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

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.