Interior continuity, continuity up to the boundary, and Harnack's inequality for double‐phase elliptic equations with nonlogarithmic conditions

Oleksandr V. Hadzhy, Igor I. Skrypnik and Mykhailo V. Voitovych

Mathematische Nachrichten, 2023, vol. 296, issue 9, 3892-3914

Abstract: We prove continuity and Harnack's inequality for bounded solutions to elliptic equations of the type div|∇u|p−2∇u+a(x)|∇u|q−2∇u=0,a(x)≥0,|a(x)−a(y)|≤A|x−y|αμ(|x−y|),x≠y,div|∇u|p−2∇u1+ln(1+b(x)|∇u|)=0,b(x)≥0,|b(x)−b(y)|≤B|x−y|μ(|x−y|),x≠y,div|∇u|p−2∇u+c(x)|∇u|q−2∇u1+ln(1+|∇u|)β=0,c(x)≥0,β≥0,|c(x)−c(y)|≤C|x−y|q−pμ(|x−y|),x≠y,$$\begin{eqnarray*} \hspace*{13pc}&&{\rm div}{\left(|\nabla u|^{p-2}\,\nabla u+a(x)|\nabla u|^{q-2}\,\nabla u\right)}=0, \quad a(x)\ge 0,\\ &&\quad |a(x)-a(y)|\le A|x-y|^{\alpha }\mu (|x-y|), \quad x\ne y, \\ &&{\rm div}{\left(|\nabla u|^{p-2}\,\nabla u {\left[1+\ln (1+b(x)\, |\nabla u|) \right]} \right)}=0, \quad b(x)\ge 0, \\ &&\quad |b(x)-b(y)|\le B|x-y|\,\mu (|x-y|),\quad x\ne y,\\ &&{\rm div}{\left(|\nabla u|^{p-2}\,\nabla u+ c(x)|\nabla u|^{q-2}\,\nabla u {\left[1+\ln (1+|\nabla u|) \right]}^{\beta } \right)}=0,\\ &&c(x)\ge 0, \, \beta \ge 0, |c(x)-c(y)|\le C|x-y|^{q-p}\,\mu (|x-y|), \quad x\ne y, \end{eqnarray*}$$under the precise choice of μ.

Date: 2023
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