# American Institute of Mathematical Sciences

October  2019, 39(10): 6103-6129. doi: 10.3934/dcds.2019266

## Standing waves for Schrödinger-Poisson system with general nonlinearity

 1 School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China 2 School of Mathematics and Statistics, Hunan University of Technology and Business, Changsha, 410205 Hunan, China

* Corresponding author

Received  March 2019 Published  July 2019

Fund Project: This work was supported by the NNSF (Nos. 11571370, 11601145, 11701173), by the Natural Science Foundation of Hunan Province (Nos. 2017JJ3130, 2017JJ3131), by the Excellent youth project of Education Department of Hunan Province (17B143, 18B342), and by the Project funded by China Postdoctoral Science Foundation (2019M652790).

In this paper we consider the following Schrödinger-Poisson system with general nonlinearity
 $\begin{eqnarray*} \left\{ \begin{array}{ll} -\varepsilon^2\Delta u+V(x)u+\psi u = f(u),\,\, x\in\mathbb{R}^3,\\ -\varepsilon^2\Delta\psi = u^2,\,\,u>0,\,\, u\in H^1(\mathbb{R}^3),\\ \end{array} \right. \end{eqnarray*}$
where
 $\varepsilon>0$
is a small positive parameter. Under a local condition imposed on the potential
 $V$
and general conditions on
 $f,$
we construct a family of positive semiclassical solutions. Moreover, the concentration phenomena around local minimum of
 $V$
and exponential decay of semiclassical solutions are also explored. We do not need the monotonicity of the function
 $u\rightarrow\frac{f(u)}{u^3}$
, and our results include the case
 $f(u) = |u|^{p-2}u$
for
 $3 . Since without more global information on the potential, in the proofs we apply variational methods, penalization techniques and some analytical techniques. Citation: Zhi Chen, Xianhua Tang, Ning Zhang, Jian Zhang. Standing waves for Schrödinger-Poisson system with general nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 6103-6129. doi: 10.3934/dcds.2019266 ##### References:  [1] A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404. doi: 10.1142/S021919970800282X. [2] A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108. doi: 10.1016/j.jmaa.2008.03.057. [3] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293. doi: 10.12775/TMNA.1998.019. [4] V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420. doi: 10.1142/S0129055X02001168. [5] J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal., 185 (2007), 185-200. doi: 10.1007/s00205-006-0019-3. [6] J. Byeon and L. Jeanjean, Standing waves with a critical frequency for nonlinear Schrödinger equations Ⅱ, Calc. Var. Partial Differ. Equ., 18 (2003), 207-219. doi: 10.1007/s00526-002-0191-8. [7] G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Maxwell systems, J. Differ. Equ., 248 (2010), 521-543. doi: 10.1016/j.jde.2009.06.017. [8] S. T. Chen and X. H. Tang, Ground state sign-changing solutions for a class of Schrödinger-Poisson type problems in$\mathbb{R}^3$, Z. Angew. Math. Phys., 67 (2016), Art. 102, 18 pp. doi: 10.1007/s00033-016-0695-2. [9] S. T. Chen and X. H. Tang, Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete. Contin. Dyn. Syst., 38 (2018), 2333-2348. doi: 10.3934/dcds.2018096. [10] S. T. Chen and X. H. Tang, Geometrically distinct solutions for Klein-Gordon-Maxwell systems with superlinear nonlinearities, Appl. Math. Letters, 90 (2019), 188-193. doi: 10.1016/j.aml.2018.11.007. [11] S. T. Chen and X. H. Tang, Ground state solutions of Schrödinger-Poisson systems with variable potential and convolution nonlinearity, J. Math. Anal. Appl., 473 (2019), 87-111. doi: 10.1016/j.jmaa.2018.12.037. [12] T. D'Aprile and J. C. Wei, On bound states concentrating on spheres for the Maxwell-Schrödinger equation, SIAM J. Math. Anal., 37 (2005), 321-342. doi: 10.1137/S0036141004442793. [13] G. M. Figueiredo, N. Ikoma and J. R. Santos Junior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Ration. Mech. Anal., 213 (2014), 931-979. doi: 10.1007/s00205-014-0747-8. [14] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren Math. Wiss., vol. 224, Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0. [15] X. M. He, Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations, Z. Angew. Math. Phys., 62 (2011), 869-889. doi: 10.1007/s00033-011-0120-9. [16] X. M. He and W. M. