On the compactness threshold in the critical Kirchhoff equation

TL;DR

This paper investigates the critical Kirchhoff equation with a nonlocal term, establishing a variational characterization of the compactness threshold and deriving existence and multiplicity results.

Contribution

It provides the first variational characterization of the compactness threshold for critical Kirchhoff problems with nonlocal terms.

Findings

Established a variational characterization of the compactness threshold.

Proved existence of solutions below the threshold.

Demonstrated multiplicity of solutions using the characterization.

Abstract

We study a class of critical Kirchhoff problems with a general nonlocal term. The main difficulty here is the absence of a closed-form formula for the compactness threshold. First we obtain a variational characterization of this threshold level. Then we prove a series of existence and multiplicity results based on this variational characterization.