\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 84, pp. 1--6.\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-2006/84\hfil Existence and uniqueness results] {Existence and uniqueness results of positive solutions for nonvariational quasilinear elliptic systems} \author[ D. A. Kandilakis, N. E. Sidiropoulos\hfil EJDE-2006/84\hfilneg] {Dimitrios A. Kandilakis, Nikolaos E. Sidiropoulos} % in alphabetical order \address{Dimitrios A. Kandilakis \newline Department of Sciences, Technical University of Crete, 73100 Chania, Greece} \email{dkan@science.tuc.gr} \address{Nikolaos E. Sidiropoulos \newline Department of Sciences, Technical University of Crete, 73100 Chania, Greece} \email{nsid@science.tuc.gr} \date{} \thanks{Submitted April 6, 2006. Published July 27, 2006.} \subjclass[2000]{35B50, 35D05, 35J55} \keywords{Quasilinear elliptic system; nonvariational system; \hfill\break\indent maximum principle; uniqueness; positive solution} \begin{abstract} We provide conditions for the existence and uniqueness of positive solutions to the quasilinear elliptic system \begin{gather*} -\Delta _{p}u=f(x,u,v) \\ -\Delta _{q}v=g(x,u,v) \end{gather*} with Dirichlet boundary conditions on a bounded domain $\Omega \subseteq \mathbb{R}^N$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \section{Introduction} The aim of this paper is to provide existence and uniqueness results for positive solutions of the following weakly coupled quasilinear eliptic system with homogeneous Dirichlet data $$\begin{gathered} -\Delta _{p}u=f(x,u,v) \quad \text{in }\Omega \\ -\Delta _{q}v=g(x,u,v) \quad \text{in }\Omega \\ u=v=0 \quad \text{on }\partial \Omega \end{gathered} \label{q}$$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with a smooth boundary $\partial \Omega$ and $f,g:\overline{\Omega }\times [ 0,\infty )\times [ 0,\infty )\to [0,\infty )$ are continuous functions. As usual, for $s>1$, \begin{equation*} -\Delta _{s}u:=\mathop{\rm div}(|\nabla u|^{s-2}\nabla u) \end{equation*} denotes the $s$-Laplace operator. Elliptic equations involving the $s$-Laplace operator arise in some physical models like the flow of non-Newtonian fluids: pseudo-plastic fluids correspond to $s\in (1,2)$ while dilatant fluids correspond to $s>2$. The case $s=2$ expresses Newtonian fluids \cite{atk-ali}. On the other hand, quasilinear systems like (\ref{q}) describe various nonlinear phenomena such as chemical reactions, pattern formation, population evolution where, for example, $u$ and $v$ represent the concentrations of two species in the process. As a consequence, positive solutions of (\ref{q}) are of interest. Several methods have been used to treat quasilinear equations and systems. In the scalar case, weak solutions can be obtained through variational methods which provide critical points of the corresponding energy functional, an approach which is also fruitful in the case of potential systems i.e, the nonlinearities on the right hand side are the gradient of a $C^{1}-$functional \cite{amb-az-per}, \cite{fi}, \cite{sta-zog} . However, due to the loss of the variational structure, the treatment of nonvariational systems like (\ref{q}) is more complicated and is based mostly on topological methods \cite{alv-fi}. Recently, there