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 53 (2012), 023702, 19 pp. doi: 10.1063/1.3683156. [17] Y. He and G. B. Li, Standing waves for a class of Schrödinger-Poisson equations in$\mathbb{R}^3$involving critical Sobolev exponents, Ann. Acad. Sci. Fenn. Math., 40 (2015), 729-766. doi: 10.5186/aasfm.2015.4041. [18] I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials, Adv. Nonlinear Stud., 8 (2008), 573-595. doi: 10.1515/ans-2008-0305. [19] I. Ianni, Solutions of the Schrödinger-Poisson problem concentrating on spheres, part Ⅱ: Existence, Math. Models Methods Appl. Sci., 19 (2009), 877-910. doi: 10.1142/S0218202509003656. [20] I. Ianni and G. Vaira, Solutions of the Schrödinger-Poisson problem concentrating on spheres, part Ⅰ: necessary condition, Math. Models Methods Appl. Sci., 19 (2009), 707-720. doi: 10.1142/S0218202509003589. [21] L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633-1659. doi: 10.1016/S0362-546X(96)00021-1. [22] G. B. Li and S. S. Yan, Eigenvalue problems for quasilinear elliptic equations on$\mathbb{R}^N$, Commun. Partial Differ. Equ., 14 (1989), 1291-1314. doi: 10.1080/03605308908820654. [23] P. L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33-97. doi: 10.1007/BF01205672. [24] P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part 2, Ann. Inst. H. Poincaré Anal. Non. Linéaire., 1 (1984), 223-283. doi: 10.1016/S0294-1449(16)30422-X. [25] P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990. doi: 10.1007/978-3-7091-6961-2. [26] N. S. Papageorgiou, V. D. Rǎdulescu and D. Repovs, Nonlinear Analysis-Theory and Methods,, Springer Monographs in Mathematics, Springer, BerlinCham, 2019. [27] M. del Pino and P. Felmer, Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4 (1996), 121-137. doi: 10.1007/BF01189950. [28] P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703. doi: 10.1512/iumj.1986.35.35036. [29] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631. [30] 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. [31] D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell concentration around a sphere, Math. Models Methods Appl. Sci., 15 (2005), 141-164. doi: 10.1142/S0218202505003939. [32] D. Ruiz and G. Vaira, Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of potential, Rev. Mat. Iberoamericana., 27 (2011), 253-271. doi: 10.4171/RMI/635. [33] J. Seok, On nonlinear Schrödinger-Poisson equations with general potentials, J. Math. Anal. Appl., 401 (2013), 672-681. doi: 10.1016/j.jmaa.2012.12.054. [34] J. J. Sun and S. W. Ma, Ground state solutions for some Schrödinger-Poisson systems with periodic potentials, J. Differe. Equ., 260 (2016), 2119-2149. doi: 10.1016/j.jde.2015.09.057. [35] 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. [36] X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozǎev type for Kirchhoff-type problems with general potentials, Calc. Var. PDE, 56 (2017), Art. 110, 25 pp. doi: 10.1007/s00526-017-1214-9. [37] X. H. Tang, X. Y. Lin and J. S. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Differ. Equ., 31 (2018), 369-383. doi: 10.1007/s10884-018-9662-2. [38] X. F. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 153 (1993), 229-244. doi: 10.1007/BF02096642. [39] J. Wang, L. X. Tian, J. X. Xu and F. B. Zhang, Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in$\mathbb{R}^{3}$, Calc. Var. PDE., 48 (2013), 243-273. doi: 10.1007/s00526-012-0548-6. [40] J. Wang, L. X. Tian, J. X. Xu and F. B. Zhang, Existence of multiple positive solutions for Schrödinger-Poisson systems with critical growth, Z. Angew. Math. Phys., 66 (2015), 2441-2471. doi: 10.1007/s00033-015-0531-0. [41] M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1. [42] J. Zhang, W. Zhang and X. H. 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. [43] X. Zhang and J. K. Xia, Semi-classical solutions for Schrödinger-Poisson equations with a critical frequency, J. Differ. Equ., 265 (2018), 2121-2170. doi: 10.1016/j.jde.2018.04.023. [44] 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. [45] L. G. Zhao and F. K. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164. doi: 10.1016/j.na.2008.02.116. 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] J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal., 185 (2007), 185-200. doi: 10.1007/s00205-006-0019-3. [6] J. Byeon and L. Jeanjean, Standing waves with a critical frequency for nonlinear Schrödinger equations Ⅱ, Calc. Var. Partial Differ. Equ., 18 (2003), 207-219. doi: 10.1007/s00526-002-0191-8. [7] G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Maxwell systems, J. Differ. Equ., 248 (2010), 521-543. doi: 10.1016/j.jde.2009.06.017. [8] S. T. Chen and X. H. Tang, Ground state sign-changing solutions for a class of Schrödinger-Poisson type problems in$\mathbb{R}^3$, Z. Angew. Math. Phys., 67 (2016), Art. 102, 18 pp. doi: 10.1007/s00033-016-0695-2. [9] S. T. Chen and X. H. Tang, Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete. Contin. Dyn. Syst., 38 (2018), 2333-2348. doi: 10.3934/dcds.2018096. [10] S. T. Chen and X. H. Tang, Geometrically distinct solutions for Klein-Gordon-Maxwell systems with superlinear nonlinearities, Appl. Math. Letters, 90 (2019), 188-193. doi: 10.1016/j.aml.2018.11.007. [11] S. T. Chen and X. H. Tang, Ground state solutions of Schrödinger-Poisson systems with variable potential and convolution nonlinearity, J. Math. Anal. Appl., 473 (2019), 87-111. doi: 10.1016/j.jmaa.2018.12.037. [12] T. D'Aprile and J. C. Wei, On bound states concentrating on spheres for the Maxwell-Schrödinger equation, SIAM J. Math. Anal., 37 (2005), 321-342. doi: 10.1137/S0036141004442793. [13] G. M. Figueiredo, N. Ikoma and J. R. Santos Junior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Ration. Mech. Anal., 213 (2014), 931-979. doi: 10.1007/s00205-014-0747-8. [14] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren Math. Wiss., vol. 224, Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0. [15] X. M. He, Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations, Z. Angew. Math. Phys., 62 (2011), 869-889. doi: 10.1007/s00033-011-0120-9. [16] X. M. He and W. M. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 53 (2012), 023702, 19 pp. doi: 10.1063/1.3683156. [17] Y. He and G. B. Li, Standing waves for a class of Schrödinger-Poisson equations in$\mathbb{R}^3$involving critical Sobolev exponents, Ann. Acad. Sci. Fenn. Math., 40 (2015), 729-766. doi: 10.5186/aasfm.2015.4041. [18] I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials, Adv. Nonlinear Stud., 8 (2008), 573-595. doi: 10.1515/ans-2008-0305. [19] I. Ianni, Solutions of the Schrödinger-Poisson problem concentrating on spheres, part Ⅱ: Existence, Math. Models Methods Appl. Sci., 19 (2009), 877-910. doi: 10.1142/S0218202509003656. [20] I. Ianni and G. Vaira, Solutions of the Schrödinger-Poisson problem concentrating on spheres, part Ⅰ: necessary condition, Math. Models Methods Appl. Sci., 19 (2009), 707-720. doi: 10.1142/S0218202509003589. [21] L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633-1659. doi: 10.1016/S0362-546X(96)00021-1. [22] G. B. Li and S. S. Yan, Eigenvalue problems for quasilinear elliptic equations on$\mathbb{R}^N$, Commun. Partial Differ. Equ., 14 (1989), 1291-1314. doi: 10.1080/03605308908820654. [23] P. L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33-97. doi: 10.1007/BF01205672. [24] P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part 2, Ann. Inst. H. Poincaré Anal. Non. Linéaire., 1 (1984), 223-283. doi: 10.1016/S0294-1449(16)30422-X. [25] P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990. doi: 10.1007/978-3-7091-6961-2. [26] N. S. Papageorgiou, V. D. Rǎdulescu and D. Repovs, Nonlinear Analysis-Theory and Methods,, Springer Monographs in Mathematics, Springer, BerlinCham, 2019. [27] M. del Pino and P. Felmer, Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4 (1996), 121-137. doi: 10.1007/BF01189950. [28] P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703. doi: 10.1512/iumj.1986.35.35036. [29] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631. [30] 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. [31] D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell concentration around a sphere, Math. Models Methods Appl. Sci., 15 (2005), 141-164. doi: 10.1142/S0218202505003939. [32] D. Ruiz and G. Vaira, Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of potential, Rev. Mat. Iberoamericana., 27 (2011), 253-271. doi: 10.4171/RMI/635. [33] J. Seok, On nonlinear Schrödinger-Poisson equations with general potentials, J. Math. Anal. Appl., 401 (2013), 672-681. doi: 10.1016/j.jmaa.2012.12.054. [34] J. J. Sun and S. W. Ma, Ground state solutions for some Schrödinger-Poisson systems with periodic potentials, J. Differe. Equ., 260 (2016), 2119-2149. doi: 10.1016/j.jde.2015.09.057. [35] 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. [36] X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozǎev type for Kirchhoff-type problems with general potentials, Calc. Var. PDE, 56 (2017), Art. 110, 25 pp. doi: 10.1007/s00526-017-1214-9. [37] X. H. Tang, X. Y. Lin and J. S. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Differ. Equ., 31 (2018), 369-383. doi: 10.1007/s10884-018-9662-2. [38] X. F. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 153 (1993), 229-244. doi: 10.1007/BF02096642. [39] J. Wang, L. X. Tian, J. X. Xu and F. B. Zhang, Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in$\mathbb{R}^{3}$, Calc. Var. PDE., 48 (2013), 243-273. doi: 10.1007/s00526-012-0548-6. [40] J. Wang, L. X. Tian, J. X. Xu and F. B. Zhang, Existence of multiple positive solutions for Schrödinger-Poisson systems with critical growth, Z. Angew. Math. Phys., 66 (2015), 2441-2471. doi: 10.1007/s00033-015-0531-0. [41] M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1. [42] J. Zhang, W. Zhang and X. H. 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. [43] X. Zhang and J. K. Xia, Semi-classical solutions for Schrödinger-Poisson equations with a critical frequency, J. Differ. Equ., 265 (2018), 2121-2170. doi: 10.1016/j.jde.2018.04.023. [44] 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. [45] L. G. Zhao and F. K. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164. doi: 10.1016/j.na.2008.02.116.  [1] Sitong Chen, Junping Shi, Xianhua Tang. 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