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arXiv · 2411.16257

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

Stefano Biagi·Francesco Esposito·Alberto Roncoroni·Eugenio Vecchi
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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}.

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Stefano Biagi, Francesco Esposito, Alberto Roncoroni, Eugenio Vecchi. 2024-11-25. Brezis-Nirenberg-type results for the anisotropic $p$-Laplacian. https://arxiv.org/abs/2411.16257

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