Multiple positive solutions for a nonlocal problem with fast increasing weight and critical exponent
Abstract In this paper, we are concerned with the following nonlocal problem: − ( a − ϵ ∫ R 3 K ( x ) | ∇ u | 2 d x ) div ( K ( x ) ∇ u ) = λ K ( x ) f ( x ) | u | q − 2 u + K ( x ) | u | 4 u , x ∈ R 3 , $$ -\left (a-\epsilon \displaystyle \int _{\mathbb{R}^{3}} K(x)| \nabla u|^{2}dx\right )\text{di...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2025-01-01
|
| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13661-024-01986-5 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | Abstract In this paper, we are concerned with the following nonlocal problem: − ( a − ϵ ∫ R 3 K ( x ) | ∇ u | 2 d x ) div ( K ( x ) ∇ u ) = λ K ( x ) f ( x ) | u | q − 2 u + K ( x ) | u | 4 u , x ∈ R 3 , $$ -\left (a-\epsilon \displaystyle \int _{\mathbb{R}^{3}} K(x)| \nabla u|^{2}dx\right )\text{div}(K(x)\nabla u)=\lambda K(x)f(x)|u|^{q-2}u+K(x)|u|^{4}u, \quad x\in \mathbb{R}^{3}, $$ where a , λ > 0 $a, \lambda >0$ , 1 < q < 2 $1< q<2$ , K ( x ) = exp ( | x | α / 4 ) $K(x)=\exp ({|x|^{\alpha}/4})$ with α ≥ 2 $\alpha \geq 2$ , ϵ > 0 $\epsilon >0$ is small enough, and f ( x ) ≥ 0 $f(x)\ge 0$ satisfies some integrability condition. By using the Ekeland variational principle and the concentration compactness principle, we establish the existence of two positive solutions for the problem and prove that at least one of them is a positive ground state solution. |
|---|---|
| ISSN: | 1687-2770 |