have been significant studies of (\ref{q}). Dalmasso \cite{dal} provided existence and uniqueness results for positive solutions in the semilinear case $p=q=2$ with the assumption that $f$ is a function of $v$ and $g$ is a function of $u$, that is, (\ref{q}) is the Lane-Emden system. Existence results in the case $f$ and $g$ are monomials of $u$ and $v$ are also provided in \cite{cle-fle-the}, while the quasilinear Lane-Emden system was studied by Hai \cite{hai}. In this paper we adopt the method in \cite{hai} to complement and extend corresponding results in the aforementioned papers. \section{Main Results} We make the following assumptions: \begin{itemize} \item[(H1)] $f,g:\overline{\Omega }\times [ 0,\infty )\times [ 0,\infty )\to [0,\infty )$ are continuous functions such that \\ (i) $u\to f(x,u,v)$ and $u\to g(x,u,v)$ are nondecreasing for every $x\in$ $\overline{\Omega }$ and $v\geq 0$. \\ (ii) $v\to f(x,u,v)$ and $v\to g(x,u,v)$ are nondecreasing for every $x\in$ $\overline{\Omega}$ and $u\geq 0$. \item[(H2)] For each $a>0$, \begin{equation*} \limsup_{z\to 0^{+}} \frac{h^{\frac{1}{p-1}}(z,ak^{\frac{1}{q-1}}(z,z))}{z} =\infty , \end{equation*} where $h(u,v):=\min_{x\in \overline{\Omega }} f(x,u,v)$ and $k(u,v):=\min_{x\in \overline{\Omega }}g(x,u,v)$. \item[(H3)] For each $b>0$, \begin{equation*} \liminf_{z\to +\infty }\frac{F^{\frac{1}{p-1}}(z,bG^{ \frac{1}{q-1}}(z,z))}{z}=0, \end{equation*} where $F(u,v):=\max_{x\in \overline{\Omega }} f(x,u,v)$ and $G(u,v):=\max_{x\in \overline{\Omega }}g(x,u,v)$. \item[(H4)] \begin{equation*} \liminf_{z\to +\infty }\frac{G^{\frac{1}{q-1}}(z,z)}{z} =0\quad\text{and}\quad \limsup_{z\to 0^{+}} \frac{k^{\frac{1}{q-1}}(z,z)}{z}=\infty . \end{equation*} \end{itemize} Suppose now that $D$ is a sub-domain of $\Omega$ with $\overline{D}\subset \Omega$. Let $\delta (.):=\chi _{D}(.)$, the characteristic function of $D$. The solutions $\widetilde{\varphi }$, $\widetilde{\psi }$ of the problems \begin{gather*} -\Delta _{p}\widetilde{\varphi }=\delta \quad \text{in }\Omega \\ \widetilde{\varphi }=0 \quad \text{on }\partial \Omega \end{gather*} and \begin{gather*} -\Delta _{q}\widetilde{\psi }=\delta \quad \text{in }\Omega \\ \widetilde{\psi }=0 \quad \text{on }\partial \Omega \end{gather*} will be useful in what follows. Let $\varphi$ (respectively $\psi$) denote the torsion functions relative to $\Omega$ and to the operators $-\Delta _{p}$ (respectively $-\Delta _{q})$, that is, $$\begin{gathered} -\Delta _{p}\varphi =1\quad \text{in }\Omega \\ \varphi =0 \quad \text{on }\partial \Omega \end{gathered} \label{phi}$$ and $$\begin{gathered} -\Delta _{q}\psi =1\quad \text{in }\Omega \\ \psi =0 \quad \text{on }\partial \Omega . \end{gathered} \label{psi}$$ Note that, by the strong comparison principle \cite{gue-ver}, there exist positive numbers $M$ and $m$ such that $\widetilde{\varphi }\geq M\varphi$, $\widetilde{\psi }\geq M\psi$ in $\Omega$ and $\widetilde{\varphi }$, $\widetilde{\psi },\varphi ,\psi \geq m$ on $\overline{D}$. Our existence and uniqueness results are the following. \begin{theorem} \label{thm1} Let $f,g$ satisfy (H1)-(H4). Then \eqref{q} has a positive solution $(u,v)$. \end{theorem} \begin{theorem}\label{thm2} Let $f,g$ satisfy (H1) and