# American Institute of Mathematical Sciences

September  2021, 14(9): 3055-3066. doi: 10.3934/dcdss.2020339

## Existence criteria of ground state solutions for Schrödinger-Poisson systems with a vanishing potential

 1 School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, P.R. China 2 School of Mathematics and Statistics, Guangxi Normal University, Guilin, Guangxi 541004, P.R.China

* Corresponding author: Xianhua Tang

Received  May 2019 Revised  September 2019 Published  September 2021 Early access  April 2020

Fund Project: This work is partially supported by the National Natural Science Foundation of China (No: 11571370)

In this paper, we consider the following Schrödinger-Poisson system
 $\begin{equation*} \left\{ \begin{array}{ll} -\triangle u+u+K(x)\phi(x)u = a(x)|u|^{p-2}u, \ \ \ \ x\in { \mathbb{R}}^{3},\\ -\triangle \phi = K(x)u^2, \ \ \ \ x\in { \mathbb{R}}^{3}, \end{array} \right. \end{equation*}$
where
 $p\in [4,6)$
,
 $a(x)\ge \lim_{|x|\to\infty}a(x) = a_{\infty}>0$
and
 $\lim_{|x|\to\infty}K(x) = 0$
. Lack of any symmetry property of
 $a$
and
 $K$
, we present some new sufficient conditions to guarantee the existence of a positive ground state solution of above system. Our results extend and complement the ones of [G. Cerami, G. Vaira, J. Differential Equations 248 (2010)] in which
 $p\in (4,6)$
,
 $a$
and
 $K$
need to satisfy additional integrability conditions.
Citation: Sitong Chen, Wennian Huang, Xianhua Tang. Existence criteria of ground state solutions for Schrödinger-Poisson systems with a vanishing potential. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3055-3066. doi: 10.3934/dcdss.2020339
##### References:
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Differential Equations, 248 (2010), 521-543.  doi: 10.1016/j.jde.2009.06.017.  Google Scholar [11] G. Cerami and R. Molle, Positive bound state solutions for some Schrödinger-Poisson systems, Nonlinearity, 29 (2016), 3103-3119.  doi: 10.1088/0951-7715/29/10/3103.  Google Scholar [12] S. T. Chen and X. H. Tang, Nehari type ground state solutions for asymptotically periodic Schrödinger-Poisson systems, Taiwan. J. Math., 21 (2017), 363-383.  doi: 10.11650/tjm/7784.  Google Scholar [13] S. T. Chen and X. H. Tang, Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete Contin. Dyn. Syst., 38 (2018), 2333-2348.  doi: 10.3934/dcds.2018096.  Google Scholar [14] S. T. Chen and X. H. Tang, On the planar Schrödinger-Poisson system with the axially symmetric potential, J. Differential Equations, 268 (2020), 945-976.  doi: 10.1016/j.jde.2019.08.036.  Google Scholar [15] S. T. Chen, A. Fiscella, P. Pucci and X. H. Tang, Semiclassical ground state solutions for critical Schrödinger-Poisson systems with lower perturbations, J. Differential Equations, 268 (2020), 2672-2716.  doi: 10.1016/j.jde.2019.09.041.  Google Scholar [16] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1978), 209-243.  doi: 10.1007/BF01221125.  Google Scholar [17] M. K. Kwong, Uniqueness of positive solution of $\triangle u-u+u^p = 0$ in $\mathbb{R}^N$, Arch. Rat. Math. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar [18] E. H. Lieb, Thomas-Fermi and related theories and molecules, Rev. Modern Phys., 53 (1981), 603-641.  doi: 10.1103/RevModPhys.53.603.  Google Scholar [19] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev inequality and related inequalities, Ann. of Math., 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar [20] E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 1997.  Google Scholar [21] P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1984), 33-97.  doi: 10.1007/BF01205672.  Google Scholar [22] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar [23] N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Nonlinear Analysis- Theory and Methods, Springer Monographs in Mathematics. Springer, Cham, 2019. doi: 10.1007/978-3-030-03430-6.  Google Scholar [24] D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674.  doi: 10.1016/j.jfa.2006.04.005.  Google Scholar [25] D. Ruiz, On the Schrödinger-Poisson-Slater system: Behavior of minimizers, radial and nonradial cases, Arch. Ration. Mech. Anal., 198 (2010), 349-368.  doi: 10.1007/s00205-010-0299-5.  Google Scholar [26] J. J. Sun and S. W. Ma, Ground state solutions for some Schrödinger-Poisson systems with periodic potentials, J. Differential Equations, 260 (2016), 2119-2149.  doi: 10.1016/j.jde.2015.09.057.  Google Scholar [27] X. H. Tang, Non-Nehari manifold method for asymptotically linear Schrödinger equation, J. Aust. Math. Soc., 98 (2015), 104-116.  doi: 10.1017/S144678871400041X.  Google Scholar [28] X. H. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equation, Sci. China Math., 58 (2015), 715-728.  doi: 10.1007/s11425-014-4957-1.  Google Scholar [29] X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials, Discrete Contin. Dyn. Syst., 37 (2017), 4973-5002.  doi: 10.3934/dcds.2017214.  Google Scholar [30] X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 56 (2017), Art. 110, 25 pp. doi: 10.1007/s00526-017-1214-9.  Google Scholar [31] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar [32] L. G. Zhao and F. K. Zhao, On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl., 346 (2008), 155-169.  doi: 10.1016/j.jmaa.2008.04.053.  Google Scholar

