Weighted anisotropic Sobolev inequality with extremal and associated singular problems

Abstract

We consider singular problems associated with the weighted anisotropic $p$-Laplace operator $$ H_{p,w}u=\text{div}(w(x)(H(\nabla u))^{p-1}\nabla H(\nabla u)), $$ where $H$ is a Finsler-Minkowski norm and the weight $w$ belongs to a class of $p$-admissible weights, which may vanish or blow up near the origin. We discuss existence and regularity properties of weak solutions for the mixed and exponential singular nonlinearities. In particular, the existence result for the purely singular problem leads us to the validity of a weighted anisotropic Sobolev inequality with an extremal.