\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 71, 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} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/71\hfil Positive solutions] {Existence of positive solutions for quasilinear elliptic systems involving the $p$-Laplacian} \author[X. Shang, J. Zhang\hfil EJDE-2009/71\hfilneg] {Xudong Shang, Jihui Zhang} % in alphabetical order \address{Xudong Shang \newline Department of Mathematics, Nanjing Normal University Taizhou College, Taizhou 225300, Jiangsu, China} \email{xudong-shang@163.com} \address{Jihui Zhang \newline Institute of Mathematics, School of Mathematics and Computer Sciences, Nanjing Normal University, Nanjing 210097, Jiangsu, China} \email{jihuiz@jlonline.com} \thanks{Submitted March 6, 2009. Published June 1, 2009.} \subjclass[2000]{34A34, 34B18} \keywords{$p$-Laplacian system; positive solutions; operator of type $(S)_+$} \begin{abstract} In this article, we study the existence of positive solutions for the quasilinear elliptic system \begin{gather*} -\Delta_{p}u = f(x,u,v) \quad x \in \Omega ,\\ -\Delta_{p}v = g(x,u,v) \quad x \in \Omega ,\\ u = v = 0 \quad x \in \partial\Omega. \end{gather*} Using degree theoretic arguments based on the degree map for operators of type $(S)_{+}$, under suitable assumptions on the nonlinearities, we prove the existence of positive weak solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \section{Introduction and main result} In this paper we study the existence of positive solution for the nonlinear elliptic system $$\label{e1.1} \begin{gathered} -\Delta_{p}u = f(x,u,v) \quad x \in \Omega ,\\ -\Delta_{p}v = g(x,u,v) \quad x \in \Omega ,\\ u = v = 0 \quad x \in \partial\Omega \end{gathered}$$ where $\Delta_{p}u = \text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian operator, and $\Omega$ is a smooth bounded region in $\mathbb{R}^N$ for $N\geq 1$, $p> 1$. Elliptic systems have several practical applications. For example they can describe the multiplicative chemical reaction catalyzed by grains under constant or variant temperature, a correspondence of the stable station of dynamical system determined by the reaction-diffusion system. In recent years, many publications have appeared concerning quasilinear elliptic systems which have been used in a great variety of applications, we refer the readers to \cite{zhang04, Djellit07, Djellit04, Drabek03, Hai07, Djellit03} and the references therein. Existence and multiplity results for quasilinear elliptic systems with variational have been broadly investigate. Djellit and Tas \cite{Djellit07} studied a class of quasilinear elliptic systems involving the $p$-Laplacian operator, the right hand sides of systems being closed related to the critical Sobolev exponents. Under some additional assumptions on the nonlinearities, they proved