\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 31, pp. 1--7.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/31\hfil Multiplicity of solutions] {Multiplicity of solutions for a quasilinear problem with supercritical growth} \author[G. M. Figueiredo\hfil EJDE-2006/31\hfilneg] {Giovany M. Figueiredo} \address{Giovany M. Figueiredo \newline Universidade Federal do Par\'{a}\\ CEP 66.075-110, Bel\'{e}m-Par\'{a}, Brazil} \email{giovany@ufpa.br} \date{} \thanks{Submitted October 26, 2005. Published March 16, 2006.} \subjclass[2000]{35A15, 35H30, 35B40} \keywords{Variational method; penalization method; Moser iteration method} \begin{abstract} The multiplicity and concentration of positive solutions are established for the equation $$-\epsilon^{p}\Delta_{p}u+V(z)|u|^{p-2}u=|u|^{q-2}u +\lambda|u|^{s-2}u \quad\text{in }\mathbb{R}^N,$$ where $10$, $p 0, \quad\text{for } z \in \mathbb{R}^N, \end{gathered}$\epsilon > 0$,$p < q < p^{*} \leq s$,$p^{*}=\frac{Np}{N-p}$,$\lambda \geq 0$and$\Delta_{p}u$is the p-Laplacian operator; that is, $$\Delta_{p}u=\sum_{i=1}^{N}{\frac{\partial}{\partial{x_{i}}} \Big(|\nabla{u}|^{p-2}\frac{\partial{u}}{\partial{x_{i}}}\Big)}.$$ We assume that$V$is a continuous function satisfying $$V(x)\geq V_{0}=\inf_{x \in \mathbb{R}^{N}} V(x)> 0\quad \mbox{for } x\in \mathbb{R}^{N}; \label{V1}$$ Also assume that there exists an open and bounded domain$\Omega \subset \mathbb{R}^{N}$such that $$V_0 < \min_{\partial\Omega}V. \label{V2}$$ In recent years, much attention has been paid to the existence and multiplicity of solutions for both subcritical and critical cases and to the concentration behavior of solutions for problem $$-\epsilon^{2}\Delta u +V(z)u=f(u) \quad\text{in } \mathbb{R}^{N}, \label{P*}$$ when$\epsilon$is small. Interesting results may be found, for example, in \cite{Alves8,Bartsch,Benci1,Chabr1,Cingolani,Gui,Rabinow} and their references. Cingolani \& Lazzo \cite{Lazzo1}, using Lusternik-Schnirelman category and involving the sets \begin{gather*} M=\{x\in \Omega : V(x)=V_{0}\}, \\ M_{\delta}=\{x\in \mathbb{R}^{N}:dist(x,M)\leq \delta \}, \quad \delta>0, \end{gather*} showed a result of multiplicity of positive solutions for \eqref{P*}, where$\Omega=\mathbb{R}^{N}$,$f(u)=|u|^{q-2}u$with$q \in (2, 2^{*})$, and $$V_{\infty}=\liminf_{|x|\to\infty}V(x)>V_{0}=\inf_{\mathbb{R}^{N}}V(x)>0. \label{V}$$ Recall that for a closed subset$Y$of a topological space$X$, the Lusternik-Schnirelman category, denoted by$\mathop{\rm cat}_{X}Y$, is the least number of closed and contractible sets in$X$which cover$Y$. Alves \& Souto \cite{Alves10} showed an existence and concentration result for \eqref{P*} with$f(u)=u^{q-1}+u^{2^{*}-1}$assuming that condition \eqref{V} holds. Alves \& Figueiredo \cite{AlvesGio1} (see also \cite{giovany}) proved a multiplicity result for $$-\epsilon^{p}\Delta_{p}u+V(z)|u|^{p-2}u=f(u) \ \ in \ \ \mathbb{R}^N \label{P**}$$ using again Lusternik-Schinirelman category and assuming that condition \eqref{V} holds,$2\leq p0$, there exists$\overline{\epsilon}=\overline{\epsilon}(\delta)>0$and$\lambda_{0}>0$such that \eqref{Plamb} has at least$\mathop{\rm cat}_{M_{\delta}}M$positive solutions for all$\epsilon\in(0,\overline{\epsilon})$and for all$\lambda \in [0,\lambda_{0}]$. Moreover, if$u_{\epsilon}$is a positive solution of \eqref{Plamb} and$\eta_{\epsilon}\in \mathbb{R}^{N}$a global maximum point of$u_{\epsilon}$, then $$\lim_{\epsilon\to 0}V(\eta_{\epsilon})=V_{0}.$$ \end{theorem} To solve problem \eqref{Plamb}, we first consider a truncated problem which involves only a subcritical Sobolev exponent. We show that any positive solution of truncated problem is a positive solution of \eqref{Plamb}. Hereafter, we will work with the following problem equivalent to \eqref{Plamb}, which is obtained under change of variable$z=\epsilon x$\label{Plambst} \begin{gathered} -\Delta_{p}u + V(\epsilon x)|u|^{p-2}u=|u|^{q-2}u + \lambda|u|^{s-2}u \quad\text{in } \mathbb{R}^N \\ u \in W^{1,p}(\mathbb{R}^N) \quad\text{with } 1 0,\quad \forall x \in \mathbb{R}^{N} . \end{gathered} \section{Truncated Problem} First of all, we have to note that because$f$has supercritical growth we cannot use directly variational techniques because of the lack of compactness of the Sobolev immersions. So we construct a suitable truncation of$f$in order to use variational methods or more precisely, the Mountain Pass Theorem. This truncation was used in \cite{rabino} (see also \cite{Chabr} and \cite{giovany}). Let$K>0$, be a constant to be determined later, and$\widehat{f}_{K}:\mathbb{R} \to \mathbb{R}$given by $\widehat{f}_{K}(t) = \begin{cases} 0 &\text{if } t < 0 \\ t^{q-1}+\lambda t^{s-1} &\text{if } 0 \leq t < K \\ (1+\lambda K^{s - q})t^{q-1} &\text{if } t\geq K. \end{cases}$ Consider$\alpha,\gamma\in\mathbb{R}$such that$\alpha<1<\gamma$and$\eta \in C^{1}([\alpha K ,\gamma K])$with$\alpha$and$\gamma$independent of$K$and$\eta$satisfying \begin{gather*} \eta(t)\leq\widehat{f}_{K}(t)\quad \mbox{for all } t\in [\alpha K ,\gamma K], \\ \eta(\alpha K)=\widehat{f}_{K}(\alpha K),\quad \eta(\gamma K)=\widehat{f}_{K}(\gamma K), \\ \eta'(\alpha K)=\widehat{f}'_{K}(\alpha K), \quad \eta'(\gamma K)=\widehat{f}'_{K}(\gamma K), \\ t\mapsto\frac{\eta(t)}{t^{p-1}} \mbox{ is increasing for all }t\in [\alpha K,\gamma K]. \end{gather*} Now using the functions$\eta$and$\widehat{f}_{K}$, we define $f_{K}(t) =\begin{cases} \eta(t) &\text{if } t \in [\alpha K ,\gamma K], \\ \widehat{f}_{K}(t) &\text{if } t \not\in [\alpha K ,\gamma K] \end{cases}$ and the truncated problem $$\label{Tlamb} \begin{gathered} -\Delta_{p}u + V(\epsilon x)|u|^{p-2}u = f_{K}(u) \\ u \in W^{1,p}(\mathbb{R}),\quad u > 0 \quad\text{in } \mathbb{R}^{N}. \end{gathered}$$ It is easy to check that$f_{K} \in C^{1}(\mathbb{R})$, and