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)=μ\quad &\text{in}& \quad Ω, &u=0 \quad &\text{on}& \quad \partialΩ, \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.

25 pages