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

  • The Anh Bui
  • Xuan Thinh Duong
Article
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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.

Keywords

Nonlinear p(x)-Laplacian type equation Measure data Reifenberg domain Weighted generalized Lebesgue spaces 

Mathematics Subject Classification

35B65 35J60 35J99 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsMacquarie UniversitySydneyAustralia

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