that \begin{gather*} f_{K}(t)=0,\quad \mbox{for all } t < 0, \label{fK1} \\ f_{K}(t)\leq(1+\lambda K^{s-q})t^{q-1}\quad \mbox{for all } t\geq 0,\label{fK2} \\ F_{K}(t)\leq\frac{1}{q}(1+\lambda K^{s-q})t^{q}\quad \mbox{for all } t\geq 0, \quad F_{K}(t)=\int^{t}_{0}f_{K}(\xi)d\xi, \label{fK3} \end{gather*} there exists$\theta \in \mathbb{R}$such that$p<\theta$and $$0 < \theta F_{K}(t) \leq f_{K}(t)t \quad \mbox{for all } t>0, \label{fK4}$$ the function \begin{gather} t\mapsto \frac{f_{K}(t)}{t^{p-1}} \mbox{ is increasing for all } t>0, \label{fK5} \\ f'_{K}(t)t^{2}-(p-1)f_{K}(t)t\geq (q -p )t^{q}. \label{fK6} \end{gather} \begin{remark}\label{final} \rm Note that if$u_{\epsilon,\lambda}$is a positive solution of \eqref{Tlamb} such that there exists$K_{0} > 0$, where for each$K \geq K_{0}$, there exists$\lambda_{0}(K)>0$such that$|u_{\epsilon,\lambda}|_{L^{\infty}(\mathbb{R}^{N})}\leq \alpha K$for all$\epsilon \in (0, \bar{\epsilon})$and for all$\lambda \in [0, \lambda_{0}]$, then$u_{\epsilon,\lambda}$is a positive solution of \eqref{Plambst}. \end{remark} \section{Multiplicity and Concentration of positive solutions for Truncated Problem} The result below is related to the multiplicity and concentration of solutions for \eqref{Tlamb} and its proof can be found in \cite[Theorem 1.1]{AlvesGio2} or \cite{giovany}. \begin{theorem}\label{super11} Suppose that$V$verify \eqref{V1}\eqref{V2}. Then, for any$\delta>0$, there exists$\overline{\epsilon}=\overline{\epsilon}(\delta,\lambda,K)>0$such that$(T_\lambda)$has at least$\mathop{\rm cat}_{M_{\delta}}M$positive solutions for all$\epsilon\in(0,\overline{\epsilon})$and for each$\lambda>0$. Moreover, if$u_{\epsilon,\lambda}$is a positive solution of \eqref{Tlamb} and$\eta_{\epsilon}\in \mathbb{R}^{N}$a global maximum point of$u_{\epsilon,\lambda}$, then $$\lim_{\epsilon\to 0}V(\eta_{\epsilon})=V_{0}.$$ \end{theorem} \section{Multiplicity of positive solutions for \eqref{Plambst}} We recall that the weak solutions of \eqref{Tlamb} are the critical points of the functional $$I_{\epsilon,\lambda}(u)=\frac{1}{p}\int_{\mathbb{R}^{N}}|\nabla u|^{p}+ \frac{1}{p}\int_{\mathbb{R}^{N}}V(\epsilon x)|u|^{p}-\int_{\mathbb{R}^{N}}F_{K}(u),$$ which is well defined for$u \in W_{\epsilon}$, where $$W_{\epsilon}=\{u \in W^{1,p}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}V(\epsilon x)|u|^{p}<\infty\}$$ endowed with the norm $$\|u\|^{p}_{\epsilon}= \int_{\mathbb{R}^{N}}|\nabla u|^{p}+ \int_{\mathbb{R}^{N}}V(\epsilon x)|u|^{p}.