Infinitely many geometrically distinct solutions for periodic Schrödinger–Poisson systems
Abstract This paper is dedicated to studying the following Schrödinger–Poisson system: {−△u+V(x)u+K(x)ϕ(x)u=f(x,u),x∈R3,−△ϕ=K(x)u2,x∈R3, $$ \textstyle\begin{cases} -\triangle u+V(x)u+K(x)\phi (x)u=f(x, u), \quad x\in {\mathbb {R}}^{3}, \\ -\triangle \phi =K(x)u^{2}, \quad x\in {\mathbb {R}}^{3}, \en...
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doaj-f2eccfbb77e44d0e9298b5b8ab69e3f32020-11-25T02:04:21ZengSpringerOpenBoundary Value Problems1687-27702019-03-012019111610.1186/s13661-019-1177-1Infinitely many geometrically distinct solutions for periodic Schrödinger–Poisson systemsJing Chen0Ning Zhang1School of Mathematics and Computing Sciences, Hunan University of Science and TechnologySchool of Mathematics and Statistics, Central South UniversityAbstract This paper is dedicated to studying the following Schrödinger–Poisson system: {−△u+V(x)u+K(x)ϕ(x)u=f(x,u),x∈R3,−△ϕ=K(x)u2,x∈R3, $$ \textstyle\begin{cases} -\triangle u+V(x)u+K(x)\phi (x)u=f(x, u), \quad x\in {\mathbb {R}}^{3}, \\ -\triangle \phi =K(x)u^{2}, \quad x\in {\mathbb {R}}^{3}, \end{cases} $$ where V(x) $V(x)$, K(x) $K(x)$, and f(x,u) $f(x, u)$ are periodic in x. By using the non-Nehari manifold method, we establish the existence of ground state solutions for the above problem under some weak assumptions. Moreover, when f is odd in u, we prove that the above problem admits infinitely many geometrically distinct solutions. Our results improve and complement some related literature.http://link.springer.com/article/10.1186/s13661-019-1177-1Schrödinger–Poisson systemNehari manifoldGround stateGeometrically distinct solutions |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Jing Chen Ning Zhang |
spellingShingle |
Jing Chen Ning Zhang Infinitely many geometrically distinct solutions for periodic Schrödinger–Poisson systems Boundary Value Problems Schrödinger–Poisson system Nehari manifold Ground state Geometrically distinct solutions |
author_facet |
Jing Chen Ning Zhang |
author_sort |
Jing Chen |
title |
Infinitely many geometrically distinct solutions for periodic Schrödinger–Poisson systems |
title_short |
Infinitely many geometrically distinct solutions for periodic Schrödinger–Poisson systems |
title_full |
Infinitely many geometrically distinct solutions for periodic Schrödinger–Poisson systems |
title_fullStr |
Infinitely many geometrically distinct solutions for periodic Schrödinger–Poisson systems |
title_full_unstemmed |
Infinitely many geometrically distinct solutions for periodic Schrödinger–Poisson systems |
title_sort |
infinitely many geometrically distinct solutions for periodic schrödinger–poisson systems |
publisher |
SpringerOpen |
series |
Boundary Value Problems |
issn |
1687-2770 |
publishDate |
2019-03-01 |
description |
Abstract This paper is dedicated to studying the following Schrödinger–Poisson system: {−△u+V(x)u+K(x)ϕ(x)u=f(x,u),x∈R3,−△ϕ=K(x)u2,x∈R3, $$ \textstyle\begin{cases} -\triangle u+V(x)u+K(x)\phi (x)u=f(x, u), \quad x\in {\mathbb {R}}^{3}, \\ -\triangle \phi =K(x)u^{2}, \quad x\in {\mathbb {R}}^{3}, \end{cases} $$ where V(x) $V(x)$, K(x) $K(x)$, and f(x,u) $f(x, u)$ are periodic in x. By using the non-Nehari manifold method, we establish the existence of ground state solutions for the above problem under some weak assumptions. Moreover, when f is odd in u, we prove that the above problem admits infinitely many geometrically distinct solutions. Our results improve and complement some related literature. |
topic |
Schrödinger–Poisson system Nehari manifold Ground state Geometrically distinct solutions |
url |
http://link.springer.com/article/10.1186/s13661-019-1177-1 |
work_keys_str_mv |
AT jingchen infinitelymanygeometricallydistinctsolutionsforperiodicschrodingerpoissonsystems AT ningzhang infinitelymanygeometricallydistinctsolutionsforperiodicschrodingerpoissonsystems |
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1724942900725284864 |