JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2019, Vol. 54 ›› Issue (10): 74-78.doi: 10.6040/j.issn.1671-9352.0.2018.638

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Existence of nontrival solutions for a class of Schrödinger-Poisson systems

CHEN Li-zhen1, FENG Xiao-jing2, LI Gang3   

  1. 1. School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, Shanxi, China;
    2. School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, China;
    3. School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, Jiangsu, China
  • Published:2019-10-12

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Abstract: We investigate a class of Schrödinger-Poisson systems, by means of variational method and critical point theory. Here, the Poisson term is a more general form. By adding quasi-critical growth and AR conditions to the nonlinear term, we prove the existence of nontrival solution of the system. The result supplement and promote the previous resluts on the Schrödinger-Poisson systems.

Key words: Schrö, dinger-Poisson system, variational method, mountain pass theorem

CLC Number: 

  • O176.3
[1] ILLNER R, ZWEIFEL P F, LANGE H. Global existence,uniqueness and asymptotic behaviour of solutions of the Wigner-Poisson and Schrödinger-Poisson systems[J]. Mathematical Methods in the Applied Sciences, 1994,17(5):349-376.
[2] NIER F. A stationary Schrödinger-Poisson system arising from the modelling of electronic devices[J]. Forum Mathematicum, 1990, 2(5):489-510.
[3] LI Fuyi, LI Yuhua, SHI Junping. Existence of positive solutions to Schrödinger-Poisson type systems with critical exponent[J]. Communications in Contemporary Mathematics, 2014, 16(6):1-29.
[4] LI Yuhua, LI Fuyi, SHI Junping. Existence and multiplicity of positive solutions to Schrödinger-Poisson type systems with critical nonlocal term[J]. Calculus of Variations, 2017, 56(134):1-17.
[5] MAO Anmin, YANG Lijuan, QIAN Aixia, et al. Existence and concentration of solutions of Schrödinger-Poisson system[J]. Applied Mathematics Letters, 2017, 68:8-12.
[6] 叶一蔚, 唐春雷. 带有超线性项或次线性项的Schrödinger-Poisson系统解的存在性和多重性[J]. 数学物理学报, 2015, 35A(4):668-682. YE Yiwei, TANG Chunlei. Existence and multiplicity results for Schrödinger-Poisson system with superlinear or sublinear terms[J]. Acta Mathematica Scientia, 2015, 35(A):668-682.
[7] 毛安民, 李安然. 薛定谔方程及薛定谔-麦克斯韦方程的多解[J]. 数学学报, 2012, 55(3):425-436. MAO Anmin, LI Anran. Multiplicity of solutions for a Schrödinger equation and a Schrödinger-Maxwell equation[J]. Acta Mathematica Sinica, Chinese Series, 2012, 55(3):425-436.
[8] WILLE M. Minimax theorems: progress in nonlinear differential equations and their applications[M]. Boston: Birkhäuser, 1996.
[9] BARTSCH T, WANG Zhiqiang. Existence and multiple results for some superlinear elliptic problems on RN[J]. Communication in Partial Differential Equations, 1995, 20:1725-1741.
[10] RABINOWITZ P H. Minimax methods in critical point theory with applications to differential equations[M]. Providence, RI: AMS, 1986.
[11] BARTOLO P, BENCI V, FORTUNATO D. Abstract critical point theorems and applications to some nonlinear problem with strong resonance at infinity[J]. Nonlinear Analysis, 1983, 7:981-1012.
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