the existence of at least one nontrivial solution. In \cite{ Drabek03}, the authors studied the existence and multiplicity of non-negative solutions for the quasilinear elliptic system in both bounded and unbounded domain in $\mathbb{R}^N$. Zhang \cite{zhang04} used the Leray-Schauder degree to obtain a positive solution of the nonlinear elliptic system. In this work, we show the existence of positive solutions for system \eqref{e1.1}. Using the degree theory for $(S)_{+}$ operator initiated by Browder \cite{Browder83}. Our main goal in this article is to extend the main result of \cite{zhang04} to the quasilinear case. The relevant studies about $(S)_{+}$ operators can be found in \cite{Lu05, Aman82, Zeidler90}. Through this paper, $(u,v) \in \mathbb{R}^2$. As to the nonlinearities $f,g$, we assume that they are Caratheodory functions satisfying the following growth conditions: \noindent (i) There exist $a_{i}\geq 0, c_{i} \geq 0$$(i=1,2) such that \begin{gather*} 0\leq f(x,u,v) \leq a_{1}|(u,v)|^{q-1} + c_{1}, \\ 0\leq g(x,u,v) \leq a_{2}|(u,v)|^{q-1} + c_{2} \end{gather*} where 1 0, c_{3}>0, 1 0 is satisfied. Then$$ \deg (A,B_{r}(0),0) = 1.$$\end{lemma} In this paper, we denote by$Z$the product space$W_{0}^{1,p}(\Omega)\times W_{0}^{1,p}(\Omega)$. The space$Z$will be endowed with the norm $\|z\|_{Z}^{p} = \|u\|_{W_{0}^{1,p}(\Omega)}^{p}+\|v\|_{W_{0}^{1,p}(\Omega)}^{p}, \hskip 0.5cm z=(u,v)\in Z,$ where$\|u\|_{W_{0}^{1,p}(\Omega)} = (\int_{\Omega}|\nabla u|^{p}dx)^{1/p}$. As usual, a weak solution of system \eqref{e1.1} is any$(u,v)\in Z$such that $\int_{\Omega} |\nabla u|^{p-2}\nabla u\nabla\xi dx + \int_{\Omega} |\nabla v|^{p-2}\nabla v\nabla\zeta dx - \int_{\Omega} f(x,u,v)\xi dx - \int_{\Omega} g(x,u,v)\zeta dx = 0$ for every$(\xi,\zeta)\in Z$. Next let us introduce the functionals$I_{i},F_{i}: Z\to Z^{*}(i=1,2)$as follows: \begin{gather*} \langle I_{1}(u,v),(\xi,\zeta) \rangle = \int_{\Omega} |\nabla u|^{p-2}\nabla u\nabla\xi dx, \\ \langle I_{2}(u,v),(\xi,\zeta) \rangle = \int_{\Omega} |\nabla v|^{p-2}\nabla v\nabla\zeta dx, \\ \langle F_{1}(u,v),(\xi,\zeta) \rangle = \int_{\Omega} f(x,u,v)\xi dx , \\ \langle F_{2}(u,v),(\xi,\zeta) \rangle = \int_{\Omega} g(x,u,v)\zeta dx . \end{gather*} Define the operator $A = I_{1} + I_{2} -F_{1}- F_{2}.$ \begin{lemma} \label{lemma2.4} The mapping$B = I_{1} + I_{2}$is of type$(S)_{+}$. \end{lemma} \begin{proof} Assume that$(u_{n},v_{n})\rightharpoonup (u_{0},v_{0})$in$Z$and $\limsup_{n\to +\infty} \langle B(u_{n},v_{n}), (u_{n}-u_{0},v_{n}-v_{0})\rangle \leq0.$ From the weak convergence we have that $\limsup_{n\to +\infty} \langle B(u_{0},v_{0}), (u_{n}-u_{0},v_{n}-v_{0})\rangle = 0.$ Then we get $\limsup_{n\to +\infty} \langle B(u_{n},v_{n}) - B(u_{0},v_{0}), (u_{n}-u_{0},v_{n}-v_{0})\rangle \leq 0.