\documentclass[reqno]{amsart} \usepackage{amssymb,hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 112, pp. 1--8.\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 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/112\hfil A resonance problem] {A resonance problem for the p-laplacian in $\mathbb{R}^N$} \author[G. Izquierdo B., G. L\'{o}pez G.\hfil EJDE-2005/112\hfilneg] {Gustavo Izquierdo Buenrostro \& Gabriel L\'{o}pez Garza} \address{Gustavo Izquierdo Buenrostro \hfil\break Dept. Mat. Universidad Aut\'{o}noma Metropolitana\\ M\'{e}xico} \email{iubg@xanum.uam.mx} \address{Gabriel L\'{o}pez Garza \hfill\break Dept. Mat. Universidad Aut\'{o}noma Metropolitana\\ M\'{e}xico} \email{grlzgz@xanum.uam.mx} \date{} \thanks{Submitted May 31, 2005. Published October 17, 2005.} \subjclass[2000]{35J20} \keywords{Resonance; $p$-Laplacian; improved Poincar\'{e} inequality} \begin{abstract} We show the existence of a weak solution for the problem $$-\Delta_p u=\lambda_1h(x)|u|^{p-2}u+a(x)g(u)+f(x),\quad u\in\mathcal{D}^{1,p}(\mathbb{R}^N),$$ where, $2From here and henceforth the integrals and all the spaces are taken over$\mathbb{R}^N$unless otherwise specified. The term resonance is well known in the literature, and refers to the case in which$\lambda$is an eigenvalue of the problem $$\label{eigen} \begin{gathered} -\Delta_p u=\lambda h(x)|u|^{p-2}u,\\ u\in \mathcal{D}^{1,p}. \end{gathered}$$ In \cite{All}, Allegreto et al. show that the eigenvalue problem \eqref{eigen} possesses a sequence of eigenvalues$0<\lambda_1<\lambda_2\leqslant,\dots$and a corresponding sequence of eigenfunctions$\{\varphi_j\}$, where$\varphi_1$can be chosen to be positive a.e.. Moreover, we have the Rayleigh quotient characterization: $$\label{Ray} \lambda_1=\inf\Big\{ \int |\nabla u|^p: u\in \mathcal{D}^{1,p}\mbox{ with } \int h |u|^p=1 \Big\}.$$ We consider the function$\varphi_1$to be normalized; i.e.,$\int h|\varphi_1|^p=1$and we decompose any function$u\in\mathcal{D}^{1,p}$as a direct sum $$\label{sum} \begin{gathered} u=\alpha \varphi_1+w\mbox{ where }\\ \alpha=\int h|\varphi_1|^{p-2}\varphi_1u\mbox{ and }\int h|\varphi_1|^{p-2}\varphi_1w=0. \end{gathered}$$ Hence, we introduce the spaces $$\label{spaces} \begin{gathered} V\stackrel{\text{def}}{=}\mathop{\rm span}\{\varphi_1\},\\ W \stackrel{\text{def}}{=}\big\{w\in\mathcal{D}^{1,p}:\int h|\varphi_1| ^{p-2}\varphi_1w=0\big\} \end{gathered}$$ In order to prove our main result we use some of the results introduced by Alziary, Fleckinger and Tak\'{a}\u{c} in \cite{Takac} where the cases$10,\quad 20$and$C>0$such that \label{H} 0=\int|\nabla u|^{p-2}\nabla u\cdot\nabla v -\lambda\int h|u|^{p-2}uv-\int ag(u)v-\int fv for all$u,v\in\mathcal{D}^{1,p}$. To prove theorem \ref{thm1} we use the Minimax Methods introduced by Rabinowitz \cite{R}. We recall here for the convenience of the reader some previous definitions and theorems. \noindent{\bf Palais-Smale condition.