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

TL;DR

This paper extends the concentration-compactness principle to nonlocal anisotropic p-Laplacian operators with mixed order, establishing the existence of minimizers and solutions in this complex setting.

Contribution

It introduces a novel extension of the concentration-compactness principle to nonlocal anisotropic operators with mixed differentiability orders.

Findings

Existence of minimizers for the Sobolev quotient in this class.

Proven existence of nontrivial nonnegative solutions to the critical problem.

Extension of the concentration-compactness principle to nonlocal anisotropic operators.

Abstract

In this paper, we study the existence of minimizers of the Sobolev quotient for a class of nonlocal operators with an orthotropic structure having different exponents of integrability and different orders of differentiability. Our method is based on the concentration-compactness principle which we extend to this class of operators. One consequence of our main result is the existence of a nontrivial nonnegative solution to the corresponding critical problem.