Nonexistence of nonnegative entire solutions of semilinear elliptic systems

Abstract

We consider the second order semilinear elliptic system $\Delta u=p\left(x\right)v^{\alpha },$ $\Delta v=q\left(x\right)u^{\beta },$ where $x\in {\mathbf{R}}^N,$ $N\ge 3,$ $\alpha$ and $\beta$ are positive constants, $p$ and $q$ are nonnegative continuous functions. We prove that nontrivial nonnegative entire solutions fail to exist if the functions $p$ and $q$ are of slow decay.