# 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
 $3 . Since without more global information on the potential, in the proofs we apply variational methods, penalization techniques and some analytical techniques. Citation: Zhi Chen, Xianhua Tang, Ning Zhang, Jian Zhang. Standing waves for Schrödinger-Poisson system with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6103-6129. doi: 10.3934/dcds.2019266 ##### References:  [1] A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404. doi: 10.1142/S021919970800282X. Google Scholar [2] A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108. doi: 10.1016/j.jmaa.2008.03.057. Google Scholar [3] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293. doi: 10.12775/TMNA.1998.019. Google Scholar [4] V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420. doi: 10.1142/S0129055X02001168. Google Scholar [5] J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal., 185 (2007), 185-200. doi: 10.1007/s00205-006-0019-3. Google Scholar [6] J. Byeon and L. Jeanjean, Standing waves with a critical frequency for nonlinear Schrödinger equations Ⅱ, Calc. Var. Partial Differ. Equ., 18 (2003), 207-219. doi: 10.1007/s00526-002-0191-8. Google Scholar [7] G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Maxwell systems, J. Differ. Equ., 248 (2010), 521-543. doi: 10.1016/j.jde.2009.06.017. Google Scholar [8] S. T. Chen and X. H. Tang, Ground state sign-changing solutions for a class of Schrödinger-Poisson type problems in$\mathbb{R}^3$, Z. Angew. Math. Phys., 67 (2016), Art. 102, 18 pp. doi: 10.1007/s00033-016-0695-2. Google Scholar [9] S. T. Chen and X. H. Tang, Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete. Contin. Dyn. Syst., 38 (2018), 2333-2348. doi: 10.3934/dcds.2018096. Google Scholar [10] S. T. Chen and X. H. Tang, Geometrically distinct solutions for Klein-Gordon-Maxwell systems with superlinear nonlinearities, Appl. Math. Letters, 90 (2019), 188-193. doi: 10.1016/j.aml.2018.11.007. Google Scholar [11] S. T. Chen and X. H. Tang, Ground state solutions of Schrödinger-Poisson systems with variable potential and convolution nonlinearity, J. Math. Anal. Appl., 473 (2019), 87-111. doi: 10.1016/j.jmaa.2018.12.037. Google Scholar [12] T. D'Aprile and J. C. Wei, On bound states concentrating on spheres for the Maxwell-Schrödinger equation, SIAM J. Math. Anal., 37 (2005), 321-342. doi: 10.1137/S0036141004442793. Google Scholar [13] G. M. Figueiredo, N. Ikoma and J. R. Santos Junior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Ration. Mech. Anal., 213 (2014), 931-979. doi: 10.1007/s00205-014-0747-8. Google Scholar [14] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren Math. Wiss., vol. 224, Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0. Google Scholar [15] X. M. He, Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations, Z. Angew. Math. Phys., 62 (2011), 869-889. doi: 10.1007/s00033-011-0120-9. Google Scholar [16] X. M. He and W. M. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 53 (2012), 023702, 19 pp. doi: 10.1063/1.3683156. Google Scholar [17] Y. He and G. B. Li, Standing waves for a class of Schrödinger-Poisson equations in$\mathbb{R}^3$involving critical Sobolev exponents, Ann. Acad. Sci. Fenn. Math., 40 (2015), 729-766. doi: 10.5186/aasfm.2015.4041. Google Scholar [18] I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials, Adv. Nonlinear Stud., 8 (2008), 573-595. doi: 10.1515/ans-2008-0305. Google Scholar [19] I. Ianni, Solutions of the Schrödinger-Poisson problem concentrating on spheres, part Ⅱ: Existence, Math. Models Methods Appl. Sci., 19 (2009), 877-910. doi: 10.1142/S0218202509003656. Google Scholar [20] I. Ianni and G. Vaira, Solutions of the Schrödinger-Poisson problem concentrating on spheres, part Ⅰ: necessary condition, Math. Models Methods Appl. Sci., 19 (2009), 707-720. doi: 10.1142/S0218202509003589. Google Scholar [21] L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633-1659. doi: 10.1016/S0362-546X(96)00021-1. Google Scholar [22] G. B. Li and S. S. Yan, Eigenvalue problems for quasilinear elliptic equations on$\mathbb{R}^N$, Commun. Partial Differ. Equ., 14 (1989), 1291-1314. doi: 10.1080/03605308908820654. Google Scholar [23] P. L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33-97. doi: 