Anisotropic elliptic equations involving unbounded coefficients and singular nonlinearities

TL;DR

This paper investigates the existence and regularity of solutions for a class of nonlinear singular anisotropic elliptic equations with unbounded coefficients and singular nonlinearities, extending understanding of such complex PDEs.

Contribution

It introduces new existence and regularity results for anisotropic elliptic equations with unbounded coefficients and singular nonlinearities, addressing a broader class of problems than previously studied.

Findings

Established existence of solutions under specified conditions.

Proved regularity properties of solutions.

Extended known results to equations with unbounded coefficients and singular nonlinearities.

Abstract

In this paper, we study the existence and regularity of solutions for a class of nonlinear singular elliptic equations involving unbounded coefficients and a singular right-hand side. Specifically, we are interested to problem whose simplest model is \begin{equation*} -\sum_{j=1}^N\partial_{j}\left([1+u^{q}]\vert \partial_{j} u \vert^{p_{j}-2} \partial_{j} u\right)= \frac{f}{u^{\gamma}}\text{ in $\mathcal{D},$}\quad u>0 \text{ in $\mathcal{D},$} \quad u=0 \hbox{ on}\;\; \partial\mathcal{D}, \end{equation*} where $\mathcal{D}$ is a bounded open subset of $\mathbb{R}^{N}$ with $N>2$, $ \gamma\geq0$, $q >0 $, $p_{j}>2$ for all $j=1,...,N$ and the source term $f$ belongs to $L^1(\mathcal{D})$, with $f \geq 0$ and $f \not\equiv 0$.