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 $$\begin{aligned} {\left\{ \begin{array}{ll} \mathrm{div}a(Du, x)=\mu &{}\quad \text {in} \quad \Omega ,\\ u=0 &{} \quad \text {on} \quad \partial \Omega , \end{array}\right. } \end{aligned}$$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 \(a: \mathbb {R}^n\times \mathbb {R}^n\rightarrow \mathbb {R}^n\) is modeled upon the \(p(\cdot )\)-Laplacian. We prove the estimates on weighted 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.