\documentclass[twoside]{article} \usepackage{amssymb} % font used for R in Real numbers \usepackage{amsmath} \pagestyle{myheadings} \markboth{\hfil One-dimensional elliptic equation \hfil EJDE--2000/50} {EJDE--2000/50\hfil Justino S\'anchez \& Pedro Ubilla \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol.~{\bf 2000}(2000), No.~50, pp.~1--9. \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} \\ % One-dimensional elliptic equation with concave and convex nonlinearities \thanks{ {\em Mathematics Subject Classifications:} 34B15. \hfil\break\indent {\em Key words:} m-Laplacian, concave-convex nonlinearities, exactness results, time-maps. \hfil\break\indent \copyright 2000 Southwest Texas State University and University of North Texas. \hfil\break\indent Submitted February 1, 2000. Published June 25, 2000. \hfil\break\indent (J.S.) Partially supported by FONDECYT Grant \# 1990183. \hfil\break\indent (P.U.) Partially supported by FONDECYT Grants \# 1990183 and \# 1980812, \hfil\break\indent and by a DICYT-USACH grant.} } \date{} % \author{ Justino S\'anchez \& Pedro Ubilla } \maketitle \begin{abstract} We establish the exact number of positive solutions for the boundary-value problem $$\displaylines{ -(|u'|^{m-2} u')'=\lambda u^q + u^p\quad \mbox{in }(0,1)\cr u(0)= u(1)=0\,, }$$ where $0\leq q < m- 1 < p$ and $\lambda$ is positive. \end{abstract} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \section{Introduction} We establish the exact number of positive solutions for the boundary-value problem \begin{equation} \label{P} \gathered-(|u'|^{m-2} u')'=\lambda u^q + u^p\quad \mbox{in }(0,1)\\ u(0)=u(1)=0\,, \endgathered \end{equation} where $0\leq q < m- 1 < p$ and $\lambda>0$. Problem (\ref{P}) with $m=2$ was suggested by Ambrosetti, Brezis, and Cerami in \cite{ambrosetti}. Indeed, the equation $$\gathered -\Delta u=\lambda u^q+u^p \quad\mbox{in }\Omega \\ u= 0 \quad \mbox{on }\partial\Omega \,, \endgathered $$ with $00$ such that: if $\lambda\in (0,\Lambda)$, then the latter problem has at least two positive solutions; if $\lambda=\Lambda$, then it has at least one positive solution; finally, if $\lambda>\Lambda$, then it has no positive solution. Using shooting methods, the existence and multiplicity of solutions for the quasi-linear problem \begin{equation} \label{AB} \gathered -(|u'|^{p-2}u')'=|u|^p-\mu\quad \mbox{in }(0,1)\\ u(0)=u(1)=0\\ \endgathered \end{equation} was studied recently by Addou and Benmeza\"{\i} \cite{addou}. For $\mu>0$, they determine a lower bound on the number of solutions of the problem (\ref{AB}), and their nodal properties. In the case $\mu\leq 0$, they also obtained the exact number of solutions. We note that if $\lambda=-\mu$, $m=p$ and $q=0$, then the equation studied in \cite{addou}, with $\mu<0$, turns out to be a particular case of our equation (\ref{P}). This fact inspired us to apply the techniques developed by Addou and Benmeza\"{\i} \cite{addou}. The strategy is to localize the critical points of a time mapping on a bounded interval $J$. We point out that this problem was simultaneously and independently studied by I. Addou and A. Benmeza\"{\i}. We remark that the novelty of our result is that we obtain the exact number of solutions for an equation with concave-convex nonlinearity, as