Existence of least energy nodal solution for Kirchhoff–Schrödinger–Poisson system with potential vanishing
Abstract This paper deals with the following Kirchhoff–Schrödinger–Poisson system: { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u + ϕ u = K ( x ) f ( u ) in R 3 , − Δ ϕ = u 2 in R 3 , $$ \textstyle\begin{cases} -(a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx)\Delta u+V(x)u+\phi u=K(x)...
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Online Access: | http://link.springer.com/article/10.1186/s13661-020-01408-2 |
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doaj-c7862670fcdc4b6091d76e328538c79a2020-11-25T03:54:41ZengSpringerOpenBoundary Value Problems1687-27702020-06-012020111710.1186/s13661-020-01408-2Existence of least energy nodal solution for Kirchhoff–Schrödinger–Poisson system with potential vanishingJin-Long Zhang0Da-Bin Wang1Department of Applied Mathematics, Lanzhou University of TechnologyDepartment of Applied Mathematics, Lanzhou University of TechnologyAbstract This paper deals with the following Kirchhoff–Schrödinger–Poisson system: { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u + ϕ u = K ( x ) f ( u ) in R 3 , − Δ ϕ = u 2 in R 3 , $$ \textstyle\begin{cases} -(a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx)\Delta u+V(x)u+\phi u=K(x)f(u)&\text{in } \mathbb{R}^{3}, \\ -\Delta \phi =u^{2}&\text{in } \mathbb{R}^{3}, \end{cases} $$ where a, b are positive constants, K ( x ) $K(x)$ , V ( x ) $V(x)$ are positive continuous functions vanishing at infinity, and f ( u ) $f(u)$ is a continuous function. Using the Nehari manifold and variational methods, we prove that this problem has a least energy nodal solution. Furthermore, if f is an odd function, then we obtain that the equation has infinitely many nontrivial solutions.http://link.springer.com/article/10.1186/s13661-020-01408-2Potential vanishingNehari manifoldLeast energy nodal solution |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Jin-Long Zhang Da-Bin Wang |
spellingShingle |
Jin-Long Zhang Da-Bin Wang Existence of least energy nodal solution for Kirchhoff–Schrödinger–Poisson system with potential vanishing Boundary Value Problems Potential vanishing Nehari manifold Least energy nodal solution |
author_facet |
Jin-Long Zhang Da-Bin Wang |
author_sort |
Jin-Long Zhang |
title |
Existence of least energy nodal solution for Kirchhoff–Schrödinger–Poisson system with potential vanishing |
title_short |
Existence of least energy nodal solution for Kirchhoff–Schrödinger–Poisson system with potential vanishing |
title_full |
Existence of least energy nodal solution for Kirchhoff–Schrödinger–Poisson system with potential vanishing |
title_fullStr |
Existence of least energy nodal solution for Kirchhoff–Schrödinger–Poisson system with potential vanishing |
title_full_unstemmed |
Existence of least energy nodal solution for Kirchhoff–Schrödinger–Poisson system with potential vanishing |
title_sort |
existence of least energy nodal solution for kirchhoff–schrödinger–poisson system with potential vanishing |
publisher |
SpringerOpen |
series |
Boundary Value Problems |
issn |
1687-2770 |
publishDate |
2020-06-01 |
description |
Abstract This paper deals with the following Kirchhoff–Schrödinger–Poisson system: { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u + ϕ u = K ( x ) f ( u ) in R 3 , − Δ ϕ = u 2 in R 3 , $$ \textstyle\begin{cases} -(a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx)\Delta u+V(x)u+\phi u=K(x)f(u)&\text{in } \mathbb{R}^{3}, \\ -\Delta \phi =u^{2}&\text{in } \mathbb{R}^{3}, \end{cases} $$ where a, b are positive constants, K ( x ) $K(x)$ , V ( x ) $V(x)$ are positive continuous functions vanishing at infinity, and f ( u ) $f(u)$ is a continuous function. Using the Nehari manifold and variational methods, we prove that this problem has a least energy nodal solution. Furthermore, if f is an odd function, then we obtain that the equation has infinitely many nontrivial solutions. |
topic |
Potential vanishing Nehari manifold Least energy nodal solution |
url |
http://link.springer.com/article/10.1186/s13661-020-01408-2 |
work_keys_str_mv |
AT jinlongzhang existenceofleastenergynodalsolutionforkirchhoffschrodingerpoissonsystemwithpotentialvanishing AT dabinwang existenceofleastenergynodalsolutionforkirchhoffschrodingerpoissonsystemwithpotentialvanishing |
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1724472284596404224 |