# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2022109
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## Groundstates and infinitely many solutions for the Schrödinger-Poisson equation with magnetic field

 1 School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, China 2 Department of Mathematics, University of Craiova, 200585 Craiova, Romania, China-Romania Research Center in Applied Mathematics 3 College of Science, Hunan University of Technology and Business, Changsha, 410205 Hunan, China 4 Key Laboratory of Hunan Province for Statistical Learning and Intelligent Computation, Hunan University of Technology and Business, Changsha, 410205 Hunan, China

* Corresponding author: Wen Zhang

Received  February 2022 Revised  April 2022 Early access May 2022

In this paper, we investigate the nonlinear Schrödinger-Poisson equation with magnetic field. By combining non-Nehari manifold method and some new energy estimate inequalities, we obtain the existence of a ground state solution, where the strict monotonicity condition and the Ambrosetti-Rabinowitz growth condition are not needed. Moreover, when both the potential and the nonlinearity are sign-changing, by applying the Fountain Theorem and some analytical techniques, we prove the existence of infinitely many solutions. Our results extend and improve the present ones in the literature.

Citation: Lixi Wen, Wen Zhang. Groundstates and infinitely many solutions for the Schrödinger-Poisson equation with magnetic field. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022109
##### References:
 [1] C. O. Alves, G. M. Figueiredo and M. F. Furtado, Multiple solutions for a nonlinear Schrödinger equation with magnetic fields, Comm. Partial Differential Equations, 36 (2011), 1565-1586.  doi: 10.1080/03605302.2011.593013. [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] M. Bourahma, A. Benkirane and J. Bennouna, Existence of renormalized solutions for some nonlinear elliptic equations in Orlicz spaces, Rend. Circ. Mat. Palermo, 69 (2020), 231-252.  doi: 10.1007/s12215-019-00399-z. [5] M. Chen, Q Li and S. Peng, Bound states for fractional Schrödinger-Poisson system with critical exponent, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 1819-1835.  doi: 10.3934/dcdss.2021038. [6] S. Chen, W. Huang and X. Tang, Existence criteria of ground state solutions for Schrödinger-Poisson systems with a vanishing potential, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 3055-3066.  doi: 10.3934/dcdss.2020339. [7] S. Chen and X. Tang, Ground state sign-changing solutions for a class of Schrödinger-Poisson type problems in $\mathbb{R}^3$, Z. Angew. Math. Phys., 4 (2016), Art. 102, 18 pp. doi: 10.1007/s00033-016-0695-2. [8] S. Cingolani, Semi-classical stationary states of nonlinear Schrödinger equations with an external magnetic field, J. Differential Equations, 188 (2003), 52-79.  doi: 10.1016/S0022-0396(02)00058-X. [9] S. Cingolani, S. Secchi and M. Squassina, Semiclassical limit for Schrödinger equations with magnetic field and Hartree type nonlinearities, Proc. Roy. Soc. Edinburgh, Sect. A, 140 (2010), 973-1009.  doi: 10.1017/S0308210509000584. [10] Y. Ding and S. Luan, Multiple solutions for a class of nonlinear Schrödinger equations, J. Differential Equations, 207 (2004), 423-457.  doi: 10.1016/j.jde.2004.07.030. [11] Y. Ding and A. Szulkin, Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Differential Equations, 29 (2007), 397-419.  