\documentclass[reqno]{amsart} \usepackage{hyperref}%,backref %================= Usepackages \usepackage{amsmath, amssymb, latexsym, amscd, amsthm, amsfonts, amstext} \usepackage[mathscr]{eucal} \usepackage{xspace} %\usepackage{vnfonts} %================= Style of Theorem \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem*{main}{theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} \newtheorem*{example}{Examples} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} %\usepackage[notref,notcite]{showkeys} %============================= %\renewcommand\proofname{Proof of Theorem 1.4} \renewcommand\proofname{Proof} \renewcommand\theequation{\arabic{section}.\arabic{equation}} \setlength{\hfuzz}{2pt} % =============== Symbols \def\R{\mathbb R} %\renewcommand{\baselinestretch}{1.5} %============================ \begin{document} \title[...an elliptic equation of $p$-Laplacian type...]{On an elliptic equation of $p$-Laplacian type with nonlinear boundary condition} \author[N.T. Chung]{Nguyen Thanh Chung} \address{Nguyen Thanh Chung, \newline Department of Mathematics and Informatics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Vietnam} \email{\href{mailto: N.T. Chung }{chungnt.qbu@gmail.com}} \keywords{Elliptic equation, $p$-Laplacian type, $(p-1)$-sublinear, $(p-1)$-assymptotically linear, Nonlinear boundary condition.\\ \hspace*{.3cm} {\it 2000 Mathematics Subject Classifications}: 35J65, 35J20.} \date{November 20, 2010} \begin{abstract} We consider elliptic equations of $p$-Laplacian type with the nonlinear boundary condition of the form \begin{equation*} \begin{cases} \begin{array}{rlll} -\Delta_p u +|u|^{p-2}u & = & \lambda f_1(u) +\mu g_1(u)& \text{ in } \Omega,\\ |\nabla u|^{p-2}\frac{\partial u}{\partial n} &= & \lambda f_2(u)+\mu g_2(u) & \text{ in } \partial \Omega, \end{array} \end{cases} \end{equation*} where $\Omega \subset \R^N$ ($N \geqq 3$) is a bounded domain with smooth boundary $\partial\Omega$, $\frac{\partial}{\partial n}$ is the outer unit normal derivative, $\lambda, \mu$ are parameters. The functions $f_i$, $i=1, 2$, are assumed to be $(p-1)$-sublinear while $g_i$, $i=1, 2$, are $(p-1)$-assymptotically linear at infinity. Using variational techniques, an existence result is given. \end{abstract} \maketitle \numberwithin{equation}{section} \section{Introduction} \vspace{0.3cm} Consider the elliptic equation of $p$-Laplacian type with nonlinear boundary condition \begin{equation}\label{e1.1} \begin{cases} \begin{array}{rlll} -\Delta_p u +|u|^{p-2}u& = & \lambda f_1(u) +\mu g_1(u)& \text{ in } \Omega,\\ |\nabla u|^{p-2}\frac{\partial u}{\partial n} &= & \lambda f_2(u)+\mu g_2(u) & \text{ in } \partial \Omega, \end{array} \end{cases} \end{equation} where $\Omega \subset \R^N$ ($N \geqq 3$) is a bounded domain with smooth boundary $\partial\Omega$, $\frac{\partial}{\partial n}$ is the outer unit normal derivative, $1 < p < N$, $\lambda, \mu$ are parameters. Problem (\ref{e1.1}) has been studied in many works, such as \cite{AbrdoOMed,Bonder1, Bonder2,BonMarRos,BonRos,ZhaoZhao}, in which the authors have