} Suppose that$E$is a real Banach space. A functional$I\in\mathcal{C}^1(E,\mathbb{R})$satisfies the Palais-Smale condition at level$c\in\mathbb{R}$, denoted$(PS)_c$, if any sequence$(u_n)\subset E$for which \begin{itemize} \item[(i)]$I(u_n)\to c$as$n\to\infty$and \item[(ii)]$I'(u_n)\to 0$as$n\to\infty$, \end{itemize} possesses a convergent subsequence. If$I\in\mathcal{C}^1(E,\mathbb{R})$satisfies the$(PS)_c$for every$c\in\mathbb{R}$, we say that$(u_n)$satisfies the$(PS)$condition. Any sequence for which (i) and (ii) hold is called a$(PS)_c$sequence for$I$. Now we establish a preliminary result. \begin{proposition}\label{propPS} Let$J_{\lambda}:\mathcal{D}^{1,p}\rightarrow\mathbb{R}$be defined as \ref{J} where$\lambda\in \mathbb{R}$. Suppose that$g$is a continuous function with$|g(s)|\leqslant M$for all$s\in \mathbb{R}$,$f\in L^{(p^{*})'}(\mathbb{R}^N)$,$22(see \cite{All} inequality (7) p.237 and subsequent inequalities) \label{alegreto} \begin{aligned} \int|\nabla u_n-\nabla u|^p &\leqslant C\left\{\int (|\nabla u_n|^{p-2}\nabla u_n-|\nabla u|^{p-2} \nabla u)\cdot \nabla(u_n-u) \right\}\\ &\quad \times \Big(\int|\nabla u_n|^p+ \int|\nabla u|^p \Big). \end{aligned} Thus it is sufficient to show that\lim_{n\to\infty}\int |\nabla u_n|^{p-2}\nabla u_n\cdot\nabla(u_n-u)=0$. To this aim, taking$\varphi=u_n-u, in \eqref{PS2} we have \label{PS3} \begin{aligned} &\int |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla(u_n-u)\\ &=\lambda\int h|u_n|^{p-2}u_n(u_n-u)+\int ag(u_n)(u_n-u) +\int f(u_n-u)+o(1) \end{aligned} asn\to\infty$. For the first integral in the right hand side, using the H\"{o}lder's inequality we have $$\big|\int h|u_n|^{p-2}u_n(u_n-u)\big|\leqslant \Big(\int h|u_n|^p\Big)^{1/p'}\Big( \int h|u_n-u|^p\Big)^{1/p}.$$ Noting that$h\in L^{N/p}(\mathbb{R}^N)=L^{(p^{*}/p)'}(\mathbb{R}^N)$, for$1\leqslant q< p^{*}$the functional$u\mapsto\int h|u|^q$is weakly continuous in$\mathcal{D}^{1,p}$(see \cite[Prop. 2.1 p. 826]{Ben}). Consequently, $$\label{PS4} \lim_{n\to\infty} \int h|u_n|^{p-2}u_n(u_n-u)=0.$$ For the integral$\int a g(u_n)|u_n-u|$we consider the ball$B_r(0)$. Since$a,\, g$are bounded we have $$\big|\int_{B_r(0)} ag(u_n)(u_n-u)\big|\leqslant C\int_{B_r(0)}|u_n-u|\to 0 \quad \mbox{as }n\to\infty,$$ since$u_n\to u$strongly in$L^{1}(B_r(0))$due to the Relich-Kondrachov theorem. Now, together with the assumption that$u_n\mbox{ and }g$are bounded, we obtain $$\big|\int_{\mathbb{R}^N\setminus B_r(0)}ag(u_n)|u_n-u|\big|\leqslant C\Big( \int_{\mathbb{R}^N\setminus B_r(0)}|a|^\frac{Np}{N-p}\Big) ^\frac{N+p}{Np}$$ So, by taking$r$big enough it follows that $$\limsup_{n\to\infty}\big|\int ag(u_n)|u_n-u|\big|\leqslant C\varepsilon$$ For arbitrary$\varepsilon$. Finally, since$f\in L^{(p^{*})'}(\mathbb{R}^N)$we can use similar arguments as above to show that$\lim_{n\to\infty}\int f(u_n-u)=0$. \end{proof} \section{Proof of Theorem \ref{thm1}} In this section we consider the problem $$\label{pl1} \begin{gathered} -\Delta_p u=\lambda_1h(x)|u|^{p-2}u+a(x)g(u)+f(x)\\ u\in\mathcal{D}^{1,p} \end{gathered}$$ where$h$satisfies (H). To prove the main theorem of this section we require the Saddle Point Theorem of Rabinowitz \cite{R}, which we introduce here for the convenience of the reader. \begin{theorem}[Saddle Point Theorem] \label{sadd} Let$E=V\oplus W $, where$E$is a real Banach space and$V\not= \{0\}$is