On an anisotropic p-Laplace equation with variable singular exponent

TL;DR

This paper investigates an anisotropic p-Laplacian equation with variable singular exponent, establishing existence, uniqueness, and multiplicity of solutions under different conditions, expanding understanding of nonlinear PDEs with variable exponents.

Contribution

It introduces new results on existence, uniqueness, and multiple solutions for an anisotropic p-Laplacian with variable singular exponent, including the case with purely singular nonlinearity.

Findings

Existence and uniqueness for the purely singular case.

Multiple solutions when f is constant and g is a power function.

Extension of solution theory to variable exponent anisotropic problems.

Abstract

In this article, we study the following anisotropic p-Laplacian equation with variable exponent given by \begin{equation*} (P)\left\{\begin{split} -\Delta_{H,p}u&=\frac{\la f(x)}{u^{q(x)}}+g(u)\text{ in }\Omega,\\ u&>0\text{ in }\Omega,\,u=0\text{ on }\partial\Omega, \end{split}\right. \end{equation*} under the assumption $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ with $p,N\geq 2$, $\la>0$ and $0<q \in C(\bar \Om)$. For the purely singular case that is $g\equiv 0$, we proved existence and uniqueness of solution. We also demonstrate the existence of multiple solution to $(P)$ provided $f\equiv 1$ and $g(u)=u^r$ for $r\in (p-1,p^*-1)$.