Existence and Nonexistence results for anisotropic p-Laplace equation with singular nonlinearities

TL;DR

This paper investigates the existence and nonexistence of positive solutions for an anisotropic p-Laplace equation with singular nonlinearities, providing conditions under which solutions exist or do not exist in bounded and unbounded domains.

Contribution

It establishes new existence results for exponential nonlinearities and nonexistence results for certain singular nonlinearities in anisotropic p-Laplace equations.

Findings

Existence of solutions for $f(u)=e^{1/u}$ in bounded domains.

Nonexistence of solutions for $f(u)=-e^{1/u}$ in unbounded domains.

Nonexistence for $f(u)=- (u^{- heta} + u^{-eta})$ in $ ^N$.

Abstract

Let $p_i\geq 2$ and consider the following anisotropic $p$-Laplace equation $$ -\sum_{i=1}^{N}\frac{\partial}{\partial x_i}\Big(\Big|\frac{\partial u}{\partial x_i}\Big|^{p_i-2}\frac{\partial u}{\partial x_i}\Big)=g(x)f(u),\,\,u>0\text{ in }\Omega. $$ Under suitable hypothesis on the weight function $g$ we present an existence result for $f(u)=e^\frac{1}{u}$ in a bounded smooth domain $\Omega$ and nonexistence results for $f(u)=-e^\frac{1}{u}$ or $-(u^{-\delta}+u^{-\gamma}),$ $\delta,\gamma>0$ with $\Omega=\mathbb{R}^N$ respectively.