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Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system
School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China |
$ \left\{ \begin{array}{ll} -\triangle u+V(x)u+\phi u = f(x,u), \ \ \ \ x\in { \mathbb{R} }^{2},\\ \triangle \phi = u^2, \ \ \ \ x\in { \mathbb{R} }^{2}, \end{array} \right. $ |
$ V(x) $ |
$ f(x, u) $ |
$ x $ |
$ f(x, u) $ |
$ u $ |
$ L^s( \mathbb{R} ^N) $ |
References:
[1] |
A. Ambrosetti and D. Ruiz,
Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.
doi: 10.1142/S021919970800282X. |
[2] |
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. |
[3] |
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. |
[4] |
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. |
[5] |
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. |
[6] |
I. Catto and P. L. Lions,
Binding of atoms and stability of molecules in hartree and thomas-fermi type theories, Comm. Partial Differential Equations, 18 (1993), 1149-1159.
doi: 10.1080/03605309308820967. |
[7] |
G. Cerami and J. 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. |
[8] |
J. Chen, S. T. Chen and X. H. Tang, Ground state solutions for the planar asymptotically periodic Schrödinger-Poisson system, Taiwanese J. Math., 21 (2017), 363-383.
doi: 10.11650/tjm/7784. |
[9] |
J. Chen, X. H. Tang and S. T. Chen, Existence of ground states for fractional Kirchhoff equations with general potentials via Nehari-Pohozaev manifold, Electron. J. Differ. Eq., 2018 (2018), Paper No. 142, 21 pp. |
[10] |
S. T. Chen and X. H. Tang, Ground state sign-changing solutions for a class of Schrödinger-Poisson type problems in $\mathbb R^3$, Z. Angew. Math. Phys., 67 (2016), Art. 102, 18 pp.
doi: 10.1007/s00033-016-0695-2. |
[11] |
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. |
[12] |
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. |
[13] |
S. T. Chen and X. H. Tang, Ground state solutions for generalized quasilinear Schrödinger equations with variable potentials and Berestycki-Lions nonlinearities, J. Math. Phys., 59 (2018), 081508, 18pp.
doi: 10.1063/1.5036570. |
[14] |
S. Cingolani and T. Weth,
On the planar Schrödinger-Poisson system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 169-197.
doi: 10.1016/j.anihpc.2014.09.008. |
[15] |
G. Coclite,
A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), 417-423.
|
[16] |
T. D'Aprile and D. Mugnai,
Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906.
doi: 10.1017/S030821050000353X. |
[17] |
M. Du and T. Weth,
Ground states and high energy solutions of the planar Schrödinger-Poisson system, Nonlinearity, 30 (2017), 3492-3515.
doi: 10.1088/1361-6544/aa7eac. |
[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] |
P. Markowich, C. Ringhofer and C. Schmeiser,
The concentration-compactness principle in the calculus of variations. the locally compact case. Ⅰ & Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.
doi: 10.1016/S0294-1449(16)30422-X. |
[21] |
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. |
[22] |
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. |
[23] |
E. A. B. Silva and G. F. Vieira,
Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33.
doi: 10.1007/s00526-009-0299-1. |
[24] |
J. Stubbe, Bound states of two-dimensional Schrödinger-Newton equations, arXiv: 0807.4059. |
[25] |
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. |
[26] |
X. H. Tang,
New conditions on nonlinearity for a periodic Schrödinger equation having zero as spectrum, J. Math. Anal. Appl., 413 (2014), 392-410.
doi: 10.1016/j.jmaa.2013.11.062. |
[27] |
X. H. Tang and B. T. Cheng,
Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations, 261 (2016), 2384-2402.
doi: 10.1016/j.jde.2016.04.032. |
[28] |
X. H. Tang and S. T. Chen,
Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potential, Disc. Contin. Dyn. Syst., 37 (2017), 4973-5002.
doi: 10.3934/dcds.2017214. |
[29] |
X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohožaev 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. |
[30] |
X. H. Tang, X. Y. Lin and J. S. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Differ. Equ, (2018), 1-15.
doi: 10.1007/s10884-018-9662-2. |
[31] |
Z. P. Wang and H. S. Zhou,
Positive solution for a nonlinear stationary Schrödinger-Poisson system in $\mathbb R^3$, Discrete Contin. Dyn. Syst., 18 (2007), 809-816.
