Brezis–Nirenberg type results for the anisotropic p$p$‐Laplacian

Abstract In this paper, we consider a quasilinear elliptic and critical problem with Dirichlet boundary conditions in presence of the anisotropic ‐Laplacian. The critical exponent is the usual such that the embedding is not compact. We prove the existence of a weak positive solution in presence of both a ‐linear and a ‐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 ( Comm. Pure Appl. Math . 36 (1983), 437–477).