# American Institute of Mathematical Sciences

January  2016, 15(1): 103-125. doi: 10.3934/cpaa.2016.15.103

## Concentrating ground-state solutions for a class of Schödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents

 1 School of Mathematics and Statistics, South-Central University For Nationalities, Wuhan, 430074, China 2 School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China 3 Department of Mathematics, Huazhong Normal University, Wuhan, 430079

Received  April 2015 Revised  August 2015 Published  December 2015

We are concerned with standing waves for the following Schrödinger-Poisson equation with critical nonlinearity: \begin{eqnarray} && - {\varepsilon ^2}\Delta u + V(x)u + \psi (x)u = \lambda W(x){\left| u \right|^{p - 2}}u + {\left| u \right|^4}u\;\;{\text{ in }}\mathbb{R}^3, \\ && - {\varepsilon ^2}\Delta \psi = {u^2}\;\;{\text{ in }}\mathbb{R}^3, u>0, u \in {H^1}(\mathbb{R}^3), \end{eqnarray} where $\varepsilon$ is a small positive parameter, $\lambda > 0$, $3 < p \le 4$, $V$ and $W$ are two potentials. Under proper assumptions, we prove that for $\varepsilon > 0$ sufficiently small, the above problem has a positive ground-state solution ${u_\varepsilon }$ by using a monotonicity trick and a new version of global compactness lemma. Moreover, we use another global compactness method due to [C. Gui, Commun. Partial Differential Equations 21 (1996) 787-820] to show that ${u_\varepsilon }$ concentrates around a set which is related to the set where the potential $V(x)$ attains its global minima or the set where the potential $W(x)$ attains its global maxima as $\varepsilon \to 0$. As far as we know, the existence and concentration behavior of the positive solutions to the Schrödinger-Poisson equation with critical nonlinearity $g(u): = \lambda W(x)|u{|^{p - 2}}u + |u{|^4}u$ $(3 Citation: Yi He, Lu Lu, Wei Shuai. Concentrating ground-state solutions for a class of Schödinger-Poisson equations in$\mathbb{R}^3$involving critical Sobolev exponents. Communications on Pure and Applied Analysis, 2016, 15 (1) : 103-125. doi: 10.3934/cpaa.2016.15.103 ##### References:  [1] A. Ambrosetti, On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274. doi: 10.1007/s00032-008-0094-z. [2] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal., 14 (1973), 349-381. [3] A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson equation, Commun. Contemp. Math., 10 (2008), 1-14. doi: 10.1142/S021919970800282X. [4] A. Azzollini, P. d'Avenia and A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 779-791. doi: 10.1016/j.anihpc.2009.11.012. [5] A. Azzollini and A. Pomponio, Groud state solutions for nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108. doi: 10.1016/j.jmaa.2008.03.057. [6] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293. [7] 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. [8] H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999. [9] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic problems involving critical Sobolev exponent, Commun. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405. [10] H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555. [11] G. Cerami and 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. [12] S. Cingolani and N. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Topol. Methods Nonlinear Anal., 10 (1997), 1-13. [13] D. Cao, E. S. Noussair and S. Yan, Existence and uniqueness results on single-peaked solutions of a semilinear problem, Ann. Inst. Henri Poincaré., 15 (1998), 73-111. doi: 10.1016/S0294-1449(99)80021-3. [14] 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: http://dx.doi.org/10.1017/S030821050000353X. [15] T. D'Aprile and J. Wei, On bound states concentrating on spheres for the Maxwell-Schrödinger equation, SIAM J. Math. Anal., 37 (2005), 321-342. doi: 10.1137/S0036141004442793. [16] T. D'Aprile and J. Wei, Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem, Calc. Var. Partial Differential Equations, 25 (2006), 105-137. doi: 10.1007/s00526-005-0342-9. [17] Y. Ding and X. Liu, Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities, Manuscripta Math., 140 (2013), 51-82. doi: 10.1007/s00229-011-0530-1. [18] I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0. [19] C. Gui, Existence of multi-bumb solutions for nonlinear Schrödinger equations via variational method, Comm. Partial Differential Equations, 21 (1996), 787-820. doi: 10.1080/03605309608821208. [20] X. He, Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations, Z. Angew. Math. Phys., 5 (2011), 869-889. doi: 1007/s00033-011-0120-9. [21] X. He and W. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 53 (2012), 19pp. doi: http://dx.doi.org/10.1063/1.3683156. [22] J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in$\mathbbR^N$: mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal., 35 (2010), 253-276. [23] Y. He and G. Li, The existence and concentration of weak solutions to a class of$p$-Laplacian type problems in unbounded domains, Sci. China Math., 57 (2014), 1927-1952. doi: 10.1007/s11425-014-4830-2. [24] Y. He and G. Li, Standing waves for a class of Schrödinger-Poisson equations in$\mathbbR^3$involving critical Sobolev exponents,, to appear in \emph{Annales Academi\ae Scientiarum Fennic\ae, (). [25] E. Hebey and J. Wei, Schrödinger-Poisson systems in the 3-sphere, Calc. Var. Partial Differential Equations, 47 (2013), 25-54. doi: 10.1007/s00526-012-0509-0. [26] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landsman-Lazer-type problem set on$\mathbbR^N$, Proc. Roy. Soc. Edingburgh Sect. A, 129 (1999), 787-809. doi: http://dx.doi.org/10.1017/S0308210500013147 . [27] Y. Jiang and H. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), 582-608. doi: 10.1016/j.jde.2011.05.006. [28] G. Li, Some properties of weak solutions of nonlinear scalar field equations, Ann. Acad. Sci. Fenn. A I Math., 15 (1990), 27-36. doi: 10.5186/aasfm.1990.1521. [29] Z. Liu, S. Guo and Y. Fang, Multiple semiclassical states for coupled Schrödinger-Poisson equations with critical exponential growth, J. Math. Phys., 56 (2015), 22pp. doi: http://dx.doi.org/10.1063/1.4919543. [30] G. Li and S. Yan, Eigenvalue problems for quasilinear elliptic equations on$\mathbbR^N$, Commun. Partial Differential Equations, 14 (1989), 1291-1314. doi: 10.1080/03605308908820654. [31] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032. [32] P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part II, Ann. Inst. H. Poincaré Anal. Non. Linéaire, 2 (1984) 223-283. [33] P. L. Lions, The concentration-compactness principle in the calculus of variations, The limit case, part I, Rev. Mat. H. Iberoamericano 1, 2 (1985), 145-201. doi: 10.4171/RMI/6. [34] D. Mugnai, The Schrödinger-Poisson system with positive potential, Commun. Partial Differential Equations, 36 (2011), 1099-1117. doi: 10.1080/03605302.2011.558551. [35] P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703. doi: 10.1512/iumj.1986.35.35036. [36] P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631. [37] D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674. doi: 10.1016/j.jfa.2006.04.005. [38] D. Ruiz, On the Schrödinger-Poisson-Slater System: behavior of minimizers, radial and nonradial cases, Arch. Rational Mech. Anal., 198 (2010), 349-368. doi: 10.1007/s00205-010-0299-5. [39] G. Vaira, Ground states for Schrödinger-Poisson type systems, Ricerche mat., 60 (2011), 263-297. doi: 10.1007/s11587-011-0109-x. [40] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 153 (1993), 229-244. [41] M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. [42] J. Wang, L. Tian, J. Xu and F. Zhang, Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in$\mathbbR^3$, Calc. Var. Partial Differential Equations, 48 (2013), 243-273. doi: 10.1007/s00526-012-0548-6. [43] J. Zhang, The existence and concentration of positive solutions for a nonlinear Schrödinger-Poisson system with critical growth, J. Math. Phys. 55 (2014), 14pp. doi: http://dx.doi.org/10.1063/1.4868617. [44] L. Zhao and F. 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, On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274. doi: 10.1007/s00032-008-0094-z. [2] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal., 14 (1973), 349-381. [3] A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson equation, Commun. Contemp. Math., 10 (2008), 1-14. doi: 10.1142/S021919970800282X. [4] A. Azzollini, P. d'Avenia and A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 779-791. doi: 10.1016/j.anihpc.2009.11.012. [5] A. Azzollini and A. Pomponio, Groud state solutions for nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108. doi: 10.1016/j.jmaa.2008.03.057. [6] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293. [7] 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. [8] H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999. [9] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic problems involving critical Sobolev exponent, Commun. