Calculus of Variations and Geometric Measure Theory

S. J. N. Mosconi - K. Perera - M. Squassina - Y. Yang

The Brezis-Nirenberg problem for the fractional p-Laplacian

created by squassina on 04 Aug 2015



Inserted: 4 aug 2015
Last Updated: 4 aug 2015

Pages: 24
Year: 2015


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.

Keywords: Fractional p-Laplacian, Brezis-Nirenberg problem, nontrivial solutions