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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 |
$ \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*} $ |
$ p\in [4,6) $ |
$ a(x)\ge \lim_{|x|\to\infty}a(x) = a_{\infty}>0 $ |
$ \lim_{|x|\to\infty}K(x) = 0 $ |
$ a $ |
$ K $ |
$ p\in (4,6) $ |
$ a $ |
$ K $ |
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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