Nonexistence Results for a Singular Quasilinear Anisotropic Equation

We establish the nonexistence of positive stable solutions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>u</mml:mi> <mml:mo>∈</mml:mo> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">loc</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:math> to the singular quasilinear anisotropic equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msubsup> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>B</mml:mi> <mml:mi>H</mml:mi> </mml:msubsup> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> </mml:math> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>q</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:math> , in low dimensions. Under similar assumptions, we further prove that, for the pure Finsler <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>p</mml:mi> </mml:math> -Laplacian, there exist no solutions that are stable outside a compact set. The proofs rely on several integral estimates and asymptotic analysis of the solutions.