Non-existence results for the weighted $p$-Laplace equation with singular nonlinearities

TL;DR

This paper establishes non-existence results for stable solutions of a weighted p-Laplace equation with singular nonlinearities, expanding understanding of solution behavior under specific weight functions and nonlinearities.

Contribution

It provides new non-existence theorems for stable solutions to the weighted p-Laplace equation with particular singular nonlinearities and weight functions.

Findings

No stable solutions exist for certain singular nonlinearities.

Results apply to a class of weight functions w and g.

Extends previous non-existence results to weighted and singular cases.

Abstract

In this paper we present some non existence results concerning the stable solutions to the equation $$\operatorname{div}(w(x)|\nabla u|^{p-2}\nabla u)=g(x)f(u)\;\;\mbox{in}\;\;\mathbb{R}^N;\;\;p\geq 2$$ when $f(u)$ is either $u^{-\delta}+u^{-\gamma}$, $\delta,\gamma>0$ or $\exp(\frac{1}{u})$ and for a suitable class of weight functions $w,g$.