On Critical Kirchhoff problems driven by the fractional Laplacian

TL;DR

This paper investigates a nonlocal Kirchhoff problem involving the fractional Laplacian and critical Sobolev nonlinearities, extending existing local results to the nonlocal fractional setting using variational and topological methods.

Contribution

It introduces a novel analysis of Kirchhoff problems with fractional Laplacian and critical nonlinearities, expanding the scope of known results to nonlocal operators.

Findings

Existence of solutions established for the nonlocal Kirchhoff problem.

Extension of classical results to fractional Laplacian setting.

Application of variational and topological methods to nonlocal problems.

Abstract

We study a nonlocal parametric problem driven by the fractional Laplacian operator combined with a Kirchhoff-type coefficient and involving a critical nonlinearity term in the sense of Sobolev embeddings. Our approach is of variational and topological nature. The obtained results can be viewed as a nontrivial extension to the nonlocal setting of some recent contributions already present in the literature.