finite dimensional. Suppose$I\in \mathcal{C}^1(E,{\mathbb{R}})$satisfies the (PS) condition and \begin{itemize} \item[(I1)] there is a constant$\alpha$and a bounded neighborhood$D$of 0 in$V$such that$I\big|_{\partial D}\leqslant\alpha$, and \item[(I2)] there is a constant$\beta>\alpha$such that$I|_{W}\geqslant \beta$. \end{itemize} Then,$I$possesses a critical value$c\geqslant\beta$. Moreover$c$can be characterized as $$c=\inf_{h\in\Gamma}\max_{u\in\overline{D}} I(h(u)),$$ where$\Gamma=\{h\in\mathcal{C}(\overline{D},E):h=\mbox{id on }\partial D\}$. \end{theorem} Now, we can show the existence of weak solutions for$J_{\lambda_1}$. \begin{proof}[Proof of Theorem \ref{thm1}] First, we show that the functional$J_{\lambda_1}$corresponding to problem \eqref{pl1} satisfies the$(PS)_c$condition for any$c\in\mathbb{R}$, and thereafter we verify that$J_{\lambda_1}$satisfies the other hypotheses of the Theorem \ref{sadd}. Let$(u_n)$be a$(PS)_c$sequence for the functional$J_{\lambda_1}$. We claim that$(u_n)$is bounded. For each$n\in\mathbb{N}$write $$u_n\stackrel{\text{def}}{=}v_n+w_n=\alpha_n\varphi_1+w_n\quad \mbox{with } \alpha_n\in\mathbb{R}\mbox{ and }w_n\in W.$$ Since$(u_n)$is a$(PS)_c$sequence we have$|J_{\lambda_1}(u_n)|0$. By standard calculations (see for instance \cite[p.16]{LR}), we have $$\label{t1.3} \big|\int a(G(v_n +w_n)-G(v_n))\big|\leqslant M\int a|w_m| \leqslant C_2\|w_n\|.$$ Consequently, using \eqref{t1.1}, \eqref{t1.2} and \eqref{t1.3} we have $$\label{t1.4} \big|\int aG(v_n)+\int fv_n\big| \leqslant C_1+C_2\|w_n\|+\frac{c}{p}\|w_n\|^p.$$ So, given that$\int aG(v_n)+\int fv_n\to \infty\mbox{ as }\|v_n\|=|\alpha_n|\to\infty$, we have shown that$(v_n)$is bounded if$(w_n)$is bounded. We show now that$(w_n)$is bounded. In fact, note that $$\label{t1.5} \int |\nabla u_n|^{p-2}\nabla u_n\cdot\nabla w_n \geqslant \|u_n\|^p-\int |\nabla u_n|^{p-2}u_n\cdot \nabla v_n.$$ On the other hand, since$\langle J'_{\lambda_1}(u_n),v_n \rangle\stackrel{n}{\rightarrow} 0$, there exists$m_0$such that if$n\geqslant m_0$then, $$\label{t1.6} \big|\int |\nabla u_n|^{p-2}\nabla u_n\cdot\nabla w_n- \int z_n \cdot w_n\big|\leqslant C\|w_n\|,$$ where$z_n = \lambda_1 h|u_n|^{p-2}u_n + ag(u_n)w_n+ f$. Adding and subtracting$\lambda_1\int h|u_n|^{p-2}u_n\cdot v_n\mbox{ and }\int ag(u_n)v_n+\int f(x)v_n, and substituting \eqref{t1.5} in \eqref{t1.6} \label{t1.7} \begin{aligned} \|u_n\|^{p}-\lambda_1\int h|u_n|^p &\leqslant C\|w_n\|+\langle J'_{\lambda_1}(u_n),v_m\rangle+ \int ag(u_n)v_n+\int f(x)v_n\\ &\leqslant C\|w_n\|+\langle J'_{\lambda_1}(u_n),v_m\rangle+C'|\alpha_n| \end{aligned} Again, sinceJ'_{\lambda_1}\to 0\mbox{ as }n\to\infty$, there exist$m_1$such that if$n\geqslant m_1$then$\langle J'_{\lambda_1}(u_n),v_m\rangle \leqslant C\|v_n\|=C|\alpha_n|$, taking$n\geqslant\max\{m_0,m_1\}$$$\label{t1.8} \|u_n\|^{p}-\lambda_1\int h|u_n|^p\leqslant C\|w_n\|+C'|\alpha_n|.