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 , $$ -\left (a-\epsilon \displaystyle \int _{\mathbb{R}^{3}} K(x)| \nabla u|^{2}dx\right )\text{di...
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2025-01-01
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| Series: | Boundary Value Problems |
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| Online Access: | https://doi.org/10.1186/s13661-024-01986-5 |
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| author | Xiaotao Qian Zhigao Shi |
| author_facet | Xiaotao Qian Zhigao Shi |
| author_sort | Xiaotao Qian |
| collection | DOAJ |
| description | Abstract In this paper, we are concerned with the following nonlocal problem: − ( a − ϵ ∫ R 3 K ( x ) | ∇ u | 2 d x ) div ( K ( x ) ∇ u ) = λ K ( x ) f ( x ) | u | q − 2 u + K ( x ) | u | 4 u , x ∈ R 3 , $$ -\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}, $$ where a , λ > 0 $a, \lambda >0$ , 1 < q < 2 $1< q<2$ , K ( x ) = exp ( | x | α / 4 ) $K(x)=\exp ({|x|^{\alpha}/4})$ with α ≥ 2 $\alpha \geq 2$ , ϵ > 0 $\epsilon >0$ is small enough, and f ( x ) ≥ 0 $f(x)\ge 0$ satisfies some integrability condition. By using the Ekeland variational principle and the concentration compactness principle, we establish the existence of two positive solutions for the problem and prove that at least one of them is a positive ground state solution. |
| format | Article |
| id | doaj-art-25b4e61428fa4574a8db775cce6e441d |
| institution | Kabale University |
| issn | 1687-2770 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj-art-25b4e61428fa4574a8db775cce6e441d2025-01-12T12:33:13ZengSpringerOpenBoundary Value Problems1687-27702025-01-012025111710.1186/s13661-024-01986-5Multiple positive solutions for a nonlocal problem with fast increasing weight and critical exponentXiaotao Qian0Zhigao Shi1School of Computer Science and Mathematics, Fujian University of TechnologyTeaching and research department of mathematics and physics, Fujian Jiangxia UniversityAbstract In this paper, we are concerned with the following nonlocal problem: − ( a − ϵ ∫ R 3 K ( x ) | ∇ u | 2 d x ) div ( K ( x ) ∇ u ) = λ K ( x ) f ( x ) | u | q − 2 u + K ( x ) | u | 4 u , x ∈ R 3 , $$ -\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.https://doi.org/10.1186/s13661-024-01986-5Nonlocal problemVariational methodsCritical nonlinearityConcave-convex nonlinearity |
| spellingShingle | Xiaotao Qian Zhigao Shi Multiple positive solutions for a nonlocal problem with fast increasing weight and critical exponent Boundary Value Problems Nonlocal problem Variational methods Critical nonlinearity Concave-convex nonlinearity |
| title | Multiple positive solutions for a nonlocal problem with fast increasing weight and critical exponent |
| title_full | Multiple positive solutions for a nonlocal problem with fast increasing weight and critical exponent |
| title_fullStr | Multiple positive solutions for a nonlocal problem with fast increasing weight and critical exponent |
| title_full_unstemmed | Multiple positive solutions for a nonlocal problem with fast increasing weight and critical exponent |
| title_short | Multiple positive solutions for a nonlocal problem with fast increasing weight and critical exponent |
| title_sort | multiple positive solutions for a nonlocal problem with fast increasing weight and critical exponent |
| topic | Nonlocal problem Variational methods Critical nonlinearity Concave-convex nonlinearity |
| url | https://doi.org/10.1186/s13661-024-01986-5 |
| work_keys_str_mv | AT xiaotaoqian multiplepositivesolutionsforanonlocalproblemwithfastincreasingweightandcriticalexponent AT zhigaoshi multiplepositivesolutionsforanonlocalproblemwithfastincreasingweightandcriticalexponent |