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

TL;DR

This paper establishes conditions under which the weighted p-Laplace equation has no stable solutions in the entire space, focusing on specific nonlinearities and weight functions.

Contribution

It provides new sufficient conditions on the weight function for the non-existence of stable solutions to the weighted p-Laplace equation.

Findings

No stable solutions exist under the given conditions.

Applicable to nonlinearities like negative power and exponential functions.

Results extend understanding of solution stability in weighted p-Laplace equations.

Abstract

We provide sufficient conditions on $w\in L^1_{loc}(\mathbb{R}^N)$ 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 $C^{1,\zeta}_{loc}$ solution in $\mathbb{R}^N$ where $f(x)$ is either $-x^{-\delta}$ or $e^x$ for any $0<\zeta<1$.