# American Institute of Mathematical Sciences

September  2019, 24(9): 4685-4702. doi: 10.3934/dcdsb.2018329

## 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

* Corresponding author: Xianhua Tang

Received  May 2018 Revised  August 2018 Published  September 2019 Early access  January 2019

Fund Project: This work is partially supported by the Hunan Provincial Innovation Foundation for Postgraduate (No: CX2017B041) and the National Natural Science Foundation of China (No: 11571370).

This paper is concerned with the following planar Schrödinger-Poisson system
 $\left\{ \begin{array}{ll} -\triangle u+V(x)u+\phi u = f(x,u), \ \ \ \ x\in { \mathbb{R} }^{2},\\ \triangle \phi = u^2, \ \ \ \ x\in { \mathbb{R} }^{2}, \end{array} \right.$
where
 $V(x)$
and
 $f(x, u)$
are axially symmetric in
 $x$
, and
 $f(x, u)$
is asymptotically cubic or super-cubic in
 $u$
. With a different variational approach used in [S. Cingolani, T. Weth, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 33 (2016) 169-197], we obtain the existence of an axially symmetric Nehari-type ground state solution and a nontrivial solution for the above system. The axial symmetry is more general than radial symmetry, but less used in the literature, since the embedding from the space of axially symmetric functions to
 $L^s( \mathbb{R} ^N)$
is not compact. Our results generalize previous ones in the literature, and some of new phenomena do not occur in the corresponding problem for higher space dimensions.
Citation: Sitong Chen, Xianhua Tang. Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4685-4702. doi: 10.3934/dcdsb.2018329
##### 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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