Non-existence of stable solutions for weighted $p$-Laplace equation

Abstract

We provide sufficient conditions on $w\in L_{loc}^1({\mathbb{R}}^N)$ such that the weighted $p$-Laplace equation $$-\operatorname{div}\big(w(x)|\nabla u|^{p-2}\nabla u\big)=f(u)\;\;\mbox{in}\;\;\mathbb{R}^N$$ does not admit any stable $C_{loc}^{1,\hat{I}\P }$ solution in ${\mathbb{R}}^N$ where $f(x)$ is either $-x^{-\hat{I}´}$ or $e^x$ for any $0<\hat{I}\P <1$.