The Brezis-Nirenberg problem for the fractional p-Laplacian

2015/08/04 by Mosconi, Sunra, Perera, Kanishka, Squassina, Marco +1 · 2 citations
#35B33 #35J92 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 35R11 #Secondary 35A15

paper · doi:10.48550/arxiv.1508.00700

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.

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

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