assume that there exist positive constants $r_{1},r_{2},s_{1},s_{2}$ such that \begin{equation*} \frac{f(x,s,t)}{s^{r_{1}}},\frac{g(x,s,t)}{s^{s_{1}}} \end{equation*} are nonincreasing for $x\in \overline{\Omega }$ and $t\geq 0$, and \begin{equation*} \frac{f(x,s,t)}{t^{r_{2}}},\quad \frac{g(x,s,t)}{t^{s_{2}}} \end{equation*} are nonincreasing for $x\in\overline{\Omega }$ and $s\geq 0$. If one of the following conditions is satisfied: \begin{itemize} \item[(i)] $\frac{r_{1}+r_{2}}{p-1}<1$ and $\frac{(r_{1}+r_{2})s_{1} +s_{2}(p-1)}{(p-1)(q-1)}<1$, \item[(ii)] $\frac{s_{1}+s_{2}}{q-1}<1$ and $\frac{(s_{1}+s_{2})r_{2}+r_{1}(q-1)}{(p-1)(q-1)}<1$, \item[(iii)] $\frac{s_{1}+s_{2}}{q-1}<1$ and $\frac{r_{1}+r_{2}}{p-1}$ $<1$, \item[(iv)] $\frac{r_{1}+r_{2}}{p-1}>1$ , $\frac{s_{1}+s_{2}}{q-1}<1$ and $\frac{(r_{1}+r_{2})(q-1)r_{1}+(s_{1}+s_{2})(p-1)r_{2}}{(p-1)(q-1)^{2}}<1$, \item[(v)] $\frac{r_{1}+r_{2}}{p-1}<1$ , $\frac{s_{1}+s_{2}}{q-1}>1$ and $\frac{(r_{1}+r_{2})(q-1)s_{1}+(s_{1}+s_{2})(p-1)s_{2}}{(p-1)^{2}(q-1)}<1$, \end{itemize} then (\ref{q}) admits at most one positive solution. \end{theorem} \begin{remark} \label{rmk3} \rm (i) If $f(u,v)=u^{\alpha }+v^{\beta }$ and $g(u,v)=u^{\gamma }+v^{\delta }$, $\alpha ,\beta ,\gamma ,\delta \geq 0$, then (H2)-H(4) require \begin{equation*} \alpha \max \{ |\varphi |_{\infty },|\psi |_{\infty }\} $such that \begin{gather*} F^{\frac{1}{p-1}}(R,|\psi |_{\infty }G^{\frac{1}{q-1}}(R,R))|\varphi |_{\infty }\leq R, \\ G^{\frac{1}{q-1}}(R,R)|\psi |_{\infty }\leq R. \end{gather*} We claim that$T(\overline{B}(0,R)\times \overline{B}(0,R))\subseteq \overline{B}(0,R)\times \overline{B}(0,R)$, where$\overline{B}(0,R)$denotes the closed ball centered at$0$with radius$R$in$C(\overline{\Omega })$. Indeed, let$w_{1},w_{2}\in C(\overline{\Omega })$, with$|w_{1}|_{\infty }\leq R$and$|w_{2}|_{\infty }\leq R. Then, in view of (\ref{psi}), \begin{align*} -\Delta _{q}v &=g(x,\max (w_{1,}\varepsilon \varphi ),\max (w_{2,}\varepsilon \psi ))\ \\ &\leq G(R,R)( -\Delta _{q}\psi ) =-\Delta _{q}( G^{\frac{1}{ q-1}}(R,R)\psi ) \quad \text{in }\Omega , \end{align*} which implies by strong comparison principle \cite{gue-ver} that \begin{equation*} v\leq G^{\frac{1}{q-1}}(R,R)\psi . \end{equation*} Consequently,|v|_{\infty }\leq R. On the other hand, \begin{align*} -\Delta _{p}u&=f(x,\max (w_{1,}\varepsilon \varphi ),v)\leq F(R,v) \\ &\leq F(R,G^{\frac{1}{q-1}}(R,R)\psi )\leq F(R,G^{\frac{1}{q-1}}(R,R)|\psi |_{\infty }), \end{align*} which, by the strong comparison principle, implies \begin{equation*} u\leq F^{\frac{1}{p-1}}(R,g^{\frac{1}{q-1}}(R,R)|\psi |_{\infty })|\varphi |_{\infty }\leq R, \end{equation*} and so|u|_{\infty }\leq R$, proving the claim. By the Schauder fixed point theorem,$T$has a fixed point$(u,v)$with$|u|_{\infty }\leq R$and$|v|_{\infty }\leq R$. We will show next that$|u|_{\infty }\geq \varepsilon \varphi $and$|v|_{\infty }\geq \varepsilon \psi . Since \begin{align*} -\Delta _{q}v &=g(x,\max (u,\varepsilon \varphi ),\max (v,\varepsilon \psi ))\geq g(x,\varepsilon \varphi ,\varepsilon \psi )\ \\ &\geq \begin{cases} g(x,\varepsilon m,\varepsilon m) & \text{in }\overline{D} \\ 0 & \text{in }\Omega \backslash \overline{D} \end{cases} \\ &\geq \begin{cases} k(\varepsilon