10.1007/BF01205672. Google Scholar [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. Google Scholar [25] P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990. doi: 10.1007/978-3-7091-6961-2. Google Scholar [26] N. S. Papageorgiou, V. D. Rǎdulescu and D. Repovs, Nonlinear Analysis-Theory and Methods,, Springer Monographs in Mathematics, Springer, BerlinCham, 2019. Google Scholar [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. Google Scholar [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. Google Scholar [29] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [41] M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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$, Z. Angew. Math. Phys., 67 (2016), Art. 102, 18 pp. doi: 10.1007/s00033-016-0695-2. Google Scholar [9] S. T. Chen and X. H. Tang, Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete. Contin. Dyn. Syst., 38 (2018), 2333-2348. doi: 10.3934/dcds.2018096. Google Scholar [10] S. T. Chen and X. H. Tang, Geometrically distinct solutions for Klein-Gordon-Maxwell systems with superlinear nonlinearities, Appl. Math. Letters, 90 (2019), 188-193. doi: 10.1016/j.aml.2018.11.007. Google Scholar [11] S. T. Chen and X. H. Tang, Ground state solutions of Schrödinger-Poisson systems with variable potential and convolution nonlinearity, J. Math. Anal. Appl., 473 (2019), 87-111. doi: 10.1016/j.jmaa.2018.12.037. Google Scholar [12] T. D'Aprile and J. C. Wei, On bound states concentrating on spheres for the Maxwell-Schrödinger equation, SIAM J. Math. Anal., 37 (2005), 321-342. doi: 10.1137/S0036141004442793. Google Scholar [13] G. M. Figueiredo, N. Ikoma and J. R. Santos Junior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Ration. Mech. Anal., 213 (2014), 931-979. doi: 10.1007/s00205-014-0747-8. Google Scholar [14] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren Math. Wiss., vol. 224, Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0. Google Scholar [15] X. M. He, Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations, Z. Angew. Math. Phys., 62 (2011), 869-889. doi: 10.1007/s00033-011-0120-9. Google Scholar [16] X. M. He and W. M. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 53 (2012), 023702, 19 pp. doi: 10.1063/1.3683156. Google Scholar [17] Y. He and G. B. Li, Standing waves for a class of Schrödinger-Poisson equations in$\mathbb{R}^3$involving critical Sobolev exponents, Ann. Acad. Sci. Fenn. Math., 40 (2015), 729-766. doi: 10.5186/aasfm.2015.4041. Google Scholar [18] I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials, Adv. Nonlinear Stud., 8 (2008), 573-595. doi: 10.1515/ans-2008-0305. Google Scholar [19] I. Ianni, Solutions of the Schrödinger-Poisson problem concentrating on spheres, part Ⅱ: Existence, Math. Models Methods Appl. Sci., 19 (2009), 877-910. doi: 10.1142/S0218202509003656. Google Scholar [20] I. Ianni and G. Vaira, Solutions of the Schrödinger-Poisson problem concentrating on spheres, part Ⅰ: necessary condition, Math. Models Methods Appl. Sci., 19 (2009), 707-720. doi: 10.1142/S0218202509003589. Google Scholar [21] L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633-1659. doi: 10.1016/S0362-546X(96)00021-1. Google Scholar [22] G. B. Li and S. S. Yan, Eigenvalue problems for quasilinear elliptic equations on$\mathbb{R}^N$, Commun. Partial Differ. Equ., 14 (1989), 1291-1314. doi: 10.1080/03605308908820654. Google Scholar [23] P. L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33-97. doi: 10.1007/BF01205672. Google Scholar [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. Google Scholar [25] P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990. doi: 10.1007/978-3-7091-6961-2. Google Scholar [26] N. S. Papageorgiou, V. D. Rǎdulescu and D. Repovs, Nonlinear Analysis-Theory and Methods,, Springer Monographs in Mathematics, Springer, BerlinCham, 2019. Google Scholar [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. Google Scholar [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. Google Scholar [29] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [41] M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [45] L. G. Zhao and F. K. 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