Energy properties of critical Kirchhoff problems with applications

TL;DR

This paper characterizes the energy functional's properties for critical Kirchhoff problems, providing criteria for convexity and compactness, and applying these results to elliptic problems involving Kirchhoff terms.

Contribution

It offers a complete characterization of the energy functional's regularity properties and applies these to analyze elliptic Kirchhoff problems.

Findings

Criteria for weak lower semicontinuity established

Conditions for convexity and Palais-Smale validity derived

Applications to elliptic Kirchhoff problems demonstrated

Abstract

In this paper we fully characterize the sequentially weakly lower semicontinuity of the parameter-depending energy functional associated with the critical Kirchhoff problem. We also establish sufficient criteria with respect to the parameters for the convexity and validity of the Palais-Smale condition of the same energy functional. We then apply these regularity properties in the study of some elliptic problems involving the critical Kirchhoff term.