H\"older continuity for the solutions of the p(x)-Laplace equation with general right-hand side

TL;DR

This paper proves that bounded solutions to a class of variable exponent p-Laplace equations are locally Hölder continuous when the source terms are in appropriate Lebesgue spaces, extending regularity results.

Contribution

It establishes Hölder continuity for solutions of the p(x)-Laplace equation with general right-hand sides under minimal integrability conditions.

Findings

Bounded solutions are locally Hölder continuous.

Regularity holds for general right-hand side functions in Lebesgue spaces.

Extends known regularity results to variable exponent settings.

Abstract

We show that bounded solutions of the quasilinear elliptic equation $\Delta_{p(x)} u=g+div(\textbf{F})$ are locally H\"{o}lder continuous provided that the functions $g$ and $\textbf{F}$ are in suitable Lebesgue spaces.