show all references

##### References:
 [1] A. Ambrosetti, On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.  doi: 10.1007/s00032-008-0094-z.  Google Scholar [2] A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.  doi: 10.1142/S021919970800282X.  Google Scholar [3] A. Azzollini, P. d'Avenia and A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 779-791.  doi: 10.1016/j.anihpc.2009.11.012.  Google Scholar [4] A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108.  doi: 10.1016/j.jmaa.2008.03.057.  Google Scholar [5] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.  doi: 10.12775/TMNA.1998.019.  Google Scholar [6] V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.  doi: 10.1142/S0129055X02001168.  Google Scholar [7] R. Benguria, H. Brezis and E. H. Lieb, The Thomas-Fermi-von Weizsäcker theory of atoms and molecules, Comm. Math. Phys., 79 (1981), 167-180.  doi: 10.1007/BF01942059.  Google Scholar [8] I. Catto and P.-L. Lions, Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. I: A necessary and sufficient condition for the stability of general molecular system, Comm. Partial Differential Equations, 17 (1992), 1051-1110.  doi: 10.1080/03605309208820878.  Google Scholar [9] G. Cerami, Some nonlinear elliptic problems in unbounded domains, Milan J. Math., 74 (2006), 47-77.  doi: 10.1007/s00032-006-0059-z.  Google Scholar [10] G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 248 (2010), 521-543.  doi: 10.1016/j.jde.2009.06.017.  Google Scholar [11] G. Cerami and R. Molle, Positive bound state solutions for some Schrödinger-Poisson systems, Nonlinearity, 29 (2016), 3103-3119.  doi: 10.1088/0951-7715/29/10/3103.  Google Scholar [12] S. T. Chen and X. H. Tang, Nehari type ground state solutions for asymptotically periodic Schrödinger-Poisson systems, Taiwan. J. Math., 21 (2017), 363-383.  doi: 10.11650/tjm/7784.  Google Scholar [13] S. T. Chen and X. H. Tang, Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete Contin. Dyn. Syst., 38 (2018), 2333-2348.  doi: 10.3934/dcds.2018096.  Google Scholar [14] S. T. Chen and X. H. Tang, On the planar Schrödinger-Poisson system with the axially symmetric potential, J. Differential Equations, 268 (2020), 945-976.  doi: 10.1016/j.jde.2019.08.036.  Google Scholar [15] S. T. Chen, A. Fiscella, P. Pucci and X. H. Tang, Semiclassical ground state solutions for critical Schrödinger-Poisson systems with lower perturbations, J. Differential Equations, 268 (2020), 2672-2716.  doi: 10.1016/j.jde.2019.09.041.  Google Scholar [16] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1978), 209-243.  doi: 10.1007/BF01221125.  Google Scholar [17] M. K. Kwong, Uniqueness of positive solution of $\triangle u-u+u^p = 0$ in $\mathbb{R}^N$, Arch. Rat. Math. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar [18] E. H. Lieb, Thomas-Fermi and related theories and molecules, Rev. Modern Phys., 53 (1981), 603-641.  doi: 10.1103/RevModPhys.53.603.  Google Scholar [19] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev inequality and related inequalities, Ann. of Math., 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar [20] E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 1997.  Google Scholar [21] P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1984), 33-97.  doi: 10.1007/BF01205672.  Google Scholar [22] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar [23] N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Nonlinear Analysis- Theory and Methods, Springer Monographs in Mathematics. Springer, Cham, 2019. doi: 10.1007/978-3-030-03430-6.  Google Scholar [24] D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674.  doi: 10.1016/j.jfa.2006.04.005.  Google Scholar [25] D. Ruiz, On the Schrödinger-Poisson-Slater system: Behavior of minimizers, radial and nonradial cases, Arch. Ration. Mech. Anal., 198 (2010), 349-368.  doi: 10.1007/s00205-010-0299-5.  Google Scholar [26] J. J. Sun and S. W. Ma, Ground state solutions for some Schrödinger-Poisson systems with periodic potentials, J. Differential Equations, 260 (2016), 2119-2149.  doi: 10.1016/j.jde.2015.09.057.  Google Scholar [27] X. H. Tang, Non-Nehari manifold method for asymptotically linear Schrödinger equation, J. Aust. Math. Soc., 98 (2015), 104-116.  doi: 10.1017/S144678871400041X.  Google Scholar [28] X. H. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equation, Sci. China Math., 58 (2015), 715-728.  doi: 10.1007/s11425-014-4957-1.  Google Scholar [29] X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials, Discrete Contin. Dyn. Syst., 37 (2017), 4973-5002.  doi: 10.3934/dcds.2017214.  Google Scholar [30] X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 56 (2017), Art. 110, 25 pp. doi: 10.1007/s00526-017-1214-9.  Google Scholar [31] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar [32] L. G. Zhao and F. K. Zhao, On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl., 346 (2008), 155-169.  doi: 10.1016/j.jmaa.2008.04.053.  Google Scholar
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