$$ Now fix$\gamma>0$, and suppose that$(u_n)\in \mathcal{C'}_\gamma$for all$n$. Then we have,$|\alpha_n|\leqslant (1/\gamma)\|w_n\|$and$\int h|u_n|^p\leqslant(1/\Lambda_{\gamma})\int|\nabla u_n|^p$. Thus, by Lemma \ref{Lambda} $$\label{t1.9} \big(1 -\frac{\lambda_1}{\Lambda_{\gamma}}\big)\|u_n\|^p\leqslant C\|w_n\|$$ Since the projection$u\mapsto w$is bounded in$\mathcal{D}^{1,p}$we obtain $$\label{t1.9'} \|w_n\|^p\leqslant C_\gamma\|w_n\|,$$ given that$\lambda_1/ \Lambda_\gamma<1$by Lemma \ref{Lambda}. Hence by Lemma \ref{Lambda},$\Lambda_\gamma>\lambda_1$; therefore,$(w_n)$is bounded if$(u_n)\in\mathcal{C'}_\gamma$. Now, set$\gamma_n =\|w_n\|/|\alpha_n|$and define $$\gamma \stackrel{\text{def}}{=}\liminf_n\gamma_n.$$ We have two cases: (i)$\gamma\in(0,\infty]$and (ii)$\gamma=0$. By the above argument, if$\gamma\in(0,\infty]$then$(w_n)$is bounded and the proof is concluded. If$\gamma=0$, take$\varepsilon>0$arbitrarily small, such that$\|w_n\|\leqslant\varepsilon|\alpha_n|$. Using inequality \eqref{t1.8}, Lemma \ref{Lambdatilde} with$\phi=\phi_n \stackrel{\text{def}}{=}(\|w_n\|/|\alpha_n|)\cdot w_n/\|w_n\|$, and the fact that the projection$u\mapsto \alpha$is bounded in$\mathcal{D}^{1,p}$we obtain \begin{gather*} |u_n|^p\big( 1-\frac{\lambda_1}{\tilde{\Lambda}} \big) \leqslant C\varepsilon|\alpha_n|+C'\|v_n\|,\\ |\alpha_n|^p\leqslant c_\gamma|\alpha_n|. \end{gather*} Therefore,$|\alpha_n|$is bounded, and since$\|w_n\|\leqslant\varepsilon|\alpha_n|$we have that$(u_n)$is bounded as wanted. To verify the geometric hypotheses of the Saddle Point Theorem we note that since$\lambda_1$is isolated (see \cite{All}) we have $$\label{g1} \lambda_2\stackrel{\text{def}}{=}\inf\big\{\|w\|^p:w\in W,\int h|w|^p=1 \big\},$$ which satisfies$\lambda_1<\lambda_2$. As a consequence of \eqref{g1} we have $$\label{g2} \int|\nabla w|^p\geqslant\lambda_2\int h|w|^p,\quad \forall w\in W.$$ Now, if$w\in W$, $$\label{g3} \int|\nabla w|^p-\lambda_1\int h|w|^p\geqslant \big(1-\frac{\lambda_1}{\lambda_2}\big).$$ Moreover, since$|g(s)|\leqslant M$for all$s\in\mathbb{R}$, we have that for all$w\in\mathcal{D}^{1,p}$, $$\big|\int aG(w) \big|\leqslant M\int |a||w|\leqslant C\|w\|.$$ Therefore,$J_{\lambda_1}$is bounded from below on$W$; i.e. (I2) in Theorem \ref{sadd} holds. Finally, if$v\in V$we have $$J_{\lambda_1}(v)=-\int aG(v)-\int fv.$$ Since$\int aG(v)+\int fv\to\infty$as$\|v\|\to\infty$by \eqref{alp} and, therefore, (I1) in the Saddle Point Theorem also holds. Hence,$J_{\lambda_1}$has a critical point and the proof is concluded. \end{proof} \noindent\textbf{Remark.} Suppose$\lim_{s\to\infty} g(s)=g_{\infty}$and$\lim_{s\to-\infty} g(s)=g_{-\infty}$exist. Then, if$g_{\infty}>0$and$g_{-\infty}<0$,$G(s)=\int^{s}_{0}g(t)dt\to\infty$as$|s|\to\infty$. Consequently, by L' H\^{o}spital's rule, the Lebesgue dominated convergence theorem and the fact that$\varphi_1>0$a.e. in$\mathbb{R}^N\$ we have that $$\lim_{|t|\to\infty}\frac{1}{t}\int a(x)G(t\varphi_1) =\lim_{|t|\to\infty}\int ag(t\varphi_1)\varphi_1 = \begin{cases} g_{\infty}\int a\varphi_1 &\mbox{as }t\to\infty,\\ g_{-\infty}\int a\varphi_1 &\mbox{as }t\to-\infty . \end{cases}$$ Thus, the condition \eqref{alp} in the resonance Theorem \ref{thm1} holds if $g_{\infty}\int a\varphi_1+\int f\varphi_1>0\quad \mbox{and}\quad g_{-\infty}\int a\varphi_1+\int f\varphi_1<0 ,$ or \label{LL} g_{-\infty}\int a\varphi_1<-\int f\varphi_1