Positive Radial Solutions for a Class of Singular p-Laplacian Systems in a Ball

Abstract

We prove the existence and nonexistence of positive radial solutions for the system

$$\left\{\begin{array}{ll} -\Delta_{p}u_{1}=h_{1}(u_{2})+\mu _{1}f_{1}(u_{2}) & \quad \text{in} \, B, \\ -\Delta_{p}u_{2}=h_{2}(u_{1})+\mu _{2}f_{2}(u_{1}) & \quad \text{in}\, B, \\ u_{1}=u_{2}=0 & \quad \text{on} \, \partial B, \end{array}\right.$$

where ${p > 1, \Delta _{p}u = {\rm div}(|\nabla u|^{p-2}\nabla u), \, B}$ is the open unit ball in${\mathbb{R}^{N},h_{i}, f_{i}:(0,\infty) \rightarrow \mathbb{R}}$ with f _i asymptotically p-linear at ∞, and μ _i are positive constants, i = 1, 2.