Existence and nonexistence of positive radial solutions for quasilinear systems

Abstract

The paper deals with the existence and nonexistence of positive radial solutions for the weakly coupled quasilinear system div$( | \nabla u|^{p-2}\nabla u ) + \lambda f(v)=0$, div $( | \nabla v|^{p-2}\nabla v ) + \lambda g(u)=0$ in $\B$, and $\u =v=0$ on $\partial B,$ where $p>1$, $B$ is a finite ball, $f$ and $g$ are continuous and nonnegative functions. We prove that there is a positive radial solution for the problem for various intevals of $\lambda$ in sublinear cases. In addition, a nonexistence result is given. We shall use fixed point theorems in a cone.