The Brezis-Nirenberg problem for the fractional p-Laplacian

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 not equal 2. 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.