\frac{1}{p}}} \cdot \frac{1}{u^{2(1+\frac{1}{p})}}du=-\infty$ which proves that $I_0=-\infty$. \paragraph{Proof of $I_{r}\in [-\infty , +\infty )$.} We distinguish two cases. \noindent Case $H_p'(r)\neq 0$. \quad Since $\lim_{u \rightarrow r}\frac{H_p(r)-H_p(u)}{r-u}=H_p' (r)\neq 0$ and $\lim_{u \rightarrow r}\frac{F(r)-F(u)}{r-u}=f(r)>0\,,$ it follows that $\frac{H_p(r)-H_p(u)}{(F(r)-F(u))^{1+\frac{1}{p}}} =\frac{(\frac{H_p(r)-H_p(u)}{r-u})}{(\frac{F(r)-F(u)}{r-u})^{1+\frac{1}{p}}} \cdot\frac{1}{(r-u)^{\frac{1}{p}}} \approx \frac{H_p'(r)}{(f(r))^{1+\frac{1}{p}}}\cdot \frac{1}{(r-u)^{\frac{1}{p}}}$ for all $u \in (r-\varepsilon, r)$ and for some $\varepsilon>0$. Since $\frac{1}{p}<1$ and $\frac{H_p'(r)}{(f(r))^{1+(1/p)}}\neq 0$, $\int_{r/2}^{r}\frac{H_p'(r)}{(f(r))^{1+\frac{1}{p}}}\cdot \frac{1% }{(r-u)^{\frac{1}{p}}}du\in (-\infty , +\infty )$ which proves that $I_{r}\in [-\infty , +\infty )$. \medskip \noindent Case $H_p'(r)=0$. \quad From equations (46)-(50), it follows that $\beta _p\leq r\leq \gamma _p$. First assume that $\beta _p \neq r$. Then in a left neighborhood of $r$ the integrand function is identically zero. That is, there exists $\varepsilon >0$ such that $\frac{H_p(r)-H_p(u)}{(F(r)-F(u))^{1+\frac{1}{p}}}=0$ for all $u \in (r-\varepsilon , r)$. Therefore, $\int_{r-\varepsilon}^{r}\frac{H_p(r)-H_p(u)}{(F(r)-F(u))^{1 +\frac{1}{p}} }\,du=0\,.$ So, the integral $I_{r}$ presents no singularity at $r$ and $I_{r}\in (-\infty , +\infty )$. \medskip Now assume that $r=\beta _p$. Then, by (46)-(48) $H_p(r)-H_p(u)\leq 0$ \ for all $u \in (0, r)$. Therefore, $\int_{r/2}^{r}\frac{H_p(r)-H_p(u)}{(F(r)-F(u))^{1+\frac{1}{p}}}\,du\in [-\infty , 0],$ which proves that $I_{r}\in [-\infty , +\infty )$. Therefore, $S'(r)=-\infty$. \medskip Now, we shall prove that $S'(c)=+\infty$. Since $\lim_{u \rightarrow c} \frac{H_p(c)-H_p(u)}{c-u} = H_p'(c) =-cf'(c)>0\,,$ because of (15), and $\lim_{u \rightarrow c}\frac{F(c)-F(u)}{(c-u)^{2}}=-\frac{f'(c)}{2}>0\,,$ it follows that $\frac{H_p(c)-H_p(u)}{(F(c)-F(u))^{1+\frac{1}{p}}} =\frac{(\frac{H_p(c)-H_p(u)}{c-u})} {(\frac{F(c)-F(u)}{(c-u)^{2}})^{1+\frac{1}{p}}} \cdot \frac{1}{(c-u)^{1+\frac{2}{p}}} \approx \frac{(-cf'(c))}{(-\frac{f'(c)}{2})^{1+\frac{1}{p}}} \cdot \frac{1}{(c-u)^{1+\frac{2}{p}}}$ for all $u \in (c-\varepsilon , c)$ and for some $\varepsilon >0$. Since $1+\frac{2}{p}>1$ and $\frac{(-cf'(c))}{(-\frac{f'(c)}{2})^{1+\frac{1}{p}}}>0$, it follows that $\int_0^{c}\frac{(-cf'(c))}{(-\frac{f'(c)}{2})^{1+\frac{1}{p}}} \cdot \frac{1}{(c-u)^{1+\frac{2}{p}}}du=+\infty,$ which proves that $S'(c)=+\infty$. Therefore, the present proof is complete, and the claim of the addendum is proved. \hfill$\diamondsuit$ \end{document}