well as their asymptotic behavior for small parameter $\lambda$. Finally, we should note that in \cite{ambrosetti,garcia,ubilla}, the problem of determining the exact number of solution is not solved. \section{Results and Methods Employed} We first state our main result. \begin{theorem} There exists a number $\lambda^*> 0$ such that: \begin{itemize} \item[(a)] If $\lambda > \lambda^* $, then (\ref{P}) has no solutions. \item[(b)]If $\lambda = \lambda^* $, then (\ref{P}) has exactly one positive solution. \item[(c)] If $0< \lambda < \lambda^* $, then (\ref{P}) has exactly two positive solutions, $u_\lambda$ and $v_\lambda$. \item[(d)] The solutions $u_\lambda$ and $v_\lambda$ satisfy $\lim_{\lambda\to 0} \|u_\lambda\|_\infty=0$ and $$\big(\frac{2^m}{m'}I_p^m(p+1)\big)^{1/(p-m+1)}\leq\lim_{\lambda\to 0}\|v_\lambda\|_\infty\leq\big(\frac{2^m}{m'}I_q^m(p+1) \big)^{1/(p-m+1)}\,,$$ where $I_r=\int_0^1(1-t^{r+1})^{-1/m}\,dt \,.$ \end{itemize} \end{theorem} \noindent In this article we use a shooting method. More precisely, we study the ordinary differential equation \begin{equation} \label{PE} \gathered -(|u'|^{m-2} u')'= \lambda|u|^{q-1}u+|u|^{p-1}u\quad\mbox{in }(0,1) \\ u(0)=0,\quad u'(0)=E>0\,. \endgathered\end{equation} The solution to this problem is $4T$-periodic function, with $$ T= T(\lambda, S) = (m')^{-1/m} S^{ \frac{m-1-q}{m}} \int^1_0 \big(S^{p-q} \frac{(1-t^{p+1})}{p+1} + \lambda \frac{(1-t^{q+1})}{q+1} \big)^{-1/m}dt \,,$$ where $S= S(\lambda,E)$ is the first zero of the function $E^m - m' G(\lambda,\cdot)$, $$G(\lambda,u)= \frac{\lambda u^{q+1}}{q+1} + \frac{u^{p+1}}{p+1}\,,$$ and $m'$ is defined by $1/m + 1/m' =1$. See, e.g. \cite{garcia,ubilla}. The solution $u$ to Problem (\ref{PE}) satisfies the following conditions \begin{itemize} \item $u(2kT) = 0$, with $k \in {\mathbb Z}$. \item $u(x) > 0$, for $x\in (0,2T) $ and $u(x) < 0$, for $x \in (2T,4T)$. \item Every hump of $u $ is symmetrical about the center of the interval of its definition, where we call hump of $u$ its restriction to the open interval $I=(x_1,x_2)$, with $x_1$ and $x_2$ two consecutive zeros of $u$. \item Every positive (resp. negative) hump of $u$ may be obtained by translating the first positive (resp. negative) hump. \item The derivative of each hump of $u$ vanishes once and only once. \end{itemize} \noindent Thus, when $T=1/2 $, we obtain positive solutions of Problem (\ref{P}). In order to prove Theorem 1, we need the following technical lemma. \begin{lemma} \begin{itemize} \item[(a)] $S(\lambda,\cdot)$ is an increasing function, $$S(\lambda, 0)= 0 \quad \hbox{and} \quad \lim_{E \rightarrow + \infty} S(\lambda, E) = +\infty \, . $$ \item[(b)] $\displaystyle\lim_{E\rightarrow + \infty} T(\lambda, S(\lambda, E)) = \displaystyle\lim_{E \rightarrow 0^+} T(\lambda, S(\lambda, E)) = 0 \, .$ \item[(c)] $T(\lambda, S (\lambda,\cdot)) $ has a unique maximum for each $\lambda > 0 \, .