On a class of singular superlinear elliptic systems in a ball

Abstract

We establish the existence of large positive radial solutions for the elliptic system $$ \left\{ \begin{array}{c} -\Delta u=\lambda f(v) \ \text{in} \ B\\ -\Delta v=\lambda g(u) \ \text{in} \ B\\ u=v=0 \ \text{on} \ \partial B \end{array} \right. $$ when the parameter $\lambda>0$ is small, where $B$ is the open unit ball $\mathbb{R}^N,N>2, f,g:(0,\infty) \rightarrow \mathbb{R}$ are possibly singular at 0 and $f(u) \sim u^p,g(v) \sim v^q$ at $\infty$ for some $p,q>0$ with $pq>1$. Our approach is based on fixed point theory in a cone.