used different methods to obtain the existence of solutions. In a recent paper \cite{Chung}, we have considered the situation: $g_i \equiv 0$ $(i = 1, 2)$, $f_i$, $i=1, 2$, are $(p-1)$-sublinear at infinity. We then used the three critical point theorem of G. Bonanno \cite{Bonanno} to obtain a multiplicity result for (\ref{e1.1}). A natural question is to see what happens if the problem in \cite{Chung} is affected by a certain perturbation. For this purpose, in this note, we establish an existence result for (\ref{e1.1}) in the case when $f_i: \R \to \R$, $i = 1, 2$, are $(p-1)$-sublinear and $ g_i: \R \to \R$, $i=1, 2$, are $(p-1)$-assymptotically at infinity. The proof relies essentially on the minimum principle in \cite[Theorem 2.1]{AnaChaKarZer}. In order to state the main result of this work, we would introduce the following hypotheses \begin{enumerate} \item[$(\bf f)$] $f_i$, $i= 1, 2$ are continuous and $(p-1)$-sublinear at infinity, i.e., $$ \lim_{|t| \to \infty}\frac{|f_i(t)|}{|t|^{p-1}}=0; $$ \item[$(\bf g)$] $g_i$, $i= 1, 2$ are continuous and $(p-1)$-assymptotically at infinity, i.e., $$ \lim_{|t| \to \infty}\frac{|g_i(t)|}{|t|^{p-1}}=l_i< +\infty. $$ \end{enumerate} Let $W^{1,p}(\Omega)$ be the usual Sobolev space with respect to the norm $$ \|u\|_{1,p}^p = \int_\Omega (|\nabla u|^p + |u|^p) dx $$ and $W^{1,p}_0(\Omega)$ is the closure of $C^\infty_0(\Omega)$ in $W^{1,p}(\Omega)$. For any $1 < p < N$ and $1 \leqq q \leqq p^\star = \frac{Np}{N-p}$, we denote by $S_{q,\Omega}$ the best constant in the embedding $W^{1,p}(\Omega) \hookrightarrow L^q(\Omega)$ and for all $1 \leqq q \leqq p_\star = \frac{(N-1)p}{N-p}$, we also denote by $S_{q,\partial\Omega}$ the best constant in the embedding $W^{1,p}(\Omega) \hookrightarrow L^q(\partial\Omega)$, i.e. $$ S_{q,\partial\Omega} = \inf_{u \in W^{1,p}(\Omega)\backslash W^{1,p}_0(\Omega)} \frac{\int_\Omega (|\nabla u|^p + |u|^p) dx}{\left(\int_{\partial\Omega} |u|^qd\sigma\right)^{\frac{p}{q}}}. $$ Moreover, if $1 \leqq q < p^\star$, then the embedding $W^{1,p}(\Omega) \hookrightarrow L^q(\Omega)$ is compact and if $1 \leqq q < p_\star$, then the embedding $W^{1,p}(\Omega) \hookrightarrow L^q(\partial\Omega)$ is compact. As a consequence, we have the existence of extremals, i.e. functions where the infimum is attained (see \cite{Bonder1, BonRos}). \begin{definition}\label{dn1.1} A function $u \in W^{1,p}(\Omega)$ is said to be a weak solution of problem (\ref{e1.1}) if and only if \begin{multline*} \int_\Omega \Big[|\nabla u|^{p-2}\nabla u\cdot\nabla v+|u|^{p-2}uv\Big]dx-\lambda\int_\Omega f_1(u)vdx -\lambda\int_{\partial\Omega}f_2(u)vd\sigma\\ - \mu \int_\Omega g_1(u)vdx-\mu\int_{\partial\Omega}g_2(u)vd\sigma=0 \end{multline*} for all $v \in W^{1,p}(\Omega)$. \end{definition} \begin{theorem}\label{dl1.2} Assume conditions $(\bf f)$ and $(\bf g)$ are fulfilled. Moreover, there exists $s_0>0$ such that $$ F_1(s_0):=\int_0^{s_0}f_1(t)dt> 0 \text{ and } F_2(s_0):= \int_0^{s_0}f_2(t)dt >0. $$ Then for each $\lambda \in \R$ large enough, there exists $\overline\mu>0$, such that problem (\ref{e1.1}) has at least a non-trivial weak solution $u$ in $W^{1,p}(\Omega)$ for every $\mu \in (0,\overline\mu)$. \end{theorem} \section{Existence of solutions} For $\lambda, \mu \in \R$, let us define the functional $J_{\lambda,\mu}: W^{1,p}(\Omega) \to \R$ associated to problem (\ref{e1.1}) by the formula \begin{align}\label{2.1} \begin{split} J_{\lambda,\mu}(u) & = \frac{1}{p}\int_\Omega \Big[|\nabla u|^p+|u|^p\Big]dx-\lambda \int_\Omega F_1(u)dx - \lambda\int_{\partial\Omega}F_2(u)d\sigma \\ & \hspace{2cm} -\mu\int_\Omega G_1(u)dx -\mu\int_{\partial\Omega}G_2(u) d\sigma \\ & = \Lambda(u)-I_{\lambda,\mu}(u), \end{split} \end{align} where \begin{align}\label{2.2} \begin{split} \Lambda(u) & = \frac{1}{p}\int_\Omega \Big[|\nabla u|^p+|u|^p\Big]dx, \\ I_{\lambda,\mu}(u) & = \lambda\int_\Omega F_1(u)dx+\lambda\int_{\partial\Omega}F_2(u) d\sigma +\mu\int_\Omega G_1(u)dx +\mu\int_{\partial\Omega}G_2(u)d\sigma \end{split} \end{align} for all $u \in W^{1,p}(\Omega)$. Then, a simple computation shows that $J_{\lambda,\mu}$ is of $C^1$ class and \begin{multline*} DJ_{\lambda,\mu}(u)(v)=\int_\Omega \Big[|\nabla u|^{p-2}\nabla u\cdot\nabla v+|u|^{p-2}uv \Big]dx-\lambda\int_\Omega f_1(u)vdx -\lambda\int_{\partial\Omega}f_2(u)vd\sigma\\ - \mu \int_\Omega g_1(u)vdx-\mu\int_{\partial\Omega}g_2(u)vd\sigma=0 \end{multline*} for all $u,v \in W^{1,p}(\Omega)$. Thus, weak solutions of problem (\ref{e1.1}) are exactly the critical points of $J_{\lambda,\mu}$. \begin{lemma}\label{bd2.1} For every $\lambda \in \R$, there exists $\overline\mu>0$, depending on $\lambda$, such that for every $\mu \in (0, \overline\mu)$, the functional $J_{\lambda,\mu}$ is coercive. \end{lemma} \begin{proof} Firstly, we have $$ S_{p,\Omega}\|u\|_{L^p(\Omega)} \leqq \|u\|_{1,p} \text{ and } S_{p,\partial\Omega}\|u\|_{L^p( \partial\Omega)} \leqq \|u\|_{1,p} $$ for all $u \in W^{1,p}(\Omega)$. Let us fix $\lambda \in \R$, arbitrary. By $(\bf f)$, there exist $\delta_i = \delta_i(\lambda)$, $ i=1,2$, such that $$ |f_1(t)| \leqq S^p_{p,\Omega} \frac{1}{2(1+|\lambda|)}|t|^{p-1}, \quad \forall |t| \geqq \delta_1 $$ and $$ |f_2(t)| \leqq S^p_{p,\partial\Omega} \frac{1}{2(1+|\lambda|)}|t|^{p-1}, \quad \forall |t| \geqq \delta_2. $$ Integrating the above inequalities, we have \begin{equation}\label{e2.3} |F_1(t)| \leqq S^p_{p,\Omega} \frac{1}{2p(1+|\lambda|)}|t|^p +\max_{|s|\leqq \delta_1}|f_1(s)||t|, \quad \forall t \in \R \end{equation} and \begin{equation}\label{e2.4} |F_2(t)| \leqq S^p_{p,\partial\Omega} \frac{1}{2p(1+|\lambda|)}|t|^p +\max_{|s|\leqq \delta_2}|f_2(s)||t|, \quad \forall t \in \R. \end{equation} Since $g_i$, $i=1,2$ are $(p-1)$-asymptotically linear at infinity, there exist two constants $m_i >0$, $i = 1, 2$, such that $$ |g_1(t)| \leqq m_1pS_{p,\Omega}^p |t|^{p-1}+m_1, $$ $$ |g_2(t)| \leqq m_2pS_{p,\partial\Omega}^p |t|^{p-1}+m_2 $$ for all $t \in \R$. It implies that \begin{equation}\label{e2.5} |G_1(t)| \leqq m_1S_{p,\Omega}^p |t|^p+m_1|t|, \end{equation} and \begin{equation}\label{e2.6} |G_2(t)| \leqq m_2S_{p,\partial\Omega}^p |t|^p+m_2|t| \end{equation} for