$ By the monotonicity property of$B, we obtain $\limsup_{n\to +\infty} \langle B(u_{n},v_{n}) - B(u_{0},v_{0}), (u_{n}-u_{0},v_{n}-v_{0})\rangle = 0.$ This implies \begin{displaymath} \begin{aligned} &\lim_{n\to +\infty} \int_{\Omega}(|\nabla u_{n}|^{p-2}\nabla u_{n}- |\nabla u_{0}|^{p-2}\nabla u_{0})(\nabla u_{n}-\nabla u_{0})dx \\ &+ \int_{\Omega}(|\nabla v_{n}|^{p-2}\nabla v_{n}- |\nabla v_{0}|^{p-2}\nabla v_{0})(\nabla v_{n}-\nabla v_{0})dx = 0. \end{aligned} \end{displaymath} Observe that for allx,y\in \mathbb{R}^N, |x-y|^{p}\leq \begin{cases} (|x|^{p-2}x-|y|^{p-2}y)(x-y) & \text{if } p\geq 2, \\ [(|x|^{p-2}x-|y|^{p-2}y)(x-y)]^{p/2}(|x|+|y|)^{(2-p)p/2} & \text{if } 10 such that $$\label{e3.1} \deg (A,B_{R}^{K},0) = 0 \quad \text{for all } R\geq R_{0}.$$ Let $$\label{e3.2} \langle L(u,v),(\xi,\zeta)\rangle = \int_{\Omega}((k+\epsilon) u^{p-1}\xi + (k+\epsilon) v^{p-1}\zeta) dx, \quad \forall (\xi,\zeta) \in Z$$ where k is a real number, 0<\epsilon<\epsilon^{'}. Since L is a completely continuous operator, the homotopy H_{t}:[0,1]\times K\to Z^{*} defined by \begin{align*} \langle H_{t}(u,v),(\xi,\zeta) \rangle &= \int_{\Omega} (|\nabla u|^{p-2}\nabla u \nabla\xi + |\nabla v|^{p-2}\nabla v \nabla\zeta) dx \\ &\quad - (1-t)\int_{\Omega} (f(x,u,v)\xi + g(x,u,v)\zeta )dx \\ &\quad - t \int_{\Omega} ((k+\epsilon)u^{p-1}\xi + (k+\epsilon)v^{p-1}\zeta) dx, \end{align*} where the value of k will be fixed later. Clearly H_{t} is of type (S)_{+}. We claim that there exists R_{0} > 0 such that \[ H_{t}(u,v)\not= 0 \quad \text{for all } t\in[0,1],\; (u,v)\in\partial B_{R}^{K} \; R\geq R_{0}. Suppose that is not true. Then we can find sequences\{t_{n}\}\subset [0,1]$and$\{(u_{n},v_{n})\}\subset Z$such that$t_{n}\to t\in [0,1]$,$\|(u_{n},v_{n})\|_{Z}\to \inftyand \label{e3.3} \begin{aligned} &\int_{\Omega} (|\nabla u_{n}|^{p-2}\nabla u_{n} \nabla\xi + |\nabla v_{n}|^{p-2}\nabla v_{n} \nabla\zeta) dx \\ &= t_{n} \int_{\Omega} ((k+\epsilon)u_{n}^{p-1}\xi + (k+\epsilon)v_{n}^{p-1}\zeta) dx \\ &\quad + (1-t_{n})\int_{\Omega} (f(x,u_{n},v_{n})\xi + g(x,u_{n},v_{n})\zeta )dx \end{aligned} for every(\xi,\zeta) \in Z$. Put$(\omega_{n},\psi_{n}) = \frac{(u_{n},v_{n})}{\|(u_{n},v_{n})\|_{Z}}$, we may assume that there exists$(\omega_{0},\psi_{0})\in Z$satisfying \begin{gather*} (\omega_{n},\psi_{n})\rightharpoonup (\omega_{0},\psi_{0}) \quad \text{in } Z,\\ (\omega_{n},\psi_{n})\to (\omega_{0},\psi_{0}) \quad \text{in } L^{p}(\Omega)\times L^{p}(\Omega), \\ (\omega_{n},\psi_{n})\to (\omega_{0},\psi_{0}) \quad \text{a.e. } x\in \Omega. \end{gather*} Applying the test function$(\omega_{n} - \omega_{0},\psi_{n} - \psi_{0})\in Z in \eqref{e3.3}, we find \label{e3.4} \begin{aligned} &\int_{\Omega} (|\nabla \omega_{n}|^{p-2}\nabla \omega_{n} \nabla(\omega_{n}- \omega_{0})+ |\nabla \psi_{n}|^{p-2}\nabla \psi_{n} \nabla(\psi_{n}- \psi_{0}))dx \\ &=(1-t_{n})\int_{\Omega}(\frac{f(x,u_{n},v_{n})}{\|(u_{n},v_{n})\|_{Z}^{p-1}}(\omega_{n}- \omega_{0}) + \frac{g(x,u_{n},v_{n})}{\|(u_{n},v_{n})\|_{Z}^{p-1}}(\psi_{n}- \psi_{0}))dx \\ &\quad + t_{n} \int_{\Omega} ((k+\epsilon)\omega_{n}^{p-1}(\omega_{n}- \omega_{0}) + (k+\epsilon)\psi_{n}^{p-1}(\psi_{n}- \psi_{0})) dx\,. \end{aligned} We know that \begin{gather*} (1-t_{n})\int_{\Omega}(\frac{f(x,u_{n},v_{n})}{\|(u_{n},v_{n}) \|_{Z}^{p-1}}(\omega_{n}- \omega_{0}) + \frac{g(x,u_{n},v_{n})}{\|(u_{n},v_{n})\|_{Z}^{p-1}}(\psi_{n}- \psi_{0}))dx \to 0 \quad n\to\infty, \\ t_{n} \int_{\Omega} ((k+\epsilon)\omega_{n}^{p-1}(\omega_{n}- \omega_{0}) + (k+\epsilon)\psi_{n}^{p-1}(\psi_{n}- \psi_{0})) dx\to 0\quad n\to\infty. \end{gather*} By \eqref{e3.4}, we have $\lim_{n\to +\infty} \int_{\Omega} (|\nabla \omega_{n}|^{p-2}\nabla \omega_{n} \nabla(\omega_{n}- \omega_{0})+ |\nabla \psi_{n}|^{p-2}\nabla \psi_{n} \nabla(\psi_{n}- \psi_{0})) dx = 0;$ i.e., $\lim_{n\to +\infty} \langle B(\omega_{n},\psi_{n}), (\omega_{n},\psi_{n})-(\omega_{0},\psi_{0})\rangle = 0 .$ According to Lemma \ref{lemma2.4} we obtain(\omega_{n},\psi_{n})\to(\omega_{0},\psi_{0})$in$Z$as$n\to\infty$and$\|(\omega_{0},\psi_{0})\|=1$. This shows that$(\omega_{0},\psi_{0})\not= (0,0)$. Let now apply the test function$(\omega_{0},\psi_{0})\in Zin \eqref{e3.4}, we get \label{e3.5} \begin{aligned} &\int_{\Omega} (|\nabla \omega_{n}|^{p-2}\nabla \omega_{n} \nabla \omega_{0}+ |\nabla \psi_{n}|^{p-2}\nabla \psi_{n} \nabla\psi_{0}) dx \\ &=(1-t_{n})\int_{\Omega}(\frac{f(x,u_{n},v_{n})}{\|(u_{n},v_{n})\|_{Z}^{p-1}} \omega_{0} + \frac{g(x,u_{n},v_{n})}{\|(u_{n},v_{n})\|_{Z}^{p-1}} \psi_{0})dx \\ &\quad + t_{n} \int_{\Omega} ((k+\epsilon)\omega_{n}^{p-1}\omega_{0} + (k+\epsilon)\psi_{n}^{p-1}\psi_{0}) dx \end{aligned} Passing to the limit asn\to\infty$, using Fatou's lemma and hypothesis$(\text{iii})$, we obtain $$\label{e3.6} \int_{\Omega} (|\nabla \omega_{0}|^{p} + |\nabla \psi_{0}|^{p})dx \geq (k + \epsilon) \int_{\Omega} (\omega_{0}^{p} + \psi_{0}^{p}) dx.$$ Now we take$k=\frac{\|(\omega_{0}, \psi_{0})\|_{Z}^{p}}{\|(\omega_{0}, \psi_{0})\|_{L^{p}\times L^{p}}^{p}}$, by \eqref{e3.6} we have $k+\epsilon \leq \frac{\int_{\Omega}(|\nabla \omega_{0}|^{p} + |\nabla \psi_{0}|^{p})dx}{\int_{\Omega}(\omega_{0}^{p} + \psi_{0}^{p})dx} = k.$ This contradiction shows the claim. Using the homotopy invariance of the degree map, which through the homotopy$H_{t}$yields $\deg (A,B_{R}^{K},0) = \deg (H_{1},B_{R}^{K},0) \quad \text{for all } R\geq R_{0}.$ Now we computing$\deg (H_{1},B_{R}^{K},0)$. Let the homotopy$H_{t}': [0,1]\times k\to Z^{*}be defined by \begin{align*} \langle H_{t}'(u,v),(\xi,\zeta)\rangle &= \int_{\Omega} (|\nabla u|^{p-2}\nabla u \nabla\xi + |\nabla v|^{p-2}\nabla v \nabla\zeta) dx \\ &\quad - t \int_{\Omega} (m(x)\xi + m(x)\zeta) dx -\int_{\Omega}((k +\epsilon )u^{p-1}\xi + (k +\epsilon )v^{p-1}\zeta) dx \end{align*} for all(\xi,\zeta) \in Z$,$t\in [0,1]$,$m(x)\in L_{+}^{\infty}(\Omega)=\{u(x)\in L^{\infty}(\Omega)|u(x)\geq0,\forall x\in\Omega\}$. Clearly, it is a$(S)_{+}$homotopy. So we have $\deg (H_{1},B_{R}^{K},0) = \deg (H_{t}',B_{R}^{K},0).