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

Abstract

Abstract We study the local behavior of bounded local weak solutions to a class of anisotropic singular equations of the kind ∑ i = 1 s ∂ i i u + ∑ i = s + 1 N ∂ i ( A i ( x , u , ∇ u ) ) = 0 , x ∈ Ω ⊂ ⊂ R N for 1 ≤ 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 A i {A}_{i} behaves directionally as the singular p p -Laplacian, 1 < p < 2 1\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.