The Brezis–Nirenberg problem for the fractional p-Laplacian
By Sunra Mosconi, Kanishka Perera, Marco Squassina, Yang Yang. Calculus of Variations and Partial Differential Equations, 2015
182 citations recorded by the source corpus.
We obtain nontrivial solutions to the Brezis–Nirenberg problem for the fractional p-Laplacian operator, extending some results in the literature for the fractional Laplacian. The quasilinear case presents two serious new difficulties. First an explicit formula for a minimizer in the fractional Sobolev inequality is not available when p≠2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \ne 2$$\end{document}. We get around this difficulty by working with certain asymptotic estimates for minimizers recently obtained in (Brasco et al., Cal. Var. Partial Differ Equations 55:23, 2016). The second difficulty is the lack of a direct sum decomposition suitable for applying the classical linking theorem. We use an abstract linking theorem based on the cohomological index proved in (Yang and Perera, Ann. Sci. Norm. Super. Pisa Cl. Sci. doi:10.2422/2036-2145.201406_004, 2016) to overcome this difficulty.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/c1c2de8510cadb3dc85c4ed8c4e9556e1a531109
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In Applied Mathematics
- Wenjing Chen
- Marco Squassina
- M. Bhakta
- Debangana Mukherjee
- Patrick Winkert
- Binlin Zhang
- Vicentiu D. Rădulescu
In Computational Theory and Mathematics
- Wenjing Chen
- Marco Squassina
- M. Bhakta
- Debangana Mukherjee
- Patrick Winkert
- Binlin Zhang
- Vicentiu D. Rădulescu
In Mathematical Physics
Citations in the corpus
- A Brezis-Nirenberg type result for mixed local and nonlocal operators (2022, unknown): “The case of nonlinear operators of fractional type has also been considered (see [27]).”
- Critical Nonlocal Systems with Concave-Convex Powers (2016, background): “On the other hand, as in [20], we can overcome this difficulty by the optimal asymptotic behavior of minimizers, whichwas recently obtained in [4].”
- Multiplicity results for $(p,q)$ fractional elliptic equations involving critical nonlinearities (2018, background): “Therefore, we have the following estimates from [19].”