Nonlocal critical Kirchhoff problems in high dimension
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 $R^N$, $N>4$, $a,b>0$, $\lamb...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2025-05-01
|
| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2025/46/abstr.html |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | 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. |
|---|---|
| ISSN: | 1072-6691 |