Brezis-Nirenberg-type results for the anisotropic $p$-Laplacian

TL;DR

This paper extends Brezis-Nirenberg results to anisotropic $p$-Laplacian problems, establishing existence of solutions with critical exponents and analyzing anisotropic Aubin-Talenti functions.

Contribution

It generalizes classical Brezis-Nirenberg results to anisotropic $p$-Laplacian operators, including new estimates for anisotropic Aubin-Talenti functions.

Findings

Existence of positive solutions with critical exponents

Development of precise estimates for anisotropic Aubin-Talenti functions

Generalization of classical results to anisotropic setting

Abstract

In this paper we consider a quasilinear elliptic and critical problem with Dirichlet boundary conditions in presence of the anisotropic $p$-Laplacian. The critical exponent is the usual $p^{\star}$ such that the embedding $W^{1,p}_{0}(\Omega) \subset L^{p^{\star}}(\Omega)$ is not compact. We prove the existence of a weak positive solution in presence of both a $p$-linear and a $p$-superlinear perturbation. In doing this, we have to perform several precise estimates of the anisotropic Aubin-Talenti functions which can be of interest for further problems. The results we prove are a natural generalization to the anisotropic setting of the classical ones by Brezis-Nirenberg \cite{BN}.