Ground state and nodal solutions for critical Kirchhoff–Schrödinger–Poisson systems with an asymptotically 3-linear growth nonlinearity
Abstract In this paper, we consider the existence of a least energy nodal solution and a ground state solution, energy doubling property and asymptotic behavior of solutions of the following critical problem: { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u + λ ϕ u = | u | 4 u + k f ( u ) , x ∈ R 3...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2020-08-01
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| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13661-020-01421-5 |
| _version_ | 1818508996326195200 |
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| author | Chungen Liu Hua-Bo Zhang |
| author_facet | Chungen Liu Hua-Bo Zhang |
| author_sort | Chungen Liu |
| collection | DOAJ |
| description | Abstract In this paper, we consider the existence of a least energy nodal solution and a ground state solution, energy doubling property and asymptotic behavior of solutions of the following critical problem: { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u + λ ϕ u = | u | 4 u + k f ( u ) , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3 . $$ \textstyle\begin{cases} -(a+ b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx)\Delta u+V(x)u+\lambda \phi u= \vert u \vert ^{4}u+ k f(u),&x\in \mathbb{R}^{3}, \\ -\Delta \phi =u^{2},&x\in \mathbb{R}^{3}. \end{cases} $$ By nodal Nehari manifold method, for each b > 0 $b>0$ , we obtain a least energy nodal solution u b $u_{b}$ and a ground-state solution v b $v_{b}$ to this problem when k ≫ 1 $k\gg1$ , where the nonlinear function f ∈ C ( R , R ) $f\in C(\mathbb{R},\mathbb{R})$ . We also give an analysis on the behavior of u b $u_{b}$ as the parameter b → 0 $b\to 0$ . |
| first_indexed | 2024-12-10T22:39:30Z |
| format | Article |
| id | doaj.art-581bca98a67646d1bb96e1267f2a14c8 |
| institution | Directory Open Access Journal |
| issn | 1687-2770 |
| language | English |
| last_indexed | 2024-12-10T22:39:30Z |
| publishDate | 2020-08-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-581bca98a67646d1bb96e1267f2a14c82022-12-22T01:30:45ZengSpringerOpenBoundary Value Problems1687-27702020-08-012020112810.1186/s13661-020-01421-5Ground state and nodal solutions for critical Kirchhoff–Schrödinger–Poisson systems with an asymptotically 3-linear growth nonlinearityChungen Liu0Hua-Bo Zhang1Department of Mathematics, Guangzhou UniversityDepartment of Mathematics, Guangzhou UniversityAbstract In this paper, we consider the existence of a least energy nodal solution and a ground state solution, energy doubling property and asymptotic behavior of solutions of the following critical problem: { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u + λ ϕ u = | u | 4 u + k f ( u ) , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3 . $$ \textstyle\begin{cases} -(a+ b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx)\Delta u+V(x)u+\lambda \phi u= \vert u \vert ^{4}u+ k f(u),&x\in \mathbb{R}^{3}, \\ -\Delta \phi =u^{2},&x\in \mathbb{R}^{3}. \end{cases} $$ By nodal Nehari manifold method, for each b > 0 $b>0$ , we obtain a least energy nodal solution u b $u_{b}$ and a ground-state solution v b $v_{b}$ to this problem when k ≫ 1 $k\gg1$ , where the nonlinear function f ∈ C ( R , R ) $f\in C(\mathbb{R},\mathbb{R})$ . We also give an analysis on the behavior of u b $u_{b}$ as the parameter b → 0 $b\to 0$ .http://link.springer.com/article/10.1186/s13661-020-01421-5Kirchhoff–Schrödinger–Poisson systemsNodal solutionGround state solutionNehari manifold |
| spellingShingle | Chungen Liu Hua-Bo Zhang Ground state and nodal solutions for critical Kirchhoff–Schrödinger–Poisson systems with an asymptotically 3-linear growth nonlinearity Boundary Value Problems Kirchhoff–Schrödinger–Poisson systems Nodal solution Ground state solution Nehari manifold |
| title | Ground state and nodal solutions for critical Kirchhoff–Schrödinger–Poisson systems with an asymptotically 3-linear growth nonlinearity |
| title_full | Ground state and nodal solutions for critical Kirchhoff–Schrödinger–Poisson systems with an asymptotically 3-linear growth nonlinearity |
| title_fullStr | Ground state and nodal solutions for critical Kirchhoff–Schrödinger–Poisson systems with an asymptotically 3-linear growth nonlinearity |
| title_full_unstemmed | Ground state and nodal solutions for critical Kirchhoff–Schrödinger–Poisson systems with an asymptotically 3-linear growth nonlinearity |
| title_short | Ground state and nodal solutions for critical Kirchhoff–Schrödinger–Poisson systems with an asymptotically 3-linear growth nonlinearity |
| title_sort | ground state and nodal solutions for critical kirchhoff schrodinger poisson systems with an asymptotically 3 linear growth nonlinearity |
| topic | Kirchhoff–Schrödinger–Poisson systems Nodal solution Ground state solution Nehari manifold |
| url | http://link.springer.com/article/10.1186/s13661-020-01421-5 |
| work_keys_str_mv | AT chungenliu groundstateandnodalsolutionsforcriticalkirchhoffschrodingerpoissonsystemswithanasymptotically3lineargrowthnonlinearity AT huabozhang groundstateandnodalsolutionsforcriticalkirchhoffschrodingerpoissonsystemswithanasymptotically3lineargrowthnonlinearity |