Positive Solutions for Non-cooperative Singular $p$-Laplacian Systems

Abstract

We prove the existence of\ positive solutions for the $p$-Laplacian system \[ \left\{ \begin{array}{@{\,}c@{\enskip}c} -\Delta _{p}u_{1}=\lambda f_{1}(u_{2}) &\text{in}~\Omega \,, \\ -\Delta _{p}u_{2}=\lambda f_{2}(u_{1}) &\text{in}~\Omega \,, \\ \ \ \ \quad u_{1}=u_{2}=0 & \text{on}~\partial \Omega \,, \end{array} \right. \] where $\Delta _{p}u=\mbox{div}(|\nabla u|^{p-2}\nabla u),p>1, \Omega$ is a bounded domain in $\mathbf{R}^{n}$ with smooth boundary $\partial \Omega ,f_{i}:(0,\infty) \rightarrow \mathbf{R}$ are possibly singular at 0 and are not required to be positive or nondecreasing, and $\lambda $ is a large parameter.