Existence and Nonexistence of Positive Solutions for a Weighted Quasilinear Elliptic System

Abstract

This paper deals with the existence and nonexistence of solutions for the following weighted quasilinear elliptic system, $\left({\mathcal{S}}_{{\mu }_1,{\mu }_2}^{q_1,q_2}\right)\left\{\begin{array}{cc}-\text{div}\left(q_1\left(x\right){\left|\nabla u\right|}^{p-2}\nabla u\right)={\mu }_1{\left|u\right|}^{p-2}u+\left(\alpha +1\right){\left|u\right|}^{\alpha -1}u{\left|v\right|}^{\beta +1}\hfill & \text{in\hspace{0.17em}}\mathrm{\Omega }\hfill \\ -\text{div}\left(q_2\left(x\right){\left|\nabla v\right|}^{p-2}\nabla v\right)={\mu }_2{\left|v\right|}^{p-2}v+\left(\beta +1\right){\left|u\right|}^{\alpha +1}{\left|v\right|}^{\beta -1}v\hfill & \text{in\hspace{0.17em}}\mathrm{\Omega }\hfill \\ u>0,\hspace{1em}v>0\hfill & \text{in\hspace{0.17em}}\mathrm{\Omega }\hfill \\ u=v=0\hfill & \text{on\hspace{0.17em}}\partial \mathrm{\Omega },\hfill \end{array}\right)$ where $\mathrm{\Omega }\subset {\mathrm{\Re }}^N\left(N\ge 3\right),2\le p<N$ , $q_1$ , $q_2\in W^{1,p}\left(\mathrm{\Omega }\right)\cap C\left(\overline{\mathrm{\Omega }}\right)$ , $\alpha ,\beta \ge 0$ , ${\mu }_1$ , ${\mu }_2$ , $\ge 0$ and $\alpha ,\beta >0$ satisfy $\alpha +\beta =p^{\ast }-2$ with $p^{\ast }=pN/N-p$ is the critical Sobolev exponent. By means of variational methods we prove the existence of positive solutions which depends on the behavior of the weights $q_1$ , $q_2$ near their minima and the dimension $N$ . Moreover, we use the well known Pohozaev identity for prove the nonexistence result.