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.