On the local behavior of local weak solutions to some singular anisotropic elliptic equations
Simone Ciani, Igor I. Skrypnik, Vincenzo Vesprı
Abstract
Abstract We study the local behavior of bounded local weak solutions to a class of anisotropic singular equations of the kind <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:munderover> <m:mrow> <m:mrow> <m:mo>∑</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> </m:munderover> <m:msub> <m:mrow> <m:mo>∂</m:mo> </m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:munderover> <m:mrow> <m:mrow> <m:mo>∑</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mi>s</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:munderover> <m:msub> <m:mrow> <m:mo>∂</m:mo> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mo>∇</m:mo> </m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mspace width="1.0em"/> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>⊂</m:mo> <m:mspace width="-0.3em"/> <m:mo>⊂</m:mo> <m:mspace width="0.33em"/> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> <m:mspace width="1.0em"/> <m:mstyle> <m:mspace width="0.1em"/> <m:mtext>for</m:mtext> <m:mspace width="0.1em"/> </m:mstyle> <m:mspace width="0.33em"/> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>s</m:mi> <m:mo>≤</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>N</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> </m:math> \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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:math> {A}_{i} behaves directionally as the singular <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p -Laplacian, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>1</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mn>2</m:mn> </m:math> 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.