$ \item[(d)] $\lambda \rightarrow T(\lambda, S^*_\lambda)$ is a decreasing function that satisfies $$ \lim_{\lambda\rightarrow 0^+} T(\lambda, S^*_\lambda) = +\infty\qquad\mbox{and}\qquad \lim_{\lambda \rightarrow + \infty} T(\lambda, S^*_\lambda) = 0 $$ where $S^*_\lambda = S(\lambda, E^*(\lambda)) $ and $E^*(\lambda) $ is the unique critical point of the function $T(\lambda,S(\lambda,\cdot)) $. \item[(e)] For each $\lambda$ sufficiently small, the solutions $S_1$ and $S_2$ of $T(\lambda,S)=1/2$ satisfy $$\gathered S_1\leq\big(\lambda\big(\frac{m-1-q}{p-m+1}\big)\frac{(p+1)}{(q+1)}\big) ^{1/(p-q)}\,, \\ \frac{2^m}{m'}I^m_p(p+1)\leq S_2^{p-m+1}\leq\frac{2^m}{m'}I_q^m(p+1) \,. \endgathered $$ \end{itemize}\end{lemma} \section{Proof of the Main Results} \paragraph{Proof of Theorem 1.} By Lemma 1 and the continuity of the function $\lambda \to T(\lambda, S^*_\lambda)$, there exists $\lambda^*$ which satisfies $T(\lambda^*,S^*_{\lambda^*})=1/2 $ and such that: \begin{itemize} \item If $\lambda > \lambda^* $, then $T(\lambda, S(\lambda, E)) < \frac{1}{2} $, for each $E > 0$. \item If $\lambda = \lambda^* $, then ${\displaystyle\max_{E>0}}\, T(\lambda, S (\lambda,E))= 1/2$. \item If $\lambda < \lambda^* $, then ${\displaystyle\max_{E>0}}\, T(\lambda, S (\lambda,E))>1/2$ and for each $\lambda$ sufficiently small we have that $$\|u_\lambda\|_\infty\leq\left(\lambda\left(\frac{m-1-q}{p-m+1}\right) \frac{(p+1)}{(q+1)}\right)^{1/(p-q)}$$ and $$\left(\frac{2^m}{m'}I^m_p(p+1)\right)^{1/(p-m+1)} \leq\|v_\lambda\|_\infty \leq\left(\frac{2^m}{m'}I_q^m(p+1)\right)^{1/(p-m+1)}\,.$$ \end{itemize} \noindent From these three statements and Lemma 1, Theorem 1 follows. \paragraph{Proof of Lemma 1.} The proof of (a) and (b) can be found in \cite{garcia} and \cite{ubilla}. Concerning (c), using statements (a) and (b) it suffices to show that the function $S\to T(\lambda,S)$ has a unique critical point for each $\lambda>0$.\\ On the other hand, it is easy to prove that $$ \frac{\partial T}{\partial S}= (m')^{-\frac{1}{m}}\int^S_0\frac{A(\lambda, S) - A(\lambda,\eta)}{m S (G(\lambda, S) - G(\lambda,\eta))^{\frac{m+1}{m}}}\,d\eta %\leqno{(9)} $$ where $A(\lambda, u) = \left(\frac{m-1-q}{q+1}\right)\lambda u^{q+1} - \left({\frac{p-m+1}{p+1}}\right)u^{p+1}$. Direct computations show that \begin{equation} \frac{\partial T}{\partial S} > 0 \,, \quad \mbox{for } S\in [0, \rho_1] \quad\mbox{and}\quad \displaystyle\frac{\partial T}{\partial S} < 0 \,, \quad \mbox{for } S \in [\rho_2 ,+ \infty) \label{10} \end{equation} where \begin{equation} \rho_1 = \left({\lambda\left({\frac{m-1-q}{p-m+1}}\right)}\right) ^{1/(p-q)}\quad\mbox{and}\quad \rho_2 = \left(\lambda \frac{(m-1-q)(p+1)}{(p-m+1)(q+1)} \right)^{1/(p-q)} \,. \label{11} \end{equation} Moreover, $A(\lambda,0)=A(\lambda,\rho_2)=0$ and $\rho_2>\rho_1$. Thus the critical points of $T(\lambda, \cdot)$ belong to the interval $J:=[\rho_1,\rho_2]$. By the same arguments as in Lemma 7 of \cite{addou}, it is not difficult to verify that $$ \frac{\partial^2 T}{\partial S^2} =(m')^{-1/m} \int^1_0 \frac{S(1-\eta^{p+1})^2P(x(\eta))}{m^2 (G(\lambda,S)-G(\lambda, \eta S))^{(2m+1)/m}} \,d\eta \,, $$ where $$x(\eta) = \left\{ \begin{array}{ll} \frac{q+1}{p+1} &\mbox{if } \eta = 1\\[5pt] \frac{1-\eta^{q+1}}{1-\eta^{p+1}} &\mbox{if }\eta \in [0,1)\\ \end{array} \right.$$ and $$P(x) = \frac{(q-m+1)}{q+1} \lambda^2 S^{2q} x^2-C(m,p,q) \lambda S^{p+q} x + \frac{(p-m+1)}{p+1} S^{2p} \, .