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

Abstract

We prove continuity and Harnack's inequality for bounded solutions to elliptic equations of the type $$ \begin{aligned} {\rm div}\big(|\nabla u|^{p-2}\,\nabla u+a(x)|\nabla u|^{q-2}\,\nabla u\big)=0,& \quad a(x)\geqslant0, \\ |a(x)-a(y)|\leqslant A|x-y|^{\alpha}\mu(|x-y|),& \quad x\neq y, \\ {\rm div}\Big(|\nabla u|^{p-2}\,\nabla u \big[1+\ln(1+b(x)\, |\nabla u|) \big] \Big)=0,& \quad b(x)\geqslant0, \\ |b(x)-b(y)|\leqslant B|x-y|\,\mu(|x-y|),& \quad x\neq y, \end{aligned} $$ $$ \begin{aligned} {\rm div}\Big(|\nabla u|^{p-2}\,\nabla u+ c(x)|\nabla u|^{q-2}\,\nabla u \big[1+\ln(1+|\nabla u|) \big]^{\beta} \Big)=0,& \quad c(x)\geqslant0, \, \beta\geqslant0,\phantom{=0=0} \\ |c(x)-c(y)|\leqslant C|x-y|^{q-p}\,\mu(|x-y|),& \quad x\neq y, \end{aligned} $$ under the precise choice of $\mu$.