$$ Let us also denote by$E_{V_{0},\lambda}$the energy functional associated to the problem $$\label{PV0lamb} \begin{gathered} -\Delta_{p}u + V_{0}|u|^{p-2}u = f_{K}(u) \\ u \in W^{1,p}(\mathbb{R}),\quad u > 0 \ in \ \mathbb{R}^{N}, \end{gathered}$$ that is, $$E_{V_{0},\lambda}(u) =\frac{1}{p}\int_{\mathbb{R}^{N}}|\nabla u|^p + \frac{1}{p}\int_{\mathbb{R}^{N}}V_{0}|u|^p - \int_{\mathbb{R}^N}F_{K}(u),$$ Here we will establish a preliminary estimative for$\|u_{\epsilon,\lambda}\|_{\epsilon}$. \begin{lemma}\label{super2} For any solution$u_{\epsilon,\lambda}$of \eqref{Tlamb}, there exists$\bar{C}>0$, such that \begin{eqnarray*} \|u_{\epsilon,\lambda}\|_{\epsilon}\leq \bar{C}, \end{eqnarray*} for$\epsilon > 0$sufficiently small and uniformly in$\lambda$. \end{lemma} \begin{proof} By \cite[Theorem 1.1]{AlvesGio2} (see \cite{giovany} too), we have that all solutions$u_{\epsilon,\lambda}$from \eqref{Tlamb} verify the inequality $I_{\epsilon,\lambda}(u_{\epsilon,\lambda})\leq c_{V_{0},\lambda}+ h_{\lambda}(\epsilon),$ where$c_{V_{0},\lambda}$is the level Mountain Pass related of functional$E_{V_{0},\lambda}$and$h_{\lambda}(\epsilon)\to 0$as$\epsilon\to 0$for each$\lambda \geq 0$. In this case, we may suppose that \begin{eqnarray*} I_{\epsilon,\lambda}(u_{\epsilon,\lambda})\leq c_{V_{0},\lambda}+ 1, \end{eqnarray*} for all$\epsilon\in (0,\bar{\epsilon}(K,\lambda))$. Since$c_{V_{0},\lambda}\leq c_{V_{0},0}$, we have $$\label{G} I_{\epsilon,\lambda}(u_{\epsilon,\lambda})\leq c_{V_{0},0}+ 1,$$ for all$\epsilon\in (0,\bar{\epsilon}(K,\lambda))$and for all$\lambda \geq 0. Moreover, \begin{align*} I_{\epsilon,\lambda}(u_{\epsilon,\lambda}) &=I_{\epsilon,\lambda}(u_{\epsilon,\lambda}) - \frac{1}{\theta}I'_{\epsilon,\lambda}(u_{\epsilon,\lambda}) u_{\epsilon,\lambda}\\ &=\big(\frac{1}{p}-\frac{1}{\theta}\big) \|u_{\epsilon,\lambda}\|^{p}_{\epsilon}+ \int_{\mathbb{R}^{N}}\big[\frac{1}{\theta}f_{K}(u_{\epsilon,\lambda})u_{\epsilon,\lambda} -F_{K}(u_{\epsilon,\lambda})\big]. \end{align*} By \eqref{fK4}, $I_{\epsilon,\lambda}(u_\epsilon,\lambda)\geq \big(\frac{1}{p}-\frac{1}{\theta}\big)\|u_{\epsilon,\lambda}\|^{p}_{\epsilon}$ \noindent Therefore, by \eqref{G},\|u_{\epsilon,\lambda}\|_{\epsilon}\leq \bar{C}$, for$\epsilon\in (0,\bar{\epsilon}(K,\lambda))$and for all$\lambda \geq 0$, where $$\bar{C}=\Big[(c_{V_{0},0}+1)\big(\frac{\theta p}{\theta - p}\big)\Big]^{1/p}.$$ \end{proof} Now, we use the Moser iteration technique \cite{moser} (see also \cite{Chabr}) to prove that each solution found of \eqref{Tlamb} is a solution of \eqref{Plambst} \begin{proof}[Proof of Theorem \ref{thmD}] We use the notation$u_{\epsilon,\lambda}:=u$. For each$L>0$, we define \begin{gather*} u_{L} =\begin{cases} u &\text{if } u \leq L, \\ L &\text{if } u \geq L, \end{cases}\\ z_{L} = u_{L}^{p(\beta - 1)}u \quad \text{and} \quad w_{L} = uu_{L}^{\beta-1} \end{gather*} with$\beta > 1$to be determined later. Taking$z_{L}as a test function, we obtain \begin{align*} \int_{\mathbb{R}^N}u_{L}^{p(\beta -1)}|\nabla u|^{p} &= -p(\beta -1)\int_{\mathbb{R}^N}u_{L}^{p\beta - p-1}u|\nabla u|^{p-2}\nabla u \nabla u_{L}\\ &\quad + \int_{\mathbb{R}^N}f_{K}(u)uu_{L}^{p(\beta -1)}- \int_{\mathbb{R}^N}V({\epsilon x})|u|^{p}u_{L}^{p(\beta - 1)}. \end{align*} By \eqref{fK2}, $$\label{w1} \int_{\mathbb{R}^N}u_{L}^{p(\beta -1)}|\nabla u|^{p} \leq C_{\lambda,K} \int_{\mathbb{R}^N}u^{q }u_{L}^{p(\beta - 1)},$$ whereC_{\lambda,K}= (1+\lambda K^{s-q})$. From Sobolev imbedding, H\"{o}lder inequalities and \eqref{w1}, $%\label{3eqb} |w_{L}|^{p}_{p^{*}}\leq C_{1}\beta^{p}C_{\lambda,K}\Big(\int_{\mathbb{R}^N}u^{p^{*}} \Big)^{(q-p)/p^{*}} \Big(\int_{\mathbb{R}^N}w_{L}^{pp^{*}/[p^{*}-(q-p)]} \Big)^{[p^{*}-(q-p)]/p^{*}},$ where$p < \frac{pp^{*}}{p^{*}-(q-p)} < p^{*}$. Recalling that$\|u_{\epsilon,\lambda}\|_{\epsilon}\leq \bar{C}$, we have $|w_{L}|^{p}_{p^{*}} \leq C_{2}\beta^{p}C_{\lambda,K}\bar{C}^{(q-p)/p^*} |w_{L}|^{p}_{\alpha^{*}}$ where$\alpha^{*}=\frac{pp^{*}}{p^{*}-(q-p)}$. Note that if$u^{\beta}\in L^{\alpha^{*}}(\mathbb{R}^{N})$, using the definition of$w_{L}$and the fact that$u_{L}\leq u$, we obtain $\Big(\int_{\mathbb{R}^{N}}\mid uu_{L}^{\beta - 1}\mid^{p^{*}}\Big)^{p/p^*} \leq C_{3}\beta^{p}C_{\lambda,K}\Big(\int_{\mathbb{R}^{N}}u^{\beta \alpha^{*}}\Big)^{p/\alpha^{*}}< + \infty.$ By Fatou's Lemma on the variable$L$, we get $$\label{w3} |u|_{\beta p^{*}} \leq (C_{4}C_{\lambda,K})^{1/\beta }\beta^{1/\beta}|u|_{\beta \alpha^{*}}.$$ The assertion is obtained by iteration of estimative \eqref{w3}. Namely, let$\chi=\frac{p^{*}}{\alpha^{*}}$; i.e.,$p^{*}=\chi\alpha^{*}$. Then $|u|_{\chi^{(m+1)}\alpha^{*}} \leq C_{5}(C_{4}C_{\lambda,K})^{\sum_{i=1}^{m}\frac{\chi^{-i}}{p}} \chi^{\sum_{i=1}^{m}i\chi^{-i}}\bar{C}.$ Passing to the limit as$m \to \infty$, we have $|u|_{L^{\infty}(\mathbb{R}^{N})} \leq C_{5}(C_{4}C_{\lambda,K})^{\sigma_{1}}\chi^{\sigma_{2}}\bar{C},$ where$\sigma_{1}=\sum_{i=1}^{\infty}\frac{\chi^{-i}}{p}$and$\sigma_{2}=\sum_{i=1}^{\infty}i\chi^{-i}$. To choose$\lambda_{0}$, we consider the inequality $\Big[C_{4}(1+\lambda K^{s-q})\Big]^{\sigma_{1}}\chi^{\sigma_{2}}C_{5}\bar{C}\leq \alpha K.$ We conclude that $(1+\lambda K^{s-q})^{\sigma_{1}}\leq \frac{\alpha K C_{6}}{C_{4}^{\sigma_{1}}\chi^{\sigma_{2}}\bar{C}}.$ We choose$\lambda_{0}$verifying the inequality $\lambda_{0}\leq \Big[\frac{(\alpha K C_{6})^{\frac{1}{\sigma_{1}}}}{C_{4}\chi^\frac{\sigma_{2}}{\sigma_{1}} \bar{C}^{1/\sigma_1}} -1\Big] \frac{1}{K^{s-q}}$ and fixing$K$such that $\Big[\frac{(\alpha K C_{6})^{1/\sigma_1}}{C_{4} \chi^\frac{\sigma_{2}}{\sigma_{1}}\bar{C}^{1/\sigma_1}} -1\Big] > 0,$ we have$|u_{\lambda,\epsilon}|_{L^{\infty}(\mathbb{R}^{N})} \leq \alpha K$for all$\epsilon \in (0,\bar{\epsilon}(K,\lambda))$and all$\lambda \in [0,\lambda_{0}]$. The result follows from Remark \ref{final}. \end{proof} \subsection*{Acknowledgements} We would like to thank to Professor C. O. Alves for his help and encouragement, to the anonymous referee for several suggestions that improved this article. \begin{thebibliography}{99} \bibitem{AlvesGio1} C.O. Alves and G.M. Figueiredo, \emph {Existence and multiplicity of positive solutions to a p-Laplacian equation in$\mathbb{R}^{N}$}. To