$$ % Since $0 \leq q 0$, $x\in (x_1, x_2)$; \item $P(x) < 0$, $x\in (-\infty,x_1) \cup (x_2, + \infty)$. \end{itemize} \noindent Indeed, \begin{eqnarray*} x_2 &=& \frac{S^{p-q}}{2\lambda (p+1)(m-1-q)} \\ &&\times\big(\sqrt{\mu}-(m(p^2 + q^2) - 2(m+1) pq+(m-2)(p+q) +2(m-1))\big)\,, \end{eqnarray*} where \begin{eqnarray*} \mu &=& (m(p^2+q^2)-2(m+1)pq +(m-2)(p+q) + 2(m-1))^2\\ &&+ 4(p+1)(q+1)(p-m+1)(m-1-q) \,. \end{eqnarray*} On the other hand, using that $$\alpha(x) = (p+1) x^{q+1} - (q+1)x^{p+1}, \quad \hbox{for} \; 0 \leq x \leq 1,$$ is an increasing function, it is easy to see that $$x(\eta) \in \left[\frac{q+1}{p+1}, 1\right], \quad \hbox{for all} \; \eta \in [0,1]. %\leqno{(16)} $$ We note that the function $S\to x_2(S) $ is an increasing function that satisfies $$x_2(\rho_2) = \frac{\sqrt{\mu}-(m(p^2+q^2)-2(m+1)pq +(m-2) (p+q) + 2(m-1))}{2(q+1)(p-m+1)} \, .$$ We now claim that $$x_2 (\rho_2) < \frac{q+1}{p+1} \,. %\leqno{(17)} $$ For this, we consider the function \begin{eqnarray*} F(x)&=&((m+1) q+1) x^3 - (2(m+1) q^2 + q + (m-1))x^2 \\ &&+q((m+1)q^2-q + 2(m-1))x-q^2(m-1-q)\,. \end{eqnarray*} Let us show that the function $F$ satisfies $$ F(x)> 0, \quad \hbox{for } x>m-1\,.%\leqno{(18)} $$ This statement will follow, if we prove that $F(m-1)\geq 0$, $F'(m-1)> 0$ and $F$ is convex in $[m-1,+\infty)$. In fact, \begin{align*} F(m-1)&=(m-1)\left((m+1)q^3-(2m^2-1)q^2+m^2(m-1)q\right)-q^2(m-1-q)\\ &=m^2q^3-2m^2(m-1)q^2+m^2(m-1)^2q\\ &=m^2q(q-m+1)^2\geq 0 \end{align*} On the other hand, $$F'(x)=3((m+1)q+1)x^2-2(2(m+1)q^2+q+m-1)x+q((m+1)q^2-q+2(m-1)) \, .$$ Thus \begin{eqnarray*} F'(m-1) &=&(m-1)(3(m-1)((m+1)q+1)-2(2(m+1)q^2+q+m-1))\\ &&+(m+1)q^3 -q^2+2(m-1)q\\ &=&(m-1)(-4(m+1)q^2+(3m^2-5)q+m-1)+(m+1)q^3\\ &&-q^2+2(m-1)q\\ &=&(m+1)q^3-(4m^2-3)q^2+3(m-1)^2(m+1)q+(m-1)^2. \end{eqnarray*} We note that the point $x=m-1$ is a zero of the algebraic equation $$(m+1)x^3-(4m^2-3)x^2+3(m-1)^2(m+1)x+(m-1)^2=0 \, .$$ Thus \begin{align*} F'(m-1)&=(q-m+1)((m+1)q^2-(3m^2-2)q-m+1)\\ &=(q-m+1)((q-m+1)((m+1)q+1)-2m^2q)>0\,. \end{align*} We now prove that $F$ is convex in $[m-1,+\infty)$. Note that $$F(x)=\alpha x^3+\beta x^2+\gamma x+\delta$$ where $$\gathered \alpha =(m+1)q+1\,, \quad \beta =-(2(m+1)q^2+q+m-1)\,, \\ \gamma =q((m+1)q^2-q+2(m-1))\,, \quad \delta =-q^2(m-1-q)\,. \endgathered $$ Then it easily follows that $F''(x)>0$ if and only if $x>-\beta/(3\alpha)$. Thus, $F$ is convex in $[m-1,+\infty)$ when $$m-1\geq-\frac{\beta}{3\alpha}=\frac{2(m+1)q^2+q+m-1}{3((m+1)q+1)}$$ which is equivalent to saying that $$G(q)=2(m+1)q^2+(4-3m^2)q-2(m-1)\leq 0$$ But $G(q)\leq 0$, when $q\in(r_1,r_2)$, where $$ r_1=\frac{3m^2-4-m\sqrt{9m^2-8}}{4(m+1)} \quad\mbox{and}\quad r_2=\frac{3m^2-4+m\sqrt{9m^2-8}}{4(m+1)} \,.$$ Since $r_1<0 0$ and $\lambda > 0$. Using that $\frac{d}{dE}(T(\lambda,S(\lambda,E^*(\lambda)))=0$ and by part (a), we see that $$\frac{\partial T}{\partial \lambda} (\lambda, S(\lambda, E^*(\lambda)))< 0, \mbox{ for each } \lambda >0.$$ Therefore $T(\lambda,S_\lambda^*)$ is a decreasing function of $\lambda$. On the other hand, we know that the function $T(\lambda,S(\lambda,\cdot))$ is increasing in $(0,E_1(\lambda))$, where $E_1(\lambda)=(m'G(\lambda,\rho_1))^{1/m}$, hence $E^*(\lambda)$ is the unique maximum of the function $T(\lambda,S(\lambda,\cdot))$. Then $E^*(\lambda)\geq E_1(\lambda)$. By part (a), we have that $$S(\lambda,E^*(\lambda))\geq S(\lambda,E_1(\lambda)) \, .