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

Abstract

<p style='text-indent:20px;'>In this brief note we discuss local Hölder continuity for solutions to anisotropic elliptic equations of the type</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \label{prototype} \sum\limits_{i = 1}^s \partial_{ii} u+ \sum\limits_{i = s+1}^N \partial_i \bigg(A_i(x, u, \nabla u) \bigg) = 0, \quad x \in \Omega \subset \subset \mathbb R^N \quad \text{ for } \quad 1\leq s \leq N-1, \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where each operator <inline-formula><tex-math id="M1">\begin{document}$ A_i $\end{document}</tex-math></inline-formula> behaves directionally as the singular <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian, <inline-formula><tex-math id="M3">\begin{document}$ 1< p < 2 $\end{document}</tex-math></inline-formula> and the supercritical condition <inline-formula><tex-math id="M4">\begin{document}$ p+(N-s)(p-2)>0 $\end{document}</tex-math></inline-formula> 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.</p>