Abstract
We consider the Schrödinger-Poisson system with nonlinear term Q(x)|u|p−1u, where the value of \(\mathop {\lim}\limits_{\left| x \right| \to \infty} \,\,Q\left(x \right)\)Q(x) may not exist and Q may change sign. This means that the problem may have no limit problem. The existence of nonnegative ground state solutions is established. Our method relies upon the variational method and some analysis tricks.
Similar content being viewed by others
References
Benci V, Fortunato D. An eigenvalue problem for the Schrödinger-Maxwell equations. Topol Methods Nonlinear Anal, 1998, 11(2):283–293
Benci V, Fortunato D. Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations. Rev Math Phys, 2002, 14(4):409–420
Ambrosetti A. On Schrödinger-Poisson systems. Milan J Math, 2008, 76(1):257–274
Ambrosetti A, Ruiz D. Multiple bound states for the Schrödinger-Poisson problem. Commun Contemp Math, 2008, 10(3):391–404
Azzollini A, Pomponio A. Ground state solutions for the nonlinear Schröodinger-Maxwell equations. J Math Anal Appl, 2008, 345(1):90–108
Azzollini A, d’Avenia P, Pomponio A. On the Schrödinger-Maxwell equations under the effect of a general nonlinear term. Ann Inst H Poincaré Anal Non Linéaire, 2010, 27(2):779–791
d’Avenia P. Non-radially symmetric solutions of nonlinear Schröodinger equation coupled with Maxwell equations. Adv Nonlinear Stud, 2002, 2(2):177–192
D’Aprile T, Mugnai D. Non-existence results for the coupled Klein-Gordon-Maxwell equations. Adv Nonlinear Stud, 2004, 4(3):307–322
D’Aprile T, Mugnai D. Solitary waves for nonlinear Klein-Gordon-Maxwell and Schröodinger-Maxwell equations. Proc Roy Soc Edinburgh Sect A, 2004, 134(5):893–906
Ruiz D. The Schrödinger-Poisson equation under the effect of a nonlinear local term. J Funct Anal, 2006, 237(2):655–674
Jiang Y S, Zhou H S. Schrödinger-Poisson system with steep potential well. J Differential Equations, 2011, 251(3):582–608
Zhao L G, Liu H D, Zhao F K. Existence and concentration of solutions for Schroödinger-Pisson equations with steep well potential. J Differential Equations, 2013, 255(1):1–23
Zhao L G, Zhao F K. On the existence of solutions for the Schröodinger-Poisson equations. J Math Anal Appl, 2008, 346(1):155–169
Cerami G, Vaira G. Positive solutions for some non-autonomous Schrödinger-Poisson systems. J Differential Equations, 2010, 248(3):521–543
Vaira G. Ground states for Schroödinger-Poisson type systems. Ric Mat, 2011, 60(2):263–297
Li G B, Peng S J, Yan S S. Infinitely many positive solutions for the nonlinear Schrödinger-Poisson system. Commun Contemp Math, 2010, 12(6):1069–1092
Batista A M, Furtado M F. Positive and nodal solutions for a nonlinear Schrödinger-Poisson system with sign-changing potentials. Nonlinear Anal Real World Appl, 2018, 39: 142–156
Cerami G, Molle R. Positive bound state solutions for some Schrödinger-Poisson systems. Nonlinearity, 2016, 29(10):3103–3119
Chen J Q. Multiple positive solutions of a class of non autonomous Schrödinger-Poisson systems. Nonlinear Anal Real World Appl, 2015, 21: 13–26
Huang L R, Rocha E M, Chen J Q. Two positive solutions of a class of Schrödinger-Poisson system with indefinite nonlinearity. J Differential Equations, 2013, 255(8):2463–2483
Wang C H, Yang J. Positive solution for a nonlinear Schrödinger-Poisson system. Discrete Contin Dyn Syst, 2018, 38(11):5461–5504
Chen S J, Tang C L. High energy solutions for the superlinear Schrödinger-Maxwell equations. Nonlinear Anal, 2009, 71(10):4927–4934
Zhong X J, Tang C L. Ground state sign-changing solutions for a Schrödinger-Poisson system with a critical nonlinearity in ℝ3. Nonlinear Anal Real World Appl, 2018, 39: 166–184
Ye Y W, Tang C L. Existence and multiplicity results for the Schrödinger-Poisson system with superlinear or sublinear terms. Acta Math Sci, 2015, 35A: 668–682
Ruiz D. Semiclassical states for coupled Schrödinger-Maxwell equations: concentration around a sphere. Math Models Methods Appl Sci, 2005, 15(1):141–164
Ianni I, Vaira G. On concentration of positive bound states for the Schrödinger-Poisson problem with potentials. Adv Nonlinear Stud, 2008, 8(3):573–595
He Y, Li G B. Standing waves for a class of Schrödinger-Poisson equations in ℝ3 involving critical Sobolev exponents. Ann Acad Sci Fenn Math, 2015, 40(2):729–766
Kwong M K. Uniqueness of positive solutions of Δu − u + up = 0 in ℝN. Arch Rational Mech Anal, 1989, 105(3):243–266
Berestycki H, Lions P L. Nonlinear scalar field equations. I. Existence of a ground state. Arch Rational Mech Anal, 1983, 82(4):313–345
Yang J F, Yu X H. Existence of solutions for a semilinear elliptic equation in ℝN with sign-changing weight. Adv Nonlinear Stud, 2008, 8(2):401–412
Willem M. Minimax Theorems. Boston: Birkhäuser, 1996
Brézis H, Lieb E. A relation between pointwise convergence of functions and convergence of functionals. Proc Amer Math Soc, 1983, 88(3): 486–490
Author information
Authors and Affiliations
Corresponding author
Additional information
The second author was supported by National Natural Science Foundation of China (11471267) and the first author is supported by Graduate Student Scientific Research Innovation Projects of Chongqing (CYS17084).
Rights and permissions
About this article
Cite this article
Du, Y., Tang, C. Ground state solutions for a Schrödinger-Poisson system with unconventional potential. Acta Math Sci 40, 934–944 (2020). https://doi.org/10.1007/s10473-020-0404-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10473-020-0404-2