Harnack inequality for solutions of the $p(x)$-Laplace equation under the precise non-logarithmic Zhikov's conditions

TL;DR

This paper establishes continuity and Harnack's inequality for solutions to the variable exponent p(x)-Laplace equation under specific non-logarithmic conditions on p(x), extending regularity theory in variable exponent PDEs.

Contribution

It proves regularity results for solutions of the p(x)-Laplace equation under new non-logarithmic conditions on p(x), which were not previously addressed.

Findings

Proved continuity of solutions.

Established Harnack's inequality.

Extended regularity theory for variable exponent PDEs.

Abstract

We prove continuity and Harnack's inequality for bounded solutions to the equation $$ {\rm div}\big(|\nabla u|^{p(x)-2}\,\nabla u \big)=0, \quad p(x)= p + L\frac{\log\log\frac{1}{|x-x_{0}|}}{\log\frac{1}{|x-x_{0}|}},\quad L > 0, $$ under the precise non-logarithmic condition on the function $p(x)$.