appear in Differential and integral Equations. \bibitem{AlvesGio2} C.O. Alves and G.M. Figueiredo, \emph {Multiplicity of positive solutions for a quasilinear problem in$\mathbb{R}^{N}$via penalization method}. to appear in Advanced Nonlinear Studies. \bibitem{Alves8} C.O. Alves and Soares S. H. M. \emph {On the location and profile of spike-layer nodal solutions to nonlinear Schrödinger equations}. J. Math. Anal. Appl. 296(2004)563-577. \bibitem{Alves10} C. O. Alves and M. A. S Souto . \emph {On existence and concentration behavior of ground state solutions for a class of problems with critical growth}. Comm. Pure and Applied Analysis, vol 1, num 3(2002)417-431. \bibitem{Bartsch} T. Bartsch and Z. Q. Wang \emph { Existence and multiplicity results for some superlinear elliptic problems on$\mathbb{R}^{N}$}. Comm. Partial Differential Equations, 20(1995)1725-1741. \bibitem{Benci1} Benci V and Cerami G. \emph {Existence of positive solutions of the equation$-\Delta u + a(x)u= u^{\frac{n+2}{N-2}}$in$\mathbb{R}^{N}\$}. J. Func. Anal, 88(1990),90-117. \bibitem{Chabr} J. Chabrowski and Yang Jianfu. \emph {Existence theorems for elliptic equations involving supercritical Sobolev exponents}. Adv. Diff. Eq. 2 num. 2(1997)231-256. \bibitem{Chabr1} Chabrowski J and Yang Jianfu. \emph { Multiple semiclassical solutions of the Schrödinger equation involving a critical Sobolel exponent}. Portugal Math. 57(2000)3, 273-284. \bibitem{Lazzo1} S. Cingolani and M. Lazzo, \emph {Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations}. Topol. Methods. Nonlinear Anal. 10(1997)1-13. \bibitem{Cingolani} Cingolani S and Lazzo M, \emph {Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions}. JDE, 160(2000)118-138. \bibitem{Pino} M. del Pino M and P.L Felmer \emph {Local Mountain Pass for semilinear elliptic problems in unbounded domains}. Calc. Var. 4 (1996), 121-137. \bibitem{giovany} G.M. Figueiredo, \emph {Multiplicidade de solu\c{c}\~oes positivas para uma classe de problemas quasilineares}. Doct. dissertation, Unicamp, 2004. \bibitem{Gio2} G.M. Figueiredo, \emph {Existence, multiplicity and concentration of positive solutions for a class of quasilinear problem with critical growth}. preprint. \bibitem{Gui} Gui C. \emph { Existence of multi-bumps solutions for nonlinear Schrödinger equations via variational methods}. Comm. Partial Differential Equations (1996)787-820. \bibitem{moser} J. Moser \emph {A new proof de Giorgi's theorem concerning the regularity problem for elliptic differential equations,} Comm. Pure Apll. Math. 13 (1960), 457-468. \bibitem{rabino} P.H Rabinowitz \emph {Variatinal methds for nonlinear elliptic eigenvalue problems}. Indiana Univ. J. Maths. 23(1974),729-754. \bibitem{Rabinow} Rabinowitz P. H. \emph {On a class of nonlinear Schrödinger equations}. Z. Angew Math. Phys. 43(1992)27-42. \end{thebibliography} \end{document}