doi: 10.1007/s00526-006-0071-8. [12] T. Dutko, C. Mercuri and T. M. Tyler, Groundstates and infinitely many high energy solutions to a class of nonlinear Schrödinger-Poisson systems, Calc. Var. Partial Differential Equations, 60 (2021), Paper No. 174, 46 pp. doi: 10.1007/s00526-021-02045-y. [13] M. J. Esteban and P. L. Lions, Stationary solutions of nonlinear Schrödinger equations with an external magnetic field, In Partial Differential Equations and the Calculus of Variations, (eds. F. Colombini, A. Marino, L. Modica, S. Spagnolo. ), Progress in Nonlinear Differential Equations Application, 1 (1989), 401–449. [14] F. Güngör, The Schrödinger propagator on (0, $\infty$) for a special potential by a Lie symmetry group method, Rend. Circ. Mat. Palermo, 70 (2021), 1609-1616.  doi: 10.1007/s12215-020-00576-5. [15] M. Hayashi, A note on the nonlinear Schrödinger equation in a general domain, Nonlinear Anal., 173 (2018), 99-122.  doi: 10.1016/j.na.2018.03.017. [16] C. Ji and V. D. Rădulescu, Multiplicity and concentration of solutions to the nonlinear magnetic Schrödinger equation, Calc. Var. Partial Differential Equations, 59 (2020), 115, 28 pp. doi: 10.1007/s00526-020-01772-y. [17] Y. Li, Z. Wang and J. Zeng, Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. Henri Poincaré C Anal. Non Linéaire, 23 (2006), 829-837.  doi: 10.1016/j.anihpc.2006.01.003. [18] E. H. Lieb and M. Loss, Analysis, 2$^nd$ edition, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014. [19] Y. Liu, X. Li and C. Ji, Multiplicity of concentrating solutions for a class of magnetic Schrödinger-Poisson type equation, Adv. Nonlinear Anal., 10 (2021), 131-151.  doi: 10.1515/anona-2020-0110. [20] F. Qin, J. Wang and J. Yang, Infinitely many positive solutions for Schrödinger-Poisson systems with nonsymmetry potentials, Discrete Contin. Dyn. Syst., 41 (2021), 4705-4736.  doi: 10.3934/dcds.2021054. [21] P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631. [22] 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. [23] J. Sun and S. 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. [24] X. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58 (2015), 715-728.  doi: 10.1007/s11425-014-4957-1. [25] X. Tang and S. 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. [26] T. Tao, Why are solitons stable?, Bull. Amer. Math. Soc., 46 (2009), 1-33.  doi: 10.1090/S0273-0979-08-01228-7. [27] X. Wang, T. Lin and Z. Wang, Existence and concentration of ground states for saturable nonlinear Schrödinger equations with intensity functions in $\mathbb{R}^2$, Nonlinear Anal., 173 (2018), 19-36.  doi: 10.1016/j.na.2018.03.005. [28] Y. Wang, X. Wu and C. Tang, Infinitely many high energy radial solutions for Schroödinger-Poisson system, Appl. Math. Lett., 100 (2020), 106012, 6 pp. doi: 10.1016/j. aml. 2019.106012. [29] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. [30] A. Xia, Multiplicity and concentration results for magnetic relativistic Schrödinger equations, Adv. Nonlinear Anal., 9 (2020), 1161-1186.  doi: 10.1515/anona-2020-0044. [31] J. Zhang and W. Zhang, Semiclassical states for coupled nonlinear Schrödinger system with competing potentials, J. Geom. Anal., 32 (2022), 114, 36 pp. doi: 10.1007/s12220-022-00870-x. [32] J. Zhang, W. Zhang and X. Tang, Ground state solutions for Hamiltonian elliptic system with inverse square potential, Discrete Contin. Dyn. Syst., 37 (2017), 4565-4583.  doi: 10.3934/dcds.2017195. [33] Y. Zhang, X. Tang and V. D. Rădulescu, Small perturbations for nonlinear Schrödinger equations with magnetic potential, Milan J. Math., 88 (2020), 479-506.  doi: 10.1007/s00032-020-00322-7. [34] L. Zhao and F. Zhao, On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Phys., 346 (2008), 155-169.  doi: 10.1016/j.jmaa.2008.04.053. [35] W. Zou and M. Schechter, Critical Point Theory and Its Applications, Springer, New York, 2006.

