The concentration-compactness principle for the nonlocal anisotropic $p$-Laplacian of mixed order

Abstract: In this talk, we introduce a new class of operators with an orthotropic structure having different exponents of integrability and different orders of differentiability. We prove a robust Sobolev-type inequality and establish the existence of minimizers of the Sobolev quotient. The method for the existence of minimizers is based on the concentration-compactness principle which we extend to this class of operators. One consequence of the main result is the existence of a nontrivial nonnegative solution to the corresponding critical problem.

Mathematics

Audience: researchers in the topic

( video )

Non-local operators, probability and singularities