Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 46, pp. 1-8.

Title: Nonlocal critical Kirchhoff problems in high dimension

Author: Giovanni Anello (Univ. of Messina, Italy)

Abstract:
We study the  nonlocal critical Kirchhoff problem
$$\displaylines{
-\Big(a+b\int_\Omega |\nabla u|^2dx\Big)\Delta u
 =|u|^{2^*-2}u +\lambda f(x,u), \quad \text{in } \Omega,\cr
 u=0, \quad \text{on } \partial\Omega,
}$$
where $\Omega$ is a bounded smooth domain in $R^N$, $N>4$, $a,b>0$, 
$\lambda\in R$, $2^*:=\frac{2N}{N-2}$ is the critical exponent for the 
Sobolev embedding, and $f:\Omega\times R\to R$ is a 
Caratheodory function with subcritical growth. We establish the 
existence of global minimizers for the energy functional associated to 
this problem. In particular, we improve a recent result proved by Faraci 
and Silva [3] under more strict conditions on the nonlinearity f
and under additional conditions on a and b.

Submitted January 27, 2025. Published May 06, 2025.
Math Subject Classifications: 35J20, 35J25.
Key Words: Nonlocal problem; Kirchhoff equation; weak solution;  critical growth; 
 approximation; variational methods.