MULTIPLE SOLUTIONS FOR A NONHOMOGENEOUS SCHRÖDINGER-POISSON SYSTEM WITH CONCAVE AND CONVEX NONLINEARITIES

Lixia Wang; Shiwang Ma

doi:10.11948/2156-907X.20180132

2019 Volume 9 Issue 2

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Article navigation > Journal of Applied Analysis & Computation > 2019 > 9(2): 628-637

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Lixia Wang, Shiwang Ma. MULTIPLE SOLUTIONS FOR A NONHOMOGENEOUS SCHRÖDINGER-POISSON SYSTEM WITH CONCAVE AND CONVEX NONLINEARITIES[J]. Journal of Applied Analysis & Computation, 2019, 9(2): 628-637. doi: 10.11948/2156-907X.20180132

Citation:

Lixia Wang, Shiwang Ma. MULTIPLE SOLUTIONS FOR A NONHOMOGENEOUS SCHRÖDINGER-POISSON SYSTEM WITH CONCAVE AND CONVEX NONLINEARITIES[J]. Journal of Applied Analysis & Computation, 2019, 9(2): 628-637. doi: 10.11948/2156-907X.20180132

MULTIPLE SOLUTIONS FOR A NONHOMOGENEOUS SCHRÖDINGER-POISSON SYSTEM WITH CONCAVE AND CONVEX NONLINEARITIES

Lixia Wang^1,2, ,,
Shiwang Ma³

1.
Center for Applied Mathematics, Tianjin University, Weijin Road, 300072 Tianjin, China
2.
School of Sciences, Tianjin Chengjian University, Jinjing Road, 300384 Tianjin, China
3.
School of Mathematical Sciences and LPMC, Nankai University, Weijin Road, 300071 Tianjin, China

Corresponding author: Email address: wanglixia0311@126.com(L. Wang)
Fund Project: The authors were supported by National Natural Science Foundation of China (11801400, 11571187) and Postdoctoral Science Foundation of China (2017M611159)

Abstract

Abstract

In this paper, we consider the following nonhomogeneous Schrödinger-Poisson equation $ (*)\left\{ \begin{align} & -\Delta u+V(x)u+\phi (x)u=-k(x)|u{{|}^{q-2}}u+h(x)|u{{|}^{p-2}}u+g(x),\ \ \ \ \ x\in {{\mathbb{R}}^{3}}, \\ & -\Delta \phi ={{u}^{2}},\quad {{\lim }_{\text{ }\!\!|\!\!\text{ }x|\to +\infty }}\,\phi (x)=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in {{\mathbb{R}}^{3}}, \\ \end{align} \right. $ where $ 1< q < 2, 4 < p < 6$. Under some suitable assumptions on $V(x), k(x), h(x)$ and $g(x)$, the existence of multiple solutions is proved by using the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory.
- Schrödinger-Poisson systems /
- concave and convex nonlinearities /
- variational methods /
- Ekeland's variational principle /
- Mountain Pass Theorem
MSC: 35B33, 35J65,35Q55

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References

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