Weighted variable exponent Sobolev estimates for elliptic equations with non-standard growth and measure data

TL;DR

This paper establishes weighted variable exponent Sobolev estimates for solutions to a class of nonlinear elliptic equations with non-standard growth and measure data, extending regularity theory in variable exponent spaces.

Contribution

It introduces new weighted Sobolev estimates for variable exponent spaces for elliptic equations with measure data and non-standard growth, broadening the understanding of regularity in these complex settings.

Findings

Weighted estimates on gradients in variable exponent Lebesgue spaces

New $L^q-L^r$ regularity results for solutions

Estimates on Morrey spaces for gradients

Abstract

Consider the following nonlinear elliptic equation of $p(x)$-Laplacian type with nonstandard growth \begin{equation*} \left\{ \begin{aligned} &{\rm div} a(Du, x)=\mu \quad &\text{in}& \quad \Omega, &u=0 \quad &\text{on}& \quad \partial\Omega, \end{aligned} \right. \end{equation*} where $\Omega$ is a Reifenberg domain in $\mathbb{R}^n$, $\mu$ is a Radon measure defined on $\Omega$ with finite total mass and the nonlinearity $a: \mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}^n$ is modeled upon the $p(\cdot)$-Laplacian. We prove the estimates on weighted {\it variable exponent} Lebesgue spaces for gradients of solutions to this equation in terms of Muckenhoupt--Wheeden type estimates. As a consequence, we obtain some new results such as the weighted $L^q-L^r$ regularity (with constants $q < r$) and estimates on Morrey spaces for gradients of the solutions to this non-linear equation.