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

Abstract

We prove continuity and Harnack’s inequality for bounded solutions to the equation div(∣∇u∣p(x)-2∇u)=0,p(x)=p¯+Lloglog1∣x-x0∣log1∣x-x0∣,p¯>1,L>0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \ extrm{div}\\big (\\mid \ abla u\\mid ^{p(x)-2}\\,\ abla u \\big )&=0, \\quad p(x)= {\\bar{p}} + L\\frac{\\log \\log \\frac{1}{\\mid x-x_{0}\\mid }}{\\log \\frac{1}{\\mid x-x_{0}\\mid }}, \\\\ {\\bar{p}}&>1, \\quad L>0, \\end{aligned}$$\\end{document}under the precise non-logarithmic condition on the function p(x).