show all references

##### References:
 [1] C. O. Alves, G. M. Figueiredo and M. F. Furtado, Multiple solutions for a nonlinear Schrödinger equation with magnetic fields, Comm. Partial Differential Equations, 36 (2011), 1565-1586.  doi: 10.1080/03605302.2011.593013. [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] M. Bourahma, A. Benkirane and J. Bennouna, Existence of renormalized solutions for some nonlinear elliptic equations in Orlicz spaces, Rend. Circ. Mat. Palermo, 69 (2020), 231-252.  doi: 10.1007/s12215-019-00399-z. [5] M. Chen, Q Li and S. Peng, Bound states for fractional Schrödinger-Poisson system with critical exponent, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 1819-1835.  doi: 10.3934/dcdss.2021038. [6] S. Chen, W. Huang and X. Tang, Existence criteria of ground state solutions for Schrödinger-Poisson systems with a vanishing potential, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 3055-3066.  doi: 10.3934/dcdss.2020339. [7] S. Chen and X. Tang, Ground state sign-changing solutions for a class of Schrödinger-Poisson type problems in $\mathbb{R}^3$, Z. Angew. Math. Phys., 4 (2016), Art. 102, 18 pp. doi: 10.1007/s00033-016-0695-2. [8] S. Cingolani, Semi-classical stationary states of nonlinear Schrödinger equations with an external magnetic field, J. Differential Equations, 188 (2003), 52-79.  doi: 10.1016/S0022-0396(02)00058-X. [9] S. Cingolani, S. Secchi and M. Squassina, Semiclassical limit for Schrödinger equations with magnetic field and Hartree type nonlinearities, Proc. Roy. Soc. Edinburgh, Sect. A, 140 (2010), 973-1009.  doi: 10.1017/S0308210509000584. [10] Y. Ding and S. Luan, Multiple solutions for a class of nonlinear Schrödinger equations, J. Differential Equations, 207 (2004), 423-457.  doi: 10.1016/j.jde.2004.07.030. [11] Y. Ding and A. Szulkin, Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Differential Equations, 29 (2007), 397-419.  doi: 10.1007/s00526-006-0071-8. [12] T. Dutko, C. Mercuri and T. M. Tyler, Groundstates and infinitely many high energy solutions to a class of nonlinear Schrödinger-Poisson systems, Calc. Var. Partial Differential Equations, 60 (2021), Paper No. 174, 46 pp. doi: 10.1007/s00526-021-02045-y. [13] M. J. Esteban and P. L. Lions, Stationary solutions of nonlinear Schrödinger equations with an external magnetic field, In Partial Differential Equations and the Calculus of Variations, (eds. F. Colombini, A. Marino, L. Modica, S. Spagnolo. ), Progress in Nonlinear Differential Equations Application, 1 (1989), 401–449. [14] F. Güngör, The Schrödinger propagator on (0, $\infty$) for a special potential by a Lie symmetry group method, Rend. Circ. Mat. Palermo, 70 (2021), 1609-1616.  doi: 10.1007/s12215-020-00576-5. [15] M. Hayashi, A note on the nonlinear Schrödinger equation in a general domain, Nonlinear Anal., 173 (2018), 99-122.  doi: 10.1016/j.na.2018.03.017. [16] C. Ji and V. D. Rădulescu, Multiplicity and concentration of solutions to the nonlinear magnetic Schrödinger equation, Calc. Var. Partial Differential Equations, 59 (2020), 115, 28 pp. doi: 10.1007/s00526-020-01772-y. [17] Y. Li, Z. Wang and J. Zeng, Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. Henri Poincaré C Anal. Non Linéaire, 23 (2006), 829-837.  doi: 10.1016/j.anihpc.2006.01.003. [18] E. H. Lieb and M. Loss, Analysis, 2$^nd$ edition, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014. [19] Y. Liu, X. Li and C. Ji, Multiplicity of concentrating solutions for a class of magnetic Schrödinger-Poisson type equation, Adv. Nonlinear Anal., 10 (2021), 131-151.  doi: 10.1515/anona-2020-0110. [20] F. Qin, J. Wang and J. Yang, Infinitely many positive solutions for Schrödinger-Poisson systems with nonsymmetry potentials, Discrete Contin. Dyn. Syst., 41 (2021), 4705-4736.  doi: 10.3934/dcds.2021054. [21] P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631. [22] 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. [23] J. Sun and S. 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. [24] X. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58 (2015), 715-728.  doi: 10.1007/s11425-014-4957-1. [25] X. Tang and S. 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. [26] T. Tao, Why are solitons stable?, Bull. Amer. Math. Soc., 46 (2009), 1-33.  doi: 10.1090/S0273-0979-08-01228-7. [27] X. Wang, T. Lin and Z. Wang, Existence and concentration of ground states for saturable nonlinear Schrödinger equations with intensity functions in $\mathbb{R}^2$, Nonlinear Anal., 173 (2018), 19-36.  doi: 10.1016/j.na.2018.03.005. [28] Y. Wang, X. Wu and C. Tang, Infinitely many high energy radial solutions for Schroödinger-Poisson system, Appl. Math. Lett., 100 (2020), 106012, 6 pp. doi: 10.1016/j. aml. 2019.106012. [29] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. [30] A. Xia, Multiplicity and concentration results for magnetic relativistic Schrödinger equations, Adv. Nonlinear Anal., 9 (2020), 1161-1186.  doi: 10.1515/anona-2020-0044. [31] J. Zhang and W. Zhang, Semiclassical states for coupled nonlinear Schrödinger system with competing potentials, J. Geom. Anal., 32 (2022), 114, 36 pp. doi: 10.1007/s12220-022-00870-x. [32] J. Zhang, W. Zhang and X. Tang, Ground state solutions for Hamiltonian elliptic system with inverse square potential, Discrete Contin. Dyn. Syst., 37 (2017), 4565-4583.  doi: 10.3934/dcds.2017195. [33] Y. Zhang, X. Tang and V. D. Rădulescu, Small perturbations for nonlinear Schrödinger equations with magnetic potential, Milan J. Math., 88 (2020), 479-506.  doi: 10.1007/s00032-020-00322-7. [34] L. Zhao and F. Zhao, On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Phys., 346 (2008), 155-169.  doi: 10.1016/j.jmaa.2008.04.053. [35] W. Zou and M. Schechter, Critical Point Theory and Its Applications, Springer, New York, 2006.
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