An exterior overdetermined problem for Finsler N-Laplacian in convex cones

Abstract

Abstract We consider a partially overdetermined problem for anisotropic N -Laplace equations in a convex cone $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Σ</mml:mi> </mml:math> intersected with the exterior of a bounded domain $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> in $${\mathbb {R}}^N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:math> , $$N\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . Under a prescribed logarithmic condition at infinity, we prove a rigidity result by showing that the existence of a solution implies that $$\Sigma \cap \Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Σ</mml:mi> <mml:mo>∩</mml:mo> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:math> must be the intersection of the Wulff shape and $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Σ</mml:mi> </mml:math> . Our approach is based on a Pohozaev-type identity and the characterization of minimizers of the anisotropic isoperimetric inequality inside convex cones.