American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion
  • DCDS Home
  • This Issue
  • Next Article
    Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system
January  2017, 37(1): 605-625. doi: 10.3934/dcds.2017025

Bound state solutions of Schrödinger-Poisson system with critical exponent

Xu Zhang , Shiwang Ma , and  Qilin Xie

School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

Received  March 2016 Revised  July 2016 Published  November 2016

Fund Project: Research supported by the National Natural Science Foundation of China (No. 11571187).

In this paper, we consider the following Schrödinger-Poisson problem
$\tag{P}\label{0.1} \begin{cases}- Δ u+V(x)u+K(x)φ u=|u|^{2^*-2}u, &x∈ \mathbb{R}^3,\\-Δ φ=K(x)u^2,&x∈ \mathbb{R}^3,\end{cases}$
where
$2^*=6 $
 is the critical exponent in
$\mathbb R^3$
,
$ K∈ L^{\frac{1}{2}}(\mathbb{R}^3)$
and
$V∈ L^{\frac{3}{2}}(\mathbb{R}^3)$
are given nonnegative functions. When
$|V|_{\frac{3}{2}}+|K|_{\frac{1}{2}}$
is suitable small, we prove that problem (P) has at least one bound state solution via a linking theorem.
Keywords: Schrödinger-Poisson system, critical exponent, bound state solutions.
Mathematics Subject Classification: Primary:35J20;Secondary:35J6.
Citation: Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025
References:
[1]

A. Ambrosetti, On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.  doi: 10.1007/s00032-008-0094-z.  Google Scholar

[2]

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

[3]

A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anual. Appl., 345 (2008), 90-108.  doi: 10.1016/j.jmaa.2008.03.057.  Google Scholar

[4]

V. Benci and D. Fortunato, Some compact embedding theorems for weighted Sobolev spaces, Boll. Un. Mat. Ital. B (5), 13 (1976), 832-843.   Google Scholar

[5]

V. Benci and C. Cerami, Existence of positive solutions of the equation $-Δ u+a(x)u=u^{(N+2)/(N-2)}$ in $\mathbb R^3$, J. Funct. Anal., 88 (1990), 90-117.  doi: 10.1016/0022-1236(90)90120-A.  Google Scholar

[6]

V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.   Google Scholar

[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.  Google Scholar

[8]

H. Brezis, Analyse Fonctionnelle, Masson, Parias, 1983.  Google Scholar

[9]

G. M. Coclite, A multiplicity result for the Schrödinger-Maxwell equations with negative potential, Ann. Polon. Math., 79 (2002), 21-30.   Google Scholar

[10]

T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.   Google Scholar

[11]

T. D'Aprile and J. C. Wei, On bound states concentrating on spheres for the Maxwell-Schrödinger equations, SIAM J. Math. Anal., 37 (2005), 321-342.  doi: 10.1137/S0036141004442793.  Google Scholar

[12]

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, 19pp. doi: 10.1063/1.3683156.  Google Scholar

[13]

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

[14]

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

[15]

Y. S. Jiang and H. S. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential. Equations., 251 (2011), 582-608.  doi: 10.1016/j.jde.2011.05.006.  Google Scholar

[16]

G. B. Li, S. J. Peng and C. H. Wang, Multi-bump solutions for the nonlinear Schrödinger-Poisson system J. Maht. phys. 52 (2011), 053505, 19pp. doi: 10.1063/1.3585657.  Google Scholar

[17]

P. L. Lions, The concentration-compactness method in the calculus of variations. The locally compact case, parts 1 and 2, Ann. Inst. H. Poincaré Anual. Non Linéair, 1 (1984), 109-145,223-283.   Google Scholar

[18]

Z. S. Liu and S. J. Guo, On ground state solutions for the Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 412 (2014), 435-448.  doi: 10.1016/j.jmaa.2013.10.066.  Google Scholar

[19]

D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell equations: Concentration around a sphere, Math. Models Methods Appl. Sci., 15 (2005), 141-164.  doi: 10.1142/S0218202505003939.  Google Scholar

[20]

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.  Google Scholar

[21]

