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

2024/11/25 by Stefano Biagi, Biagi, Stefano, Francesco Esposito +5
Computer Science · Mathematics · #35A15 #35B33 #35J62 #35J92 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2411.16257

openalex publication_date 2024/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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^⋆ such that the embedding W^1,p₀(Ω) ⊂ L^p^⋆(Ω) 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 \citeBN.

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