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

We prove continuity and Harnack’s inequality for bounded solutions to the equation $$\mathrm{d}\mathrm{i}\mathrm{v}\left(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p(x)-2}\mathrm{\nabla }u\right)=0,p(x)=p+L\frac{\log \log \frac{1}{\mathrm{\mid }x-x_0\mathrm{\mid }}}{\log \frac{1}{\mathrm{\mid }x-x_0\mathrm{\mid }}},L>0,$$ under the precise non-logarithmic condition on the function $p(x)$.