A. Salvatore, Multiple solitary waves for a non-homogeneous Schrödinger-Maxwell system in $\mathbb{R}^3$, Adv. Nonlinear strud., 6 (2006), 157-169.  doi: 10.1515/ans-2006-0203.  Google Scholar

[22]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.  doi: 10.1007/BF01174186.  Google Scholar

[23]

J. Wang, J. X. Xu, F. B. Zhang and X. M. Chen, Existence of multi-bump solutions for a semilinear Schrödinger-Poisson system, Nonlinearity, 26 (2013), 1377-1399.  doi: 10.1088/0951-7715/26/5/1377.  Google Scholar

[24]

M. Willem, Analyse Harmomque Réelle, Hermann, Parias, 1995.  Google Scholar

[25]

M. Willem, Minimax Theorems Birkhäuser, Boston, Basel, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[26]

Z. P. Wang and H. S. Zhou, Positive solutions for a nonlinear stationary Schrödinger-Poisson system in $\mathbb{R}^3 $, Discrete Contin. Dyn. Syst., 18 (2007), 809-816.  doi: 10.3934/dcds.2007.18.809.  Google Scholar

[27]

J. Zhang, On the Schrödinger-Poisson equations with a general nonlinearity in the critical growth, Nonlinea Anal., 75 (2012), 6391-6401.  doi: 10.1016/j.na.2012.07.008.  Google Scholar

[28]

J. Zhang, On ground state and nodal solutions of Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 428 (2015), 387-404.  doi: 10.1016/j.jmaa.2015.03.032.  Google Scholar

[29]

L. G. Zhao, H. Liu and F. K. Zhao, Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Differential. Equations, 255 (2013), 1-23.  doi: 10.1016/j.jde.2013.03.005.  Google Scholar

[30]

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

[31]

L. G. Zhao and F. K. Zhao, On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anual. Appl., 346 (2008), 155-169.  doi: 10.1016/j.jmaa.2008.04.053.  Google Scholar

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.  Google Scholar

[2]

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

[3]

A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anual. Appl., 345 (2008), 90-108.  doi: 10.1016/j.jmaa.2008.03.057.  Google Scholar

[4]

V. Benci and D. Fortunato, Some compact embedding theorems for weighted Sobolev spaces, Boll. Un. Mat. Ital. B (5), 13 (1976), 832-843.   Google Scholar

[5]

V. Benci and C. Cerami, Existence of positive solutions of the equation $-Δ u+a(x)u=u^{(N+2)/(N-2)}$ in $\mathbb R^3$, J. Funct. Anal., 88 (1990), 90-117.  doi: 10.1016/0022-1236(90)90120-A.  Google Scholar

[6]

V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.   Google Scholar

[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.  Google Scholar

[8]

H. Brezis, Analyse Fonctionnelle, Masson, Parias, 1983.  Google Scholar

[9]

G. M. Coclite, A multiplicity result for the Schrödinger-Maxwell equations with negative potential, Ann. Polon. Math., 79 (2002), 21-30.   Google Scholar

[10]

T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.   Google Scholar

[11]

T. D'Aprile and J. C. Wei, On bound states concentrating on spheres for the Maxwell-Schrödinger equations, SIAM J. Math. Anal., 37 (2005), 321-342.  doi: 10.1137/S0036141004442793.  Google Scholar

[12]

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, 19pp. doi: 10.1063/1.3683156.  Google Scholar

[13]

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

[14]

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

[15]

Y. S. Jiang and H. S. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential. Equations., 251 (2011), 582-608.  doi: 10.1016/j.jde.2011.05.006.  Google Scholar

[16]

G. B. Li, S. J. Peng and C. H. Wang, Multi-bump solutions for the nonlinear Schrödinger-Poisson system J. Maht. phys. 52 (2011), 053505, 19pp. doi: 10.1063/1.3585657.  Google Scholar

[17]

P. L. Lions, The concentration-compactness method in the calculus of variations. The locally compact case, parts 1 and 2, Ann. Inst. H. Poincaré Anual. Non Linéair, 1 (1984), 109-145,223-283.   Google Scholar

[18]

Z. S. