Nonexistence of positive solutions for a class of semilinar elliptic systems in a ball

Abstract

We prove the nonexistence of positive solutions to the system

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta u=\lambda f(v)\text { in }B, \\ -\Delta v=\mu g(u) \text { in }B, \\ u=v=0\text { on }\partial B, \end{array} \right. \end{aligned}$$

where $B$ is the open unit ball in $\mathbb {R}^{N}$,$\ N>1,\ \lambda ,\mu$ are positive constants bounded away from $0$ with $\lambda \mu$ large, $f,g$ are smooth functions with $f(0),g(0)<0,\ f\circ (cg)$ and $g\circ (cf)$ growing at least linearly at $\infty$.