$$ Thus $$ T(\lambda,S(\lambda,E^*(\lambda))\leq (m')^{-1/m}\rho_1^{\frac{m-1-p}{m}}\int_0^1\left({\frac{1-t^{p+1}}{p+1}} \right)^{-1/m}dt. $$ Since $\displaystyle\lim_{\lambda\to+\infty}\rho_1= \lim_{\lambda\to+\infty}\left({\lambda\frac{(m-1-q)}{(p-m+1)}}\right) ^{1/(p-q)}=+\infty$, it follows that $$ \lim_{\lambda\to+\infty}T(\lambda,S^*_\lambda)=0 \,. $$ From the fact that $\rho_1=S(\lambda,E_1(\lambda))$, then after some computations, we obtain that \begin{eqnarray*} T(\lambda,S(\lambda,E_1(\lambda)) &=&\lambda^{\frac{m-1-p}{m(p-q)}}(m')^{-\frac{1}{m}} \left({\frac{m-1-q}{p-m+1}}\right)^\frac{m-1-q}{m(p-q)} \\ &&\times \int_0^1\left({\left({\frac{m-1-q}{p-m+1}}\right) \left({\frac{1-t^{p+1}}{p+1}}\right)+\frac{1-t^{q+1}}{q+1}}\right) ^{-\frac{1}{m}}\,dt\,. \end{eqnarray*} Hence $$ \lim_{\lambda\to 0^+}T(\lambda,S(\lambda,E^*(\lambda)) \geq \lim_{\lambda\to 0^+} T(\lambda,S(\lambda,E_1(\lambda)) =+\infty \, , $$ which proves (d). In order to prove (e), using the concavity of the function $T(\lambda,\cdot)$ and assertions (\ref{10}) and (\ref{11}), we obtain that $S_1<\rho_2$ which prove the first inequality of (e). Using the hypothesis that $T(\lambda,S)=1/2$, we have $$\frac{2^m}{m'}I_p^m S^p\leq\frac{S^{p+1}}{p+1}+\lambda\frac{S^{q+1}}{q+1}\leq\frac{2^m}{m'}I_q^m S^m.$$ Since $S_2>\rho_1$, the above inequalities imply $$\frac{2^m}{m'}I_p^m(p+1)\leq S_2^{p-m+1}\leq\frac{2^m}{m'}I_q^m(p+1).$$ Hence the proof of Lemma 1 is now complete. \paragraph{Remark 1.} If $\lambda=0$, then it is easy to prove that the time mapping function is a decreasing function of $S$. Therefore (\ref{P}) has a unique positive solution. \paragraph{Remark 2.} Using the methods developed in this article, it is possible to find the exact number of solutions with $k$ nodes in the interval $(0,1)$ for the problem $$\displaylines{ -(|u'|^{m-2}u')'=\lambda|u|^{q-1}u+|u|^{p-1}u, \quad\mbox{in }(0,1) \cr u(0)=u(1)=0\,. }$$ In this case the same time mapping $T(\lambda,S)$ may be used. \paragraph{Remark 3.} We conjecture that the same techniques developed in \cite{addou} may be applied to determine a lower bound on the number of solutions together with their nodal properties for the problem (1) , with $\lambda < 0 $. \paragraph{Acknowledgments} The authors thank the referee for his careful reading of the manuscript and his valuable suggestions. \begin{thebibliography}{9} \bibitem{addou} I. Addou and A. Benmeza\"{\i}, {\em Boundary Value Problems for the one dimensional p-Laplacian with even super-linear Nonlinearity}, Elect. J. of Differential Equations, Vol. {\bf 9} (1999), 1--29. \bibitem{ambrosetti} A. Ambrosetti, H. Brezis and G. Cerami, {\em Combined effects of concave and convex nonlinearities in some elliptic problems}, J. Funct. Anal. 122, {\bf 2} (1994), 519--543. \bibitem{garcia} M. Garc\'{\i}a-Huidobro and P. Ubilla, {\em Multiplicity of solutions for a class of nonlinear second order equations}, Nonlinear Analysis. T.M.A., Vol. 28, {\bf 9} (1997), 1509--1520. \bibitem{ubilla} P. Ubilla, {\em Multiplicity results for the 1-dimensional generalized p-Laplacian}, \linebreak J. Math. Anal. and Appl. {\bf 190} (1995), 611--623. \end{thebibliography} \noindent{\sc Justino S\'anchez \& Pedro Ubilla} \\ Universidad de Santiago de Chile\\ Casilla 307, Correo 2, Santiago, Chile\\ e-mail: pubilla@fermat.usach.cl \end{document}