A note on the point-wise behaviour of bounded solutions for a non-standard elliptic operator

Abstract

In this brief note we discuss local Hölder continuity for solutions to anisotropic elliptic equations of the type ${\sum }_{i=1}^s{\partial }_{ii}u+{\sum }_{i=s+1}^N{\partial }_i(A_i(x,u,\nabla u))=0,$ for $x\in \Omega \subset \subset {\mathbb{R}}^N$ and $1\le s\le N-1$, where each operator $A_i$ behaves directionally as the singular $p$-Laplacian, $1<p<2$ and the supercritical condition $p+(N-s)(p-2)>0$ holds true. We show that the Harnack inequality can be proved without the continuity of solutions and that in turn this implies Hölder continuity of solutions.

17 pages