Weighted anisotropic Sobolev inequality with extremal and associated singular problems

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 weight $w$ belongs to class of $p$-admissible weights, which may vanish or blow up near origin. discuss existence regularity properties weak solutions for mixed exponential nonlinearities. In particular, result purely problem leads us validity Sobolev inequality an extremal.