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 $$\begin{array}{rl}\mathrm{div}(|\nabla u{|}^{p-2}\nabla u+a(x)|\nabla u{|}^{q-2}\nabla u)=0,& a(x)⩾0,\\ |a(x)-a(y)|⩽A|x-y{|}^{\alpha }\mu (|x-y|),& x\ne y,\\ \mathrm{div}(|\nabla u{|}^{p-2}\nabla u[1+\ln (1+b(x)|\nabla u|)])=0,& b(x)⩾0,\\ |b(x)-b(y)|⩽B|x-y|\mu (|x-y|),& x\ne y,\end{array}$$ $$\begin{array}{rl}\mathrm{div}(|\nabla u{|}^{p-2}\nabla u+c(x)|\nabla u{|}^{q-2}\nabla u[1+\ln (1+|\nabla u|){]}^{\beta })=0,& c(x)⩾0,\beta ⩾0,\\ |c(x)-c(y)|⩽C|x-y{|}^{q-p}\mu (|x-y|),& x\ne y,\end{array}$$ under the precise choice of $\mu$.

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