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

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* such that the embedding W-1,W-p (0) (Omega) subset of L-p*(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 (Comm. Pure Appl. Math. 36 (1983), 437-477).