m,\varepsilon m) & \text{in }\overline{D} \\ 0 & \text{in }\Omega \backslash \overline{D}, \end{cases} \end{align*} it follows from the strong comparison principle and (\ref{mk}) that \begin{equation*} v\geq k^{\frac{1}{q-1}}(\varepsilon m,\varepsilon m)\widetilde{\psi }\geq Mk^{\frac{1}{q-1}}(\varepsilon m,\varepsilon m)\psi \geq \varepsilon \psi . \end{equation*} Consequently, \begin{align*} -\Delta _{p}u &= f(x,\max (u,\varepsilon \varphi ),v)\\ &\geq f(x,\max(u,\varepsilon \varphi ),k^{\frac{1}{q-1}} (\varepsilon m,\varepsilon m) \widetilde{\psi }) \\ &\geq \begin{cases} f(x,\max (u,\varepsilon \varphi ),k^{\frac{1}{q-1}}(\varepsilon m,\varepsilon m)m) & \text{in }\overline{D} \\ 0 & \text{in }\Omega \backslash \overline{D} \end{cases} \\ &\geq \begin{cases} h(\varepsilon m,mk^{\frac{1}{q-1}}(\varepsilon m,\varepsilon m)m) & \text{in }\overline{D} \\ 0 & \text{in }\Omega \backslash \overline{D}, \end{cases} \end{align*} and so \begin{equation*} u\geq h^{\frac{1}{p-1}}(\varepsilon m,k^{\frac{1}{q-1}}(\varepsilon m,\varepsilon m)m)\overset{\symbol{126}}{\varphi }\geq Mh^{\frac{1}{p-1} }(\varepsilon m,mk^{\frac{1}{q-1}}(\varepsilon m,\varepsilon m))\geq \varepsilon \varphi . \end{equation*} The proof is complete. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2}] We will provide the proof only for the cases (i), (iii) and (iv). Let(u,v)$and$(u_{1},v_{1})$be positive solutions of (1). As in \cite{bre-kam}, we define \begin{equation*} \Delta =\big\{ \delta _{1}\in (0,1]:u\geq \varepsilon u_{1}\ \text{and } v\geq \varepsilon v_{1}\text{ in }\overline{\Omega }\text{ for }\varepsilon \in [ 0,\delta _{1}]\big\} . \end{equation*} Clearly$\Delta \neq \emptyset $. Let$\delta =\sup \Delta $. We will show that$\delta =1$. So assume that$\delta <1$. Let (i) hold. Then \begin{equation*} -\Delta _{p}u\geq f(x,\delta u_{1},v)\geq \delta ^{r_{1}}f(x,u_{1},v)\geq \delta ^{r_{1}}\delta ^{r_{2}}f(x,u_{1},v_{1})=\delta ^{r_{1}+r_{2}}f(x,u_{1},v_{1}), \end{equation*} and since \begin{equation*} -\Delta _{p}(\delta ^{\frac{^{r_{1}+r_{2}}}{p-1}}u_{1})=\delta ^{r_{1}+r_{2}}f(x,u_{1},v_{1}), \end{equation*} it follows that $$u\geq \delta ^{\frac{^{r_{1}+r_{2}}}{p-1}}u_{1}. \label{z}$$ Using (\ref{z}) in the equation for$v$in (\ref{q}), we get \begin{equation*} -\Delta _{q}v\geq g(x,\delta ^{\frac{^{r_{1}+r_{2}}}{p-1}}u_{1},v)\geq g(x,\delta ^{\frac{^{r_{1}+r_{2}}}{p-1}}u_{1},\delta v_{1})\geq \delta ^{ \frac{(^{r_{1}+r_{2}})s_{1}}{p-1}}g(x,u_{1},\delta v_{1}). \end{equation*} Therefore, \begin{equation*} -\Delta _{q}v\geq \delta ^{\frac{(r_{1}+r_{2})s_{1}}{p-1}}\delta ^{s_{2}}g(x,u_{1},v_{1})\geq \delta ^{\frac{(r_{1}+r_{2})s_{1}+s_{2}(p-1)}{ (p-1)}}g(x,u_{1},v_{1}), \end{equation*} and so $$v\geq \delta ^{\frac{(r_{1}+r_{2})s_{1}+s_{2}(p-1)}{(p-1)(q-1)}}v_{1}, \label{x}$$ contradicting the definition of$\delta $. In the case (iii) we have \begin{equation*} -\Delta _{q}v\geq g(x,\delta u_{1},v)\geq \delta ^{s_{1}}g(x,u_{1},v)\geq \delta ^{s_{1}}\delta ^{s_{2}}g(x,u_{1},v_{1})=\delta ^{s_{1}+s_{2}}g(x,u_{1},v_{1}). \end{equation*} Since \begin{equation*} -\Delta _{q}(\delta ^{\frac{^{s_{1}+s_{2}}}{q-1}}v_{1})=\delta ^{s_{1}+s_{2}}g(x,u_{1},v_{1}), \end{equation*} it follows that $$v\geq \delta ^{\frac{^{s_{1}+s_{2}}}{q-1}}v_{1}. \label{c}$$ In view of inequalities (\ref{z}) and (\ref{c}) we have a contradiction with the definition of$\delta $. Assume now that (iv) holds. Working as in (\ref{z}) we get $$v\geq \delta ^{\frac{^{s_{1}+s_{2}}}{q-1}}v_{1}. \label{v}$$ Using (\ref{z}) and (\ref{v})\ in the equation for$uyields \begin{align*} -\Delta _{p}u&\geq f(x,\delta ^{\frac{^{r_{1}+r_{2}}}{p-1}}u_{1},\delta ^{ \frac{^{s_{1}+s_{2}}}{q-1}}v_{1}) \\ &\geq \delta ^{\frac{(r_{1}+r_{2})(q-1)r_{1}+(s_{1}+s_{2})(p-1)r_{2}}{ (q-1)(p-1)}}f(x,u_{1},v_{1})\\ &=-\delta ^{\frac{ (r_{1}+r_{2})(q-1)r_{1}+(s_{1}+s_{2})(p-1)r_{2}}{(q-1)(p-1)}}\Delta _{p}u_{1}. \end{align*} Thus, \begin{equation*} u\geq \delta ^{\frac{(r_{1}+r_{2})(q-1)r_{1}+(s_{1}+s_{2})(p-1)r_{2}}{ (p-1)^{2}(q-1)}}u_{1}. \end{equation*} On the other hand, \begin{align*} -\Delta _{q}v &\geq g(x,\delta ^{\frac{^{r_{1}+r_{2}}}{p-1}}u_{1},\delta ^{ \frac{^{s_{1}+s_{2}}}{q-1}}v_{1}) \\ &\geq \delta ^{\frac{(r_{1}+r_{2})(q-1)r_{1}+(s_{1}+s_{2})(p-1)r_{2}}{ (q-1)(p-1)}}g(x,u_{1},v_{1}) \\ &= -\delta ^{\frac{(r_{1}+r_{2})(q-1)r_{1}+(s_{1}+s_{2})(p-1)r_{2}}{ (q-1)(p-1)}}\Delta _{q}v_{1}. \end{align*} Consequently, \begin{equation*} v\geq \delta ^{\frac{(r_{1}+r_{2})(q-1)r_{1}+(s_{1}+s_{2})(p-1)r_{2}}{ (p-1)(q-1)^{2}}}v_{1}, \end{equation*} contradicting the definition of\delta $. Thus$\delta =1$, i.e.,$v\geq v_{1}$and$u\geq u_{1}$. Similarly,$v\leq v_{1}$and$u\leq u_{1}$. Consequently,$u=u_{1}$and$ v=v_{1}$. \end{proof} \begin{thebibliography}{00} \bibitem{alv-fi} C.O. Alves, D.G. deFigueiredo, \emph{Nonvariational elliptic systems}, Discr. Contin. Dyn. Systems 8 (2) (2002), 289-302. \bibitem{amb-az-per} A. Ambrosetti, J. G. Azorero, I. Peral; \emph{Existence and multiplicity results for some nonlinear elliptic equations: a survey}, Rendiconti di Matematica, SerieVII, 20 (2000), 167-198. \bibitem{atk-ali} C. Atkinson, K. El-Ali, \emph{Some boundary value problems for the Bingham model}, J. Non-Newtonian fluid Mech. 41 (1992), 339-363. \bibitem{bre-kam} H. Brezis, S. Kamin, \emph{Sublinear elliptic equations in$\mathbb{R}^N$}, Manuscripta Math. 74 (1992), 87-106. \bibitem{cle-fle-the} P. Clement, J. Fleckinger, E. Mitidieri, F. de Thelin, \emph{Existence of positive solutions for a nonvariational quasilinear elliptic system}, J. Differential Equations 166 (2000), 455-477. \bibitem{dal} R. Dalmasso, \emph{Existence and uniqueness of positive solutions of semilinear elliptic systems}, Nonlinear Analysis TMA 39 (2000), 559-568. \bibitem{fi} D. G. de Figueiredo, \emph{Semilinear elliptic systems}, in Nonlinear Functional Analysis and Applications to Differential Equations, World Scientific (1997), 125-152 (A. Ambrosetti, K.-C. Chang, I. Ekeland, eds). \bibitem{gue-ver} M. Guedda, L. Veron, \emph{Quasilinear elliptic equations involving critical Sobolev exponents}, Nonlinear Analysis TMA 13 (1989), 879-902. \bibitem{hai} D.D. Hai, \emph{Existence and uniqueness of solutions for quasilinear elliptic systems}, Proc. AMS 133 no 1 (2004), 223-228. \bibitem{sta-zog} N. M. Stavrakakis and N. Zographopoulos, \emph{Multiplicity and regularity results for some quasilinear elliptic systems on$\mathbb{R}^N\$}, Nonlinear Analysis TMA 50 (2002), 55-69. \end{thebibliography} \end{document}