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405. [10] H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555. [11] G. Cerami and 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. [12] S. Cingolani and N. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Topol. Methods Nonlinear Anal., 10 (1997), 1-13. [13] D. Cao, E. S. Noussair and S. Yan, Existence and uniqueness results on single-peaked solutions of a semilinear problem, Ann. Inst. Henri Poincaré., 15 (1998), 73-111. doi: 10.1016/S0294-1449(99)80021-3. [14] 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: http://dx.doi.org/10.1017/S030821050000353X. [15] T. D'Aprile and J. Wei, On bound states concentrating on spheres for the Maxwell-Schrödinger equation, SIAM J. Math. Anal., 37 (2005), 321-342. doi: 10.1137/S0036141004442793. [16] T. D'Aprile and J. Wei, Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem, Calc. Var. Partial Differential Equations, 25 (2006), 105-137. doi: 10.1007/s00526-005-0342-9. [17] Y. Ding and X. Liu, Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities, Manuscripta Math., 140 (2013), 51-82. doi: 10.1007/s00229-011-0530-1. [18] I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0. [19] C. Gui, Existence of multi-bumb solutions for nonlinear Schrödinger equations via variational method, Comm. Partial Differential Equations, 21 (1996), 787-820. doi: 10.1080/03605309608821208. [20] X. He, Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations, Z. Angew. Math. Phys., 5 (2011), 869-889. doi: 1007/s00033-011-0120-9. [21] X. He and W. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 53 (2012), 19pp. doi: http://dx.doi.org/10.1063/1.3683156. [22] J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in$\mathbbR^N$: mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal., 35 (2010), 253-276. [23] Y. He and G. Li, The existence and concentration of weak solutions to a class of$p$-Laplacian type problems in unbounded domains, Sci. China Math., 57 (2014), 1927-1952. doi: 10.1007/s11425-014-4830-2. [24] Y. He and G. Li, Standing waves for a class of Schrödinger-Poisson equations in$\mathbbR^3$involving critical Sobolev exponents,, to appear in \emph{Annales Academi\ae Scientiarum Fennic\ae, (). [25] E. Hebey and J. Wei, Schrödinger-Poisson systems in the 3-sphere, Calc. Var. Partial Differential Equations, 47 (2013), 25-54. doi: 10.1007/s00526-012-0509-0. [26] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landsman-Lazer-type problem set on$\mathbbR^N$, Proc. Roy. Soc. Edingburgh Sect. A, 129 (1999), 787-809. doi: http://dx.doi.org/10.1017/S0308210500013147 . [27] Y. Jiang and H. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), 582-608. doi: 10.1016/j.jde.2011.05.006. [28] G. Li, Some properties of weak solutions of nonlinear scalar field equations, Ann. Acad. Sci. Fenn. A I Math., 15 (1990), 27-36. doi: 10.5186/aasfm.1990.1521. [29] Z. Liu, S. Guo and Y. Fang, Multiple semiclassical states for coupled Schrödinger-Poisson equations with critical exponential growth, J. Math. Phys., 56 (2015), 22pp. doi: http://dx.doi.org/10.1063/1.4919543. [30] G. Li and S. Yan, Eigenvalue problems for quasilinear elliptic equations on$\mathbbR^N$, Commun. Partial Differential Equations, 14 (1989), 1291-1314. doi: 10.1080/03605308908820654. [31] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032. [32] P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part II, Ann. Inst. H. Poincaré Anal. Non. Linéaire, 2 (1984) 223-283. [33] P. L. Lions, The concentration-compactness principle in the calculus of variations, The limit case, part I, Rev. Mat. H. Iberoamericano 1, 2 (1985), 145-201. doi: 10.4171/RMI/6. [34] D. Mugnai, The Schrödinger-Poisson system with positive potential, Commun. Partial Differential Equations, 36 (2011), 1099-1117. doi: 10.1080/03605302.2011.558551. [35] P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703. doi: 10.1512/iumj.1986.35.35036. [36] P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631. [37] D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674. doi: 10.1016/j.jfa.2006.04.005. [38] D. Ruiz, On the Schrödinger-Poisson-Slater System: behavior of minimizers, radial and nonradial cases, Arch. Rational Mech. Anal., 198 (2010), 349-368. doi: 10.1007/s00205-010-0299-5. [39] G. Vaira, Ground