Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity

TL;DR

This paper investigates the existence and behavior of non-negative solutions to a class of stationary Kirchhoff problems involving a fractional operator and critical nonlinearity, addressing degeneracy issues with variational methods.

Contribution

It extends previous results by analyzing degenerate Kirchhoff problems with fractional operators and critical nonlinearities using variational techniques.

Findings

Existence of non-negative solutions established.

Asymptotic behavior characterized.

Extension of prior results in the literature.

Abstract

This paper deals with the existence and the asymptotic behavior of non-negative solutions for a class of stationary Kirchhoff problems driven by a fractional integro-differential operator $\mathcal L_K$ and involving a critical nonlinearity. The main feature, as well as the main difficulty, of the analysis is the fact that the Kirchhoff function $M$ can be zero at zero, that is the problem is degenerate. The adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature.