# On a critical Kirchhoff-type problem

## TL;DR

This paper investigates a Kirchhoff-type differential equation involving critical Sobolev exponents, establishing conditions for the energy functional's mathematical properties crucial for solution existence.

## Contribution

It provides new sufficient conditions for the lower semicontinuity and Palais-Smale property of the associated energy functional in Kirchhoff problems with critical exponents.

## Key findings

- Established conditions for weak lower semicontinuity.
- Proved Palais-Smale property under certain conditions.
- Contributed to the mathematical understanding of Kirchhoff problems with critical Sobolev exponents.

## Abstract

In the present paper, we study a Kirchhoff type problem involving the critical Sobolev exponent. We give sufficient conditions for the sequentially weakly lower semicontinuity and the Palais Smale property of the energy functional associated to the problem.