On the local behavior of local weak solutions to some singular anisotropic elliptic equations

Abstract

We study the local behavior of bounded local weak solutions to a class of anisotropic singular equations of the kind ∑i=1s∂iiu+∑i=s+1N∂i(Ai(x,u,∇u))=0,x∈Ω⊂⊂RNfor1≤s≤(N−1),\mathop{\sum }\limits_{i=1}^{s}{\partial }_{ii}u+\mathop{\sum }\limits_{i=s+1}^{N}{\partial }_{i}({A}_{i}\left(x,u,\nabla u))=0,\hspace{1.0em}x\in \Omega \subset \hspace{-0.3em}\subset \hspace{0.33em}{{\mathbb{R}}}^{N}\hspace{1.0em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}1\le s\le \left(N-1), where each operator Ai{A}_{i} behaves directionally as the singular pp-Laplacian, 1<p<21\lt p\lt 2. Throughout a parabolic approach to expansion of positivity we obtain the interior Hölder continuity and some integral and pointwise Harnack inequalities.

Keywords

• anisotropic p-laplacian
• singular parabolic equations
• hölder continuity
• intrinsic scaling
• expansion of positivity
• intrinsic harnack inequality
• 35j75
• 35k92
• 35b65