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AIMS Mathematics

2021, Volume 6, Issue 7: 7833-7844. doi: 10.3934/math.2021455
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Research article

Existence of axially symmetric solutions for a kind of planar Schrödinger-Poisson system

  • 1.
    School of Science, Guilin University of Technology, Guilin, Guangxi 541004, China
  • 2.
    School of Science, Guilin University of Aerospace Technology, Guilin, Guangxi 541004, China
  • Received: 13 March 2021 Accepted: 10 May 2021 Published: 18 May 2021

  • MSC : 35J20, 35J62, 35Q55

  • In this paper, we study the following kind of Schrödinger-Poisson system in R2

    {−Δu+V(x)u+ϕu=K(x)f(u),   x∈R2,Δϕ=u2,                                       x∈R2,

    where f∈C(R,R), V(x) and K(x) are both axially symmetric functions. By constructing a new variational framework and using some new analytic techniques, we obtain an axially symmetric solution for the above planar system. Our result improves and extends the existing works.

    Citation: Qiongfen Zhang, Kai Chen, Shuqin Liu, Jinmei Fan. Existence of axially symmetric solutions for a kind of planar Schrödinger-Poisson system[J]. AIMS Mathematics, 2021, 6(7): 7833-7844. doi: 10.3934/math.2021455

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  • In this paper, we study the following kind of Schrödinger-Poisson system in R2

    {−Δu+V(x)u+ϕu=K(x)f(u),   x∈R2,Δϕ=u2,                                       x∈R2,

    where f∈C(R,R), V(x) and K(x) are both axially symmetric functions. By constructing a new variational framework and using some new analytic techniques, we obtain an axially symmetric solution for the above planar system. Our result improves and extends the existing works.




    [1] S. Chen, X. Tang, On the planar Schrödinger-Poisson system with the axially symmetric potential, J. Differ. Equations, 268 (2020), 945–976. doi: 10.1016/j.jde.2019.08.036
    [2] R. Benguria, H. Brezis, E. Lieb, The Thomas-Fermi-von Weizsäcker theory of atoms and molecules, Comm. Math. Phys., 79 (1981), 167–180. doi: 10.1007/BF01942059
    [3] A. Paredes, D. Olivieri, M. Humberto, From optics to dark matter: A review on nonlinear Schrödinger-Poisson systems, Physica D, 403 (2020), 132301. doi: 10.1016/j.physd.2019.132301
    [4] J. Chen, S. Chen, X. Tang, Ground state solutions for asymptotically periodic Schrödinger-Poisson systems in R2, Electron. J. Differ. Equations, 192 (2018), 1–18.
    [5] X. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equation, Sci. China Math., 58 (2015), 715–728.
    [6] F. Bernini, D. Mugnai, On a logarithmic Hartree equation, Adv. Nonlinear Anal., 9 (2020), 850–865.
    [7] S. Cingolani, T. Weth, On the planar Schrödinger-Poisson system, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 33 (2016), 169–197. doi: 10.1016/j.anihpc.2014.09.008
    [8] M. Du, T. Weth, Ground states and high energy solutions of the planar Schrödinger-Poisson system, Nonlinearity, 30 (2017), 3492–3515. doi: 10.1088/1361-6544/aa7eac
    [9] S. Chen, J. Shi, X. Tang, Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity, Discrete Contin. Dyn. Syst., Ser. A, 39 (2019), 5867–5889. doi: 10.3934/dcds.2019257
    [10] S. Chen, X. Tang, Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system, Discrete Contin. Dyn. Syst., Ser. B, 24 (2019), 4685–4702.
    [11] L. Wen, S. Chen, V. D. Rădulescu, Axially symmetric solutions of Schrödinger-Poisson system with zero mass potential in R2, Appl. Math. Lett., 104 (2020), 106244. doi: 10.1016/j.aml.2020.106244
    [12] Z. Chen, W. Zou, Ground states for a system of Schrödinger equations with critical exponent, J. Func. Anal., 262 (2012), 3091–3107. doi: 10.1016/j.jfa.2012.01.001
    [13] S. Chen, X. Tang, Berestycki-Lions conditions on ground state solutions for a nonlinear Schrödinger equation with variable potentials, Adv. Nonlinear Anal., 9 (2020), 496–515.
    [14] J. Peng, S. Chen, X. Tang, Semiclassical solutions for linearly coupled Schrödinger equations without compactness, Complex Var. Elliptic Equations, 64 (2019), 548–556. doi: 10.1080/17476933.2018.1450395
    [15] X. Tang, New super-quadratic conditions for asymptotically periodic Schrödinger equations, Can. Math. Bull., 60 (2017), 422–435. doi: 10.4153/CMB-2016-090-2
    [16] X. Tang, X. Lin, J. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Differ. Equations, 31 (2019), 369–383. doi: 10.1007/s10884-018-9662-2
    [17] G. Che, H. Chen, Existence of multiple nontrivial solutions for a class of quasilinear Schrödinger equations on RN, Bull. Belg. Math. Soc. Simon Stevin, 25 (2018), 39–53.
    [18] X. Tang, Non-Nehari manifold method for asymptotically linear Schrödinger equation, J. Aust. Math. Soc., 98 (2015), 104–116. doi: 10.1017/S144678871400041X
    [19] B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equations in RN, Ann. Mat. Pura Appl., 183 (2002), 73–83.
    [20] J. Fan, Y. Jiang, Q. Zhang, Semiclassical Solutions for a Kind of Coupled Schrödinger Equations, Adv. Math. Phys., 2020 (2020), 4378691.
    [21] X. Zhang, Existence and multiplicity of solutions for a class of elliptic boundary value problems, J. Math. Anal. Appl., 410 (2014), 213–226. doi: 10.1016/j.jmaa.2013.08.001
    [22] Q. Zhang, C. Gan, T. Xiao, Z. Jia, Some results of nontrivial solutions for Klein-Gordon-Maxwell systems with local super-quadratic conditions, J. Geom. Anal., 31 (2021), 5372–5394. doi: 10.1007/s12220-020-00483-2
    [23] L. Wang, X. Zhang, H. Fang, Multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces, Taiwan. J. Math., 21 (2017), 881–912.
    [24] Q. Zhang, C. Gan, T. Xiao, Z. Jia, An improved result for Klein-Gordon-Maxwell systems with steep potential well, Math. Methods Appl. Sci., 2020 (2020).
    [25] E. Lieb, M. Loss, Analysis, 2nd Ed., Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 2001.

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