2007, 18(4): 809-816. doi: 10.3934/dcds.2007.18.809

Positive solution for a nonlinear stationary Schrödinger-Poisson system in $R^3$

1. 

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O.Box 71010, Wuhan 430071, China, China

Received  July 2006 Revised  March 2007 Published  May 2007

In this paper, we are concerned with the following problem

(P) $ -\Delta u + V(x)u+\lambda \phi (x) u =f(x,u), x\in \mathbb{R}^3$
$ -\Delta\phi = u^2, \lim_{|x|\rightarrow +\infty}\phi(x)=0,$

where $\lambda >0$ is a parameter, the potential $V(x)$ may not be radially symmetric, and $f(x,s)$ is asymptotically linear with respect to $s$ at infinity. Under some simple assumptions on $V$ and $f$, we prove that the problem (P) has a positive solution for $\lambda$ small and has no any nontrivial solution for $\lambda$ large.

Citation: Zhengping Wang, Huan-Song Zhou. Positive solution for a nonlinear stationary Schrödinger-Poisson system in $R^3$. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 809-816. doi: 10.3934/dcds.2007.18.809
[1]

Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1345-1360. doi: 10.3934/dcdsb.2013.18.1345

[2]

Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003

[3]

Addolorata Salvatore. Sign--changing solutions for an asymptotically linear Schrödinger equation. Conference Publications, 2009, 2009 (Special) : 669-677. doi: 10.3934/proc.2009.2009.669

[4]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912

[5]

Alireza Khatib, Liliane A. Maia. A positive bound state for an asymptotically linear or superlinear Schrödinger equation in exterior domains. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2789-2812. doi: 10.3934/cpaa.2018132

[6]

Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074

[7]

Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345

[8]

Claudianor O. Alves, Giovany M. Figueiredo, Marcelo F. Furtado. Multiplicity of solutions for elliptic systems via local Mountain Pass method. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1745-1758. doi: 10.3934/cpaa.2009.8.1745

[9]

Xiang-Dong Fang. A positive solution for an asymptotically cubic quasilinear Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (1) : 51-64. doi: 10.3934/cpaa.2019004

[10]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[11]

Oliver Knill. A deterministic displacement theorem for Poisson processes. Electronic Research Announcements, 1997, 3: 110-113.

[12]

Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159

[13]

Shiwang Ma. Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2361-2380. doi: 10.3934/cpaa.2013.12.2361

[14]

François Genoud. Orbitally stable standing waves for the asymptotically linear one-dimensional NLS. Evolution Equations & Control Theory, 2013, 2 (1) : 81-100. doi: 10.3934/eect.2013.2.81

[15]

Yuanhong Wei, Xifeng Su. On a class of non-local elliptic equations with asymptotically linear term. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6287-6304. doi: 10.3934/dcds.2018154

[16]

Paolo Gidoni, Alessandro Margheri. Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 585-605. doi: 10.3934/dcds.2019024

[17]

Antonio Azzollini, Pietro d’Avenia, Valeria Luisi. Generalized Schrödinger-Poisson type systems. Communications on Pure & Applied Analysis, 2013, 12 (2) : 867-879. doi: 10.3934/cpaa.2013.12.867

[18]

Cyril Joel Batkam, João R. Santos Júnior. Schrödinger-Kirchhoff-Poisson type systems. Communications on Pure & Applied Analysis, 2016, 15 (2) : 429-444. doi: 10.3934/cpaa.2016.15.429

[19]

Daniele Garrisi, Vladimir Georgiev. Orbital stability and uniqueness of the ground state for the non-linear Schrödinger equation in dimension one. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4309-4328. doi: 10.3934/dcds.2017184

[20]

César E. Torres Ledesma. Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well. Communications on Pure & Applied Analysis, 2016, 15 (2) : 535-547. doi: 10.3934/cpaa.2016.15.535

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (89)

Other articles
by authors

[Back to Top]