We consider the Schrödinger-Poisson system with nonlinear term $Q(x)|u|^{p-1}u$, where the value of $\displaystyle\lim_{|x|\rightarrow\infty} Q(x)$ 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.
Yao DU
,
Chunlei TANG
. GROUND STATE SOLUTIONS FOR A SCHRÖDINGER-POISSON SYSTEM WITH UNCONVENTIONAL POTENTIAL[J]. Acta mathematica scientia, Series B, 2020
, 40(4)
: 934
-944
.
DOI: 10.1007/s10473-020-0404-2
[1] Benci V, Fortunato D. An eigenvalue problem for the Schrödinger-Maxwell equations. Topol Methods Nonlinear Anal, 1998, 11(2):283-293
[2] Benci V, Fortunato D. Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations. Rev Math Phys, 2002, 14(4):409-420
[3] Ambrosetti A. On Schrödinger-Poisson systems. Milan J Math, 2008, 76(1):257-274
[4] Ambrosetti A, Ruiz D. Multiple bound states for the Schrödinger-Poisson problem. Commun Contemp Math, 2008, 10(3):391-404
[5] Azzollini A, Pomponio A. Ground state solutions for the nonlinear Schrödinger-Maxwell equations. J Math Anal Appl, 2008, 345(1):90-108
[6] 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
[7] d'Avenia P. Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations. Adv Nonlinear Stud, 2002, 2(2):177-192
[8] D'Aprile T, Mugnai D. Non-existence results for the coupled Klein-Gordon-Maxwell equations. Adv Nonlinear Stud, 2004, 4(3):307-322
[9] D'Aprile T, Mugnai D. Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations. Proc Roy Soc Edinburgh Sect A, 2004, 134(5):893-906
[10] Ruiz D. The Schrödinger-Poisson equation under the effect of a nonlinear local term. J Funct Anal, 2006, 237(2):655-674
[11] Jiang Y S, Zhou H S. Schrödinger-Poisson system with steep potential well. J Differential Equations, 2011, 251(3):582-608
[12] Zhao L G, Liu H D, Zhao F K. Existence and concentration of solutions for Schrödinger-Pisson equations with steep well potential. J Differential Equations, 2013, 255(1):1-23
[13] Zhao L G, Zhao F K. On the existence of solutions for the Schrödinger-Poisson equations. J Math Anal Appl, 2008, 346(1):155-169
[14] Cerami G, Vaira G. Positive solutions for some non-autonomous Schrödinger-Poisson systems. J Differential Equations, 2010, 248(3):521-543
[15] Vaira G. Ground states for Schrödinger-Poisson type systems. Ric Mat, 2011, 60(2):263-297
[16] 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
[17] 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
[18] Cerami G, Molle R. Positive bound state solutions for some Schrödinger-Poisson systems. Nonlinearity, 2016, 29(10):3103-3119
[19] Chen J Q. Multiple positive solutions of a class of non autonomous Schrödinger-Poisson systems. Nonlinear Anal Real World Appl, 2015, 21:13-26
[20] 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
[21] Wang C H, Yang J. Positive solution for a nonlinear Schrödinger-Poisson system. Discrete Contin Dyn Syst, 2018, 38(11):5461-5504
[22] Chen S J, Tang C L. High energy solutions for the superlinear Schrödinger-Maxwell equations. Nonlinear Anal, 2009, 71(10):4927-4934
[23] Zhong X J, Tang C L. Ground state sign-changing solutions for a Schrödinger-Poisson system with a critical nonlinearity in $\mathbb{R}^3$. Nonlinear Anal Real World Appl, 2018, 39:166-184
[24] 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
[25] Ruiz D. Semiclassical states for coupled Schrödinger-Maxwell equations:concentration around a sphere. Math Models Methods Appl Sci, 2005, 15(1):141-164
[26] 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
[27] He Y, Li G B. Standing waves for a class of Schrödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents. Ann Acad Sci Fenn Math, 2015, 40(2):729-766
[28] Kwong M K. Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbb{R}^N$. Arch Rational Mech Anal, 1989, 105(3):243-266
[29] Berestycki H, Lions P L. Nonlinear scalar field equations. I. Existence of a ground state. Arch Rational Mech Anal, 1983, 82(4):313-345
[30] 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
[31] Willem M. Minimax Theorems. Boston:Birkhäuser, 1996
[32] Brézis H, Lieb E. A relation between pointwise convergence of functions and convergence of functionals. Proc Amer Math Soc, 1983, 88(3):486-490