states for Schrödinger-Poisson type systems, Ricerche mat., 60 (2011), 263-297. doi: 10.1007/s11587-011-0109-x. [40] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 153 (1993), 229-244. [41] M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. [42] J. Wang, L. Tian, J. Xu and F. Zhang, Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in$\mathbbR^3$, Calc. Var. Partial Differential Equations, 48 (2013), 243-273. doi: 10.1007/s00526-012-0548-6. [43] J. Zhang, The existence and concentration of positive solutions for a nonlinear Schrödinger-Poisson system with critical growth, J. Math. Phys. 55 (2014), 14pp. doi: http://dx.doi.org/10.1063/1.4868617. [44] L. Zhao and F. Zhao, On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl., 346 (2008), 155-169. doi: 10.1016/j.jmaa.2008.04.053.  [1] Caixia Chen, Aixia Qian. Multiple positive solutions for the Schrödinger-Poisson equation with critical growth. Mathematical Foundations of Computing, 2022, 5 (2) : 113-128. doi: 10.3934/mfc.2021036 [2] Daniele Cassani, Luca Vilasi, Jianjun Zhang. Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1737-1754. doi: 10.3934/cpaa.2021039 [3] Xiaoping Chen, Chunlei Tang. Least energy sign-changing solutions for Schrödinger-Poisson system with critical growth. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2291-2312. doi: 10.3934/cpaa.2021077 [4] Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025 [5] Mengyao Chen, Qi Li, Shuangjie Peng. Bound states for fractional Schrödinger-Poisson system with critical exponent. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1819-1835. doi: 10.3934/dcdss.2021038 [6] Lun Guo, Wentao Huang, Huifang Jia. Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in$ \mathbb{R} ^{3} $. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1663-1693. doi: 10.3934/cpaa.2019079 [7] Rong Cheng, Jun Wang. Existence of ground states for Schrödinger-Poisson system with nonperiodic potentials. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2021317 [8] Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in$ \mathbb{R}^2 $. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447 [9] Kaimin Teng, Xian Wu. Concentration of bound states for fractional Schrödinger-Poisson system via penalization methods. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1157-1187. doi: 10.3934/cpaa.2022014 [10] Qian Shen, Na Wei. Stability of ground state for the Schrödinger-Poisson equation. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2805-2816. doi: 10.3934/jimo.2020095 [11] Xia Sun, Kaimin Teng. Positive bound states for fractional Schrödinger-Poisson system with critical exponent. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3735-3768. doi: 10.3934/cpaa.2020165 [12] Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, 2021, 29 (3) : 2475-2488. doi: 10.3934/era.2020125 [13] Yao Du, Jiabao Su, Cong Wang. On the critical Schrödinger-Poisson system with$ p $-Laplacian. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1329-1342. doi: 10.3934/cpaa.2022020 [14] Antonio Azzollini, Pietro d’Avenia, Valeria Luisi. Generalized Schrödinger-Poisson type systems. Communications on Pure and Applied Analysis, 2013, 12 (2) : 867-879. doi: 10.3934/cpaa.2013.12.867 [15] Claudianor O. Alves, Minbo Yang. Existence of positive multi-bump solutions for a Schrödinger-Poisson system in$\mathbb{R}^{3}\$. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 5881-5910. doi: 10.3934/dcds.2016058 [16] Lirong Huang, Jianqing Chen. Existence and asymptotic behavior of bound states for a class of nonautonomous Schrödinger-Poisson system. Electronic Research Archive, 2020, 28 (1) : 383-404. doi: 10.3934/era.2020022 [17] 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 [18] Sitong Chen, Wennian Huang, Xianhua Tang. Existence criteria of ground state solutions for Schrödinger-Poisson systems with a vanishing potential. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3055-3066. doi: 10.3934/dcdss.2020339 [19] Xueqin Peng, Gao Jia. Existence and asymptotical behavior of positive solutions for the Schrödinger-Poisson system with double quasi-linear terms. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2325-2344. doi: 10.3934/dcdsb.2021134 [20] Vincenzo Ambrosio. Concentration phenomena for critical fractional Schrödinger systems. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2085-2123. doi: 10.3934/cpaa.2018099

2020 Impact Factor: 1.916