Interior continuity, continuity up to the boundary and Harnack's inequality for double-phase elliptic equations with non-logarithmic conditions

We prove continuity and Harnack’s inequality for bounded solutions to elliptic equations of the type $$\begin{array}{rl}{\displaystyle \mathrm{d}\mathrm{i}\mathrm{v}(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u+a(x)\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{q-2}\mathrm{\nabla }u)=0,}& {\displaystyle a(x)⩾0,}\\ {\displaystyle \mathrm{\mid }a(x)-a(y)\mathrm{\mid }⩽A\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha }\mu (\mathrm{\mid }x-y\mathrm{\mid }),}& {\displaystyle x\mathrm{\ne }y,}\\ {\displaystyle \mathrm{d}\mathrm{i}\mathrm{v}\left(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u\right[1+\ln (1+b(x)\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid })\left]\right)=0,}& {\displaystyle b(x)⩾0,}\\ {\displaystyle \mathrm{\mid }b(x)-b(y)\mathrm{\mid }⩽B\mathrm{\mid }x-y\mathrm{\mid }\mu (\mathrm{\mid }x-y\mathrm{\mid }),}& {\displaystyle x\mathrm{\ne }y,}\end{array}$$ $$\begin{array}{rl}{\displaystyle \mathrm{d}\mathrm{i}\mathrm{v}(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u+c(x)\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{q-2}\mathrm{\nabla }u[1+\ln (1+\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }){]}^{\beta })=0,}& {\displaystyle c(x)⩾0,\beta ⩾0,\phantom{=0=0}}\\ {\displaystyle \mathrm{\mid }c(x)-c(y)\mathrm{\mid }⩽C\mathrm{\mid }x-y{\mathrm{\mid }}^{q-p}\mu (\mathrm{\mid }x-y\mathrm{\mid }),}& {\displaystyle x\mathrm{\ne }y,}\end{array}$$ under the precise choice of $\mu$.