all $t \in \R$. Hence, for any $u \in W^{1,p}(\Omega)$, we deduce that \begin{align*} \begin{split} J_{\lambda,\mu}(u) &\geqq \Lambda(u) - |I_{\lambda,\mu}(u)| \\ & \geqq \frac{1}{p}\|u\|^p_{1,p}- \frac{|\lambda|}{2p(1+|\lambda|)}\|u\|^p_{1,p}-\frac{|\lambda|}{S_{ p,\Omega}}|\Omega|^\frac{1}{p'}_N\|u\|_{1,p}\max_{|s|\leqq \delta_1}|f_1(s)| \\ & \hspace{1cm} - \frac{|\lambda|}{2p(1+|\lambda|)}\|u\|^p_{1,p} -\frac{|\lambda|}{S_{p,\partial \Omega}}|\partial\Omega|^\frac{1}{p'}_{N-1}\|u\|_{1,p}\max_{|s|\leqq \delta_1}|f_2(s)| \\ & \hspace{1cm} - |\mu|m_1\|u\|_{1,p}^p-m_1\frac{|\mu|}{S_{p,\Omega}}|\Omega|^\frac{1}{p'}_N \|u\|_{1,p} \\ & \hspace{1cm} - |\mu|m_2\|u\|_{1,p}^p-m_2\frac{|\mu|}{S_{p,\partial\Omega}}|\partial\Omega |^\frac{1}{p'}_{N-1}\|u\|_{1,p} \\ & = \Big(\frac{1}{p(1+|\lambda| )}-|\mu|(m_1+m_2)\Big)\|u\|^p_{1,p}-\frac{|\lambda|}{S_{ p,\Omega}}|\Omega|^\frac{1}{p'}_N\|u\|_{1,p}\max_{|s|\leqq \delta_1}|f_1(s)| \\ & \hspace{1cm} -\frac{|\lambda|}{S_{p,\partial\Omega}}|\partial\Omega|^\frac{1}{p'}_{N-1}\|u\|_{ 1,p}\max_{|s|\leqq \delta_1}|f_2(s)| -m_1\frac{|\mu|}{S_{p,\Omega}}|\Omega|^\frac{1}{p'}_N\|u \|_{1,p} \\ & \hspace{1cm}-m_2\frac{|\mu|}{S_{p,\partial\Omega}}|\partial\Omega|^\frac{1}{p'}_{N-1}\|u\|_{ 1,p}, \end{split} \end{align*} where $p'= \frac{p}{p-1}$. Let $\overline\mu = \frac{1}{p(m_1+m_2)(1+|\lambda|)}$ and fix $\mu \in (0,\overline \mu)$. Since $p>1$ we have $J_{\lambda,\mu}(u) \to +\infty$ as $\|u\|_{1,p}\to \infty$. Thus, the functional $J_{\lambda,\mu}$ is coercive. \end{proof} \begin{lemma}\label{bd2.2} Let $\lambda$ and $\overline\mu$ be chosen as in the previous lemma. Then for each $\mu \in (0,\overline\mu)$, the functional $J_{\lambda,\mu}$ satisfies the Palais-Smale condition. \end{lemma} \begin{proof} Let $\{u_m\}$ be a sequence in $W^{1,p}(\Omega)$ such that \begin{equation}\label{e2.7} J_{\lambda,\mu}(u_m) \to \overline c, \quad DJ_{\lambda,\mu}(u_m) \to 0 \text{ in } W^{-1,p}( \Omega) \text{ as } m\to \infty. \end{equation} Since the functional $J_{\lambda,\mu}$ is coercive, the sequence $\{u_m\}$ is bounded in $W^{1,p}(\Omega)$. Then, there exist a subsequence still denoted by $\{u_m\}$ and a function $u \in W^{1,p}(\Omega)$, such that $\{u_m\}$ converges weakly to $u$ in $W^{1,p} (\Omega)$. Hence, $\{\|u_m-u\|_{1,p}\}$ is bounded and by (\ref{e2.7}), $DJ_{\lambda,\mu} (u_m)(u_m-u)$ converges to $0$ as $m \to \infty$. By $(\bf f)$, there exists a constant $C_1>0$ such that $$ |f_i(t)| \leqq C_1(1+|t|^{p-1}), \quad i = 1, 2 $$ for all $t \in \R$. Therefore, \begin{align*} \begin{split} 0 \leqq \int_\Omega|f_1(u_m)||u_m-u|dx & \leqq C_1\int_\Omega |u_m-u|dx+C\int_\Omega|u_m |^{p-1}|u_m-u|dx \\ & \leqq C_1\Big[|\Omega|^\frac{1}{p'}_N+\|u_m\|^{p-1}_{L^p(\Omega)}\Big]\|u_m-u\|_{L^p(\Omega)} \end{split} \end{align*} and \begin{align*} \begin{split} 0 \leqq \int_{\partial\Omega}|f_2(u_m)||u_m-u|dx & \leqq C_1\int_{\partial\Omega} |u_m-u|dx+ C_1\int_{\partial\Omega}|u_m|^{p-1}|u_m-u|dx \\ & \leqq C_1\Big[|\partial\Omega|^\frac{1}{p'}_{N-1}+\|u_m\|^{p-1}_{L^p(\partial\Omega)}\Big] \|u_m-u\|_{L^p(\partial\Omega)}. \end{split} \end{align*} Since $\{u_m\}$ converges strongly to $u$ in the spaces $L^p(\Omega)$ and $L^p(\partial \Omega)$, the above inequalities imply that \begin{equation}\label{e2.8} \lim_{m\to \infty}\int_\Omega f_1(u_m)(u_m-u)dx = 0 \end{equation} and \begin{equation}\label{e2.9} \lim_{m\to \infty}\int_{\partial\Omega} f_2(u_m)(u_m-u)dx = 0. \end{equation} On the other hand, by $(\bf g)$, there exists a constant $C_2>0$ such that $$ |g_i(t)| \leqq C_2(1+|t|^{p-1}), \quad i = 1, 2 $$ for all $t \in \R$. Therefore, the similar arguments above show that \begin{equation}\label{e2.10} \lim_{m\to \infty}\int_\Omega g_1(u_m)(u_m-u)dx = 0 \end{equation} and \begin{equation}\label{e2.11} \lim_{m\to \infty}\int_{\partial\Omega} g_2(u_m)(u_m-u)dx = 0. \end{equation} By relations (\ref{e2.8})-(\ref{e2.11}), we get \begin{equation}\label{e2.12} \lim_{m\to \infty}DI_{\lambda,\mu}(u_m)(u_m-u) = 0. \end{equation} Combining (\ref{e2.11}) and (\ref{e2.7}), it follows that \begin{equation}\label{e2.13} \lim_{m\to \infty}\Lambda(u_m)(u_m-u) = 0. \end{equation} Hence, standard arguments help us to show that the sequence $\{u_m\}$ converges strongly to $u$ in $W^{1,p}(\Omega)$. Thus, the functional $J_{\lambda,\mu}$ satisfies the Palais-Smale condition in $W^{1,p}(\Omega)$. \end{proof} \begin{proof}[Proof Theorem \ref{dl1.2}] By Lemmas \ref{bd2.1} and \ref{bd2.2}, using the minimum principle \cite[Theorem 2.1]{AnaChaKarZer}, we deduce that for each $\lambda \in \R$, there exists $\overline\mu > 0$, such that for any $\mu \in (0,\overline\mu)$, problem (\ref{e1.1}) has a weak solution $u \in W^{1,p}(\Omega)$. We will show that $u$ is not trivial for $\lambda$ large enough. Indeed, let $s_0$ be a real number such that $$ F_1(s_0):=\int_0^{s_0}f_1(t)dt> 0 \text{ and } F_2(s_0):= \int_0^{s_0}f_2(t)dt >0 $$ and let $\Omega_0\subset \Omega$ be an open subset with $|\Omega_0|_N>0$. Then, there exists $u_0 \in C^\infty_0(\Omega)$ such that $u_0(x) \equiv s_0$ on $\overline\Omega_0$ and $0 \leqq u_0(x) \leqq s_0$ in $\Omega \backslash \Omega_0$. We have \begin{align*} \begin{split} J_{\lambda,\mu}(u_0) & = \frac{1}{p}\int_\Omega \Big[|\nabla u_0|^p+|u_0|^p\Big]dx-\lambda \int_\Omega F_1(u_0)dx-\lambda\int_{\partial\Omega}F_2(u_0)dx \\ & \hspace{2cm} -\mu\int_\Omega G_1(u_0)dx-\mu\int_{\partial\Omega}G_2 (u_0)dx \\ & \leqq \frac{1}{p}\int_\Omega \Big[|\nabla u_0|^p+|u_0|^p\Big]dx-\lambda\int_{\Omega_0} F_1(u_0)dx-\lambda\int_{\partial\Omega_0}F_2(u_0)dx \\ & \hspace{2cm} -\mu\int_{\Omega_0} G_1(u_0)dx- \mu\int_{\partial\Omega_0} G_2(u_0)dx \\ & = C-\lambda\Big(F_1(s_0)|\Omega_0|_N+F_2(s_0)|\Omega_0|_{N-1}\Big), \end{split} \end{align*} where $C$ is a positive constant ($C$ depends on $\mu$). Therefore, for $\lambda>0$ large enough, we have $J_{\lambda,\mu}(u_0) < 0$. Thus, the solution $u$ is not trivial. The proof of Theorem \ref{dl1.2} is now completed. \end{proof} \bibliographystyle{amsplain} \begin{thebibliography}{99} \bibitem{AbrdoOMed}\textsc{E. A. M. Abreu, J. M. do \'{O} and E.S. 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