American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Nontrivial solutions for the fractional Laplacian problems without asymptotic limits near both infinity and zero
  • DCDS-S Home
  • This Issue
  • Next Article
    Multi-bubble nodal solutions to slightly subcritical elliptic problems with Hardy terms in symmetric domains
June  2021, 14(6): 1819-1835. doi: 10.3934/dcdss.2021038

Bound states for fractional Schrödinger-Poisson system with critical exponent

Mengyao Chen , Qi Li and  Shuangjie Peng ,

School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China

* Corresponding author: Shuangjie Peng

Received  December 2020 Revised  January 2021 Published  June 2021 Early access  March 2021

Fund Project: The first author is supported by funding for basic research business in Central Universities (innovative funding projects) (2020CXZZ069). The second author is supported by NSFC grant 1207116

This paper deals with the fractional Schrödinger-Poisson system
$ \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}(-\Delta )^su+V(x)u+K(x)\phi u = |u|^{2_{s}^{*}-2}u, & \text{in}\ {\Bbb R}^3,\\ (-\Delta)^{t}\phi = K(x)u^2, & \text{in}\ {\Bbb R}^3, \end{array} \right. \end{equation*} $
where
$ s\in (\frac{3}{4}, 1) $
,
$ t\in(0, 1) $
,
$ \varepsilon $
 is a positive parameter,
$ 2_{s}^{*} = \frac{6}{3-2s} $
 is the critical Sobolev exponent.
$ K(x)\in L^{\frac{6}{2t+4s-3}}({\Bbb R}^3) $
,
$ V(x)\in L^{\frac{3}{2s}}({\Bbb R}^3) $
 and
$ V(x) $
 is assumed to be zero in some region of
$ {\Bbb R}^3 $
, which means that the problem is of the critical frequency case. In virtue of a global compactness result in fractional Sobolev space and Lusternik-Schnirelman theory of critical points, we succeed in proving the multiplicity of bound states.
Keywords: Fractional Schrödinger-Poisson system, bound states, critical frequency, Lusternik-Schnirelman theory.
Mathematics Subject Classification: Primary: 35J47, 35J50.
Citation: 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
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, 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.

[3]

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.

[4]

V. Benci and G. Cerami, Existence of positive solutions of the equation $-\Delta u+a(x)u = u^{(N+2)/(N-2)}$ in ${\Bbb R}^N$, J. Funct. Anal., 88 (1990), 90-117.  doi: 10.1016/0022-1236(90)90120-A.

[5]

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.

[6]

G. Cerami and R. Molle, Positive bound state solutions for some Schrödinger-Poisson systems, Nonlinearity, 29 (2016), 3103-3119.  doi: 10.1088/0951-7715/29/10/3103.

[7]

J. Chabrowski and J. Yang, Multiple semiclassical solutions of the Schrödinger equation involving a critical Sobolev exponent, Portugal. Math., 57 (2000), 273-284. 

[8]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[9]

T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.  doi: 10.1515/ans-2004-0305.

[10]

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.

[11]

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.

[12]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[13]

R. L. Frank and E. H. Lieb, Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Calc. Var. Partial Differential Equations, 39 (2010), 85-99.  doi: 10.1007/s00526-009-0302-x.

[14]

R. L. Frank and E. H. Lieb, A new, rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality, Oper. Theory Adv. Appl., 219 (2012), 55-67.  doi: 10.1007/978-3-0348-0263-5_4.

[15]

L. Guo and Q. Li, Multiple bound state solutions for fractional Choquard equation with Hardy-Littlewood-Sobolev critical exponent, J. Math. Phys., 61 (2020), 121501, 20 pp. doi: 10.1063/5.0013475.

[16]

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.

[17]

G. Li, S. Peng and S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger-Poisson system, Commun. Contemp. Math., 12 (2010), 1069-1092.  doi: 10.1142/S0219199710004068.

[18]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118 (1983), 349-374.  doi: 10.2307/2007032.

[19]

Z. Liu and J. Zhang, Multiplicity and concentration of positive solutions for the fractional Schrödinger-Poisson systems with critical growth, ESAIM Control Optim. Calc. Var., 23 (2017), 1515-1542.  doi: 10.1051/cocv/2016063.

[20]

E. G. Murcia and G. Siciliano, Positive semiclassical states for a fractional Schrödinger-Poisson system, Differential Integral Equations, 30 (2017), 231-258. 

[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 and G. Vaira, Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential, Rev. Mat. Iberoam., 27 (2011), 253-271.  doi: 10.4171/RMI/635.

[23]

K. Teng, Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 261 (2016), 3061-3106.  doi: 10.1016/j.jde.2016.05.022.

[24]

K. Teng and R. P. Agarwal, Existence and concentration of positive ground state solutions for nonlinear fractional Schrödinger-Poisson system with critical growth, Math. Methods Appl. Sci., 41 (2018), 8258-8293.  doi: 10.1002/mma.5289.

[25]

M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.

[26]

Y. Yu, F. Zhao and L. Zhao, The concentration behavior of ground state solutions for a fractional Schrödinger-Poisson system, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 116, 25 pp. doi: 10.1007/s00526-017-1199-4.

[27]

J. Zhang, J. M. do Ó and M. Squassina, Fractional Schrödinger-Poisson systems with a general subcritical or critical nonlinearity, Adv. Nonlinear Stud., 16 (2016), 15-30.  doi: 10.1515/ans-2015-5024.

[28]

H. Zhang, J. Xu and F. Zhang, Multiplicity of semiclassical states for Schrödinger-Poisson systems with critical frequency, Z. Angew. Math. Phys., 71 (2020), Paper No. 5, 15 pp. doi: 10.1007/s00033-019-1226-8.