$ Similarly, we prove the claim concerning the homotopy$H_{t}'$. By the homotopy invariance of the degree map, we have $\deg (H_{1},B_{R}^{K},0) = \deg (H_{1}',B_{R}^{K},0).$ Next, we show that$\deg (H_{1}',B_{R}^{K},0)= 0$. If$\deg (H_{1}',B_{R}^{K},0)\not= 0$, there exits$(u,v)\in B_{R}^{K}such that \begin{align*} & \int_{\Omega} (|\nabla u|^{p-2}\nabla u \nabla\xi + |\nabla v|^{p-2}\nabla v \nabla\zeta) dx \\ &= \int_{\Omega} (m(x)\xi + m(x)\zeta) dx +\int_{\Omega}((k +\epsilon )u^{p-1}\xi + (k +\epsilon )v^{p-1}\zeta) dx. \end{align*} Clearly(u,v)\not = (0,0)$, let$(\xi,\zeta) = (u,v)$, then $\int_{\Omega} (|\nabla u|^{p} dx + |\nabla v|^{p} dx) \geq (k+\epsilon)\int_{\Omega}(u^{p}+v^{p}) dx,$ We take$k=\frac{\|(u, v)\|_{Z}^{p}}{\|(u, v)\|_{L^{p}\times L^{p}}^{p}}$, which provides a contradiction. Therefore $\deg (A,B_{R}^{K},0) = \deg (H_{1}',B_{R}^{K},0) =0.$ So \eqref{e3.1} holds. Then, note that $\lambda_{1} = \inf_{u\in W_{0}^{1,p}(\Omega)\backslash \{0\}} \frac{\int_{\Omega}|\nabla u|^{p}dx}{\int_{\Omega}|u|^{p}dx},$ we have$\|u\|_{W_{0}^{1,p}(\Omega)}^{p} \geq \lambda_{1}\|u\|_{L^{p}(\Omega)}^{p}. By(ii), we get \begin{align*} \langle A(u,v),(u,v)\rangle &= \|(u,v)\|_{Z}^{p}-\int_{\Omega}(f(x,u,v)u + g(x,u,v)v) dx\\ &\geq \frac{\epsilon'}{\lambda_{1}}(\|u\|^{p}+\|v\|^{p})- c_{4}(\|u\|^{\theta}+\|v\|^{\theta}), \end{align*} wherec_{4} > 0$. Since$\theta > p$, there exist$r > 0$such that $\langle A(u,v),(u,v)\rangle > 0,$ for all$(u,v) \in \partial B_{r}^{K}$, where$B_{r}^{K} = \{(u,v)\in K| \|(u,v)\|_{Z} < r\}$. In view of Lemma \ref{lemma2.3}, there exists sufficiently small$r>0$such that $\deg (A,B_{r}^{K} ,0) = 1.$ According to the \eqref{e3.1}, we can take$R > r$such that $\deg (A,B_{R}^{K},0) = 0.$ Since the domain additivity of type$(S)_{+}$, we obtain $\deg (A,B_{R}^{K}\backslash B_{r}^{K},0) = -1.$ So we are led to the existence of$(u,v)\in B_{r,R}^{K} = \{(u,v)\in K| r<\|(u,v)\|_{Z} < R\}$such that$A(u,v) = 0$. Hence, system\eqref{e1.1} has a positive solution. \end{proof} \subsection*{Acknowledgments} The authors would like to thank the anonymous referee for his/her valuable suggestions. \begin{thebibliography}{00} \bibitem{zhang04} J. Zhang; \emph{Existence results for the positive solutions of nonlinear elliptic systems}, Appl. Math. Com. 153 (2004) 833-842. \bibitem{Djellit07} A. Djellit, S. Tas; \emph{Quasilinear elliptic systems with critical Sobolev exponents in$\mathbb{R}^N$}, Nonlinear Anal. 66 (2007) 1485-1497. \bibitem{Djellit04} A. Djellit, S. 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