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 $$\mathrm{div}(|\nabla u{|}^{p(x)-2}\nabla u)=0,p(x)=p+L\frac{\log \log \frac{1}{|x-x_0|}}{\log \frac{1}{|x-x_0|}},L>0,$$ under the precise non-logarithmic condition on the function $p(x)$.

arXiv admin note: text overlap with arXiv:2012.10960