[29]

H. Zhang and F. Zhang, Multiplicity of semiclassical states for fractional Schrödinger equations with critical frequency, Nonlinear Anal., 190 (2020), 111599, 15pp. doi: 10.1016/j.na.2019.111599.

[30]

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 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, 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.

[3]

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.

[4]

V. Benci and G. Cerami, Existence of positive solutions of the equation $-\Delta u+a(x)u = u^{(N+2)/(N-2)}$ in ${\Bbb R}^N$, J. Funct. Anal., 88 (1990), 90-117.  doi: 10.1016/0022-1236(90)90120-A.

[5]

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.

[6]

G. Cerami and R. Molle, Positive bound state solutions for some Schrödinger-Poisson systems, Nonlinearity, 29 (2016), 3103-3119.  doi: 10.1088/0951-7715/29/10/3103.

[7]

J. Chabrowski and J. Yang, Multiple semiclassical solutions of the Schrödinger equation involving a critical Sobolev exponent, Portugal. Math., 57 (2000), 273-284. 

[8]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[9]

T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.  doi: 10.1515/ans-2004-0305.

[10]

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.

[11]

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.

[12]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[13]

R. L. Frank and E. H. Lieb, Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Calc. Var. Partial Differential Equations, 39 (2010), 85-99.  doi: 10.1007/s00526-009-0302-x.

[14]

R. L. Frank and E. H. Lieb, A new, rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality, Oper. Theory Adv. Appl., 219 (2012), 55-67.  doi: 10.1007/978-3-0348-0263-5_4.

[15]

L. Guo and Q. Li, Multiple bound state solutions for fractional Choquard equation with Hardy-Littlewood-Sobolev critical exponent, J. Math. Phys., 61 (2020), 121501, 20 pp. doi: 10.1063/5.0013475.

[16]

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.

[17]

G. Li, S. Peng and S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger-Poisson system, Commun. Contemp. Math., 12 (2010), 1069-1092.  doi: 10.1142/S0219199710004068.

[18]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118 (1983), 349-374.  doi: 10.2307/2007032.

[19]

Z. Liu and J. Zhang, Multiplicity and concentration of positive solutions for the fractional Schrödinger-Poisson systems with critical growth, ESAIM Control Optim. Calc. Var., 23 (2017), 1515-1542.  doi: 10.1051/cocv/2016063.

[20]

E. G. Murcia and G. Siciliano, Positive semiclassical states for a fractional Schrödinger-Poisson system, Differential Integral Equations, 30 (2017), 231-258. 

[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 and G. Vaira, Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential, Rev. Mat. Iberoam., 27 (2011), 253-271.  doi: 10.4171/RMI/635.

[23]

K. Teng, Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 261 (2016), 3061-3106.  doi: 10.1016/j.jde.2016.05.022.

[24]

K. Teng and R. P. Agarwal, Existence and concentration of positive ground state solutions for nonlinear fractional Schrödinger-Poisson system with critical growth, Math. Methods Appl. Sci., 41 (2018), 8258-8293.  doi: 10.1002/mma.5289.

[25]

M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.

[26]

Y. Yu, F. Zhao and L. Zhao, The concentration behavior of ground state solutions for a fractional Schrödinger-Poisson system, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 116, 25 pp. doi: 10.1007/s00526-017-1199-4.

[27]

J. Zhang, J. M. do Ó and M. Squassina, Fractional Schrödinger-Poisson systems with a general subcritical or critical nonlinearity, Adv. Nonlinear Stud., 16 (2016), 15-30.  doi: 10.1515/ans-2015-5024.

[28]

H. Zhang, J. Xu and F. Zhang, Multiplicity of semiclassical states for Schrödinger-Poisson systems with critical frequency, Z. Angew. Math. Phys., 71 (2020), Paper No. 5, 15 pp. doi: 10.1007/s00033-019-1226-8.

[29]

H. Zhang and F. Zhang, Multiplicity of semiclassical states for fractional Schrödinger equations with critical frequency, Nonlinear Anal., 190 (2020), 111599, 15pp. doi: 10.1016/j.na.2019.111599.

[30]

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]

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
[2]

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
[3]

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
[4]

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
[5]

Xinsheng Du, Qi Li, Zengqin Zhao, Gen Li. Bound state solutions for fractional Schrödinger-Poisson systems. Mathematical Foundations of Computing, 2022, 5 (1) : 57-66. doi: 10.3934/mfc.2021023
[6]

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
[7]

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
[8]

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
[9]

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
[10]

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
[11]

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
[12]

Chunhua Wang, Jing Yang. Positive solutions for a nonlinear Schrödinger-Poisson system. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5461-5504. doi: 10.3934/dcds.2018241
[13]

Miao-Miao Li, Chun-Lei Tang. Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1587-1602. doi: 10.3934/cpaa.2017076
[14]

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
[15]

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
[16]

Yong-Yong Li, Yan-Fang Xue, Chun-Lei Tang. Ground state solutions for asymptotically periodic modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system involving critical exponent. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2299-2324. doi: 10.3934/cpaa.2019104
[17]

Hangzhou Hu, Yuan Li, Dun Zhao. Ground state for fractional Schrödinger-Poisson equation in Coulomb-Sobolev space. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1899-1916. doi: 10.3934/dcdss.2021064
[18]

Mingzheng Sun, Jiabao Su, Leiga Zhao. Infinitely many solutions for a Schrödinger-Poisson system with concave and convex nonlinearities. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 427-440. doi: 10.3934/dcds.2015.35.427
[19]

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
[20]

Margherita Nolasco. Breathing modes for the Schrödinger-Poisson system with a multiple--well external potential. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1411-1419. doi: 10.3934/cpaa.2010.9.1411

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (290)
  • HTML views (135)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy