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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

Sitong Chen 1, , Wennian Huang 2, and  Xianhua Tang 1,,

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.
Keywords: Schrödinger-Poisson system, Vanishing potential, Ground state solution.
Mathematics Subject Classification: Primary: 35J20; Secondary: 35Q55.
Citation: Sitong Chen, Wennian Huang, Xianhua Tang. Existence criteria of ground state solutions for Schrödinger-Poisson systems with a vanishing potential. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3055-3066. doi: 10.3934/dcdss.2020339
References:
[1]

A. Ambrosetti, On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.  doi: 10.1007/s00032-008-0094-z.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

G. Cerami, Some nonlinear elliptic problems in unbounded domains, Milan J. Math., 74 (2006), 47-77.  doi: 10.1007/s00032-006-0059-z.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

E. H. Lieb, Thomas-Fermi and related theories and molecules, Rev. Modern Phys., 53 (1981), 603-641.  doi: 10.1103/RevModPhys.53.603.

[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.

[20]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 1997.

[21]

P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1984), 33-97.  doi: 10.1007/BF01205672.

[22]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

G. Cerami, Some nonlinear elliptic problems in unbounded domains, Milan J. Math., 74 (2006), 47-77.  doi: 10.1007/s00032-006-0059-z.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

E. H. Lieb, Thomas-Fermi and related theories and molecules, Rev. Modern Phys., 53 (1981), 603-641.  doi: 10.1103/RevModPhys.53.603.

[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.

[20]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 1997.

[21]

P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1984), 33-97.  doi: 10.1007/BF01205672.

[22]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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