The Brezis-Nirenberg problem for the fractional p-Laplacian

2015/08/04 by Sunra Mosconi, Kanishka Perera, Mosconi, Sunra +5 · 4 citations
Computer Science · Mathematics · #35B33 #35J92 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Primary 35R11 #Secondary 35A15 #math.AP #msc:35A15 #msc:35B33 #msc:35J92 #msc:35R11

paper · pdf · doi:10.48550/arxiv.1508.00700

24 pages

arxiv created 2015/08/04 · openalex publication_date 2015/08/04 · arxiv updated 2015/08/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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. We get around this difficulty by working with certain asymptotic estimates for minimizers recently obtained by Brasco, Mosconi and Squassina. 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 by Perera and Yang to overcome this difficulty.

Citations

• Fractional p-eigenvalues
• Nonlocal Harnack inequalities
• The Brezis-Nirenberg result for the fractional Laplacian
• Stability of variational eigenvalues for the fractional p-Laplacian

Cited by

• Brezis-Nirenberg type problem for fractional sub-Laplacian on the Heisenberg group
• Infinitely many solutions for nonlinear superposition operators of mixed fractional order involving critical exponent
• The Brezis-Nirenberg and logistic problem for the Logarithmic Laplacian
• The Brezis-Nirenberg problem for the fractional p-Laplacian involving critical Hardy-Sobolev exponents
• Brezis-Nirenberg type problems associated with nonlinear superposition operators of mixed fractional order

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