doi: 10.3934/dcds.2007.18.809. |
[32] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[33] |
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 and D. Ruiz,
Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.
doi: 10.1142/S021919970800282X. |
[2] |
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. |
[3] |
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. |
[4] |
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. |
[5] |
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. |
[6] |
I. Catto and P. L. Lions,
Binding of atoms and stability of molecules in hartree and thomas-fermi type theories, Comm. Partial Differential Equations, 18 (1993), 1149-1159.
doi: 10.1080/03605309308820967. |
[7] |
G. Cerami and J. 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. |
[8] |
J. Chen, S. T. Chen and X. H. Tang, Ground state solutions for the planar asymptotically periodic Schrödinger-Poisson system, Taiwanese J. Math., 21 (2017), 363-383.
doi: 10.11650/tjm/7784. |
[9] |
J. Chen, X. H. Tang and S. T. Chen, Existence of ground states for fractional Kirchhoff equations with general potentials via Nehari-Pohozaev manifold, Electron. J. Differ. Eq., 2018 (2018), Paper No. 142, 21 pp. |
[10] |
S. T. Chen and X. H. Tang, Ground state sign-changing solutions for a class of Schrödinger-Poisson type problems in $\mathbb R^3$, Z. Angew. Math. Phys., 67 (2016), Art. 102, 18 pp.
doi: 10.1007/s00033-016-0695-2. |
[11] |
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. |
[12] |
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. |
[13] |
S. T. Chen and X. H. Tang, Ground state solutions for generalized quasilinear Schrödinger equations with variable potentials and Berestycki-Lions nonlinearities, J. Math. Phys., 59 (2018), 081508, 18pp.
doi: 10.1063/1.5036570. |
[14] |
S. Cingolani and T. Weth,
On the planar Schrödinger-Poisson system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 169-197.
doi: 10.1016/j.anihpc.2014.09.008. |
[15] |
G. Coclite,
A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), 417-423.
|
[16] |
T. D'Aprile and D. Mugnai,
Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906.
doi: 10.1017/S030821050000353X. |
[17] |
M. Du and T. Weth,
Ground states and high energy solutions of the planar Schrödinger-Poisson system, Nonlinearity, 30 (2017), 3492-3515.
doi: 10.1088/1361-6544/aa7eac. |
[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] |
P. Markowich, C. Ringhofer and C. Schmeiser,
The concentration-compactness principle in the calculus of variations. the locally compact case. Ⅰ & Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.
doi: 10.1016/S0294-1449(16)30422-X. |
[21] |
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. |
[22] |
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. |
[23] |
E. A. B. Silva and G. F. Vieira,
Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33.
doi: 10.1007/s00526-009-0299-1. |
[24] |
J. Stubbe, Bound states of two-dimensional Schrödinger-Newton equations, arXiv: 0807.4059. |
[25] |
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. |
[26] |
X. H. Tang,
New conditions on nonlinearity for a periodic Schrödinger equation having zero as spectrum, J. Math. Anal. Appl., 413 (2014), 392-410.
doi: 10.1016/j.jmaa.2013.11.062. |
[27] |
X. H. Tang and B. T. Cheng,
Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations, 261 (2016), 2384-2402.
doi: 10.1016/j.jde.2016.04.032. |
[28] |
X. H. Tang and S. T. Chen,
Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potential, Disc. Contin. Dyn. Syst., 37 (2017), 4973-5002.
doi: 10.3934/dcds.2017214. |
[29] |
X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohožaev 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. |
[30] |
X. H. Tang, X. Y. Lin and J. S. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Differ. Equ, (2018), 1-15.
doi: 10.1007/s10884-018-9662-2. |
[31] |
Z. P. Wang and H. S. Zhou,
Positive solution for a nonlinear stationary Schrödinger-Poisson system in $\mathbb R^3$, Discrete Contin. Dyn. Syst., 18 (2007), 809-816.
doi: 10.3934/dcds.2007.18.809. |
[32] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[33] |
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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