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

In this paper, we consider a quasilinear elliptic and critical problem with Dirichlet boundary conditions in presence of the anisotropic (Formula presented.) -Laplacian. The critical exponent is the usual (Formula presented.) such that the embedding (Formula presented.) is not compact. We prove the existence of a weak positive solution in presence of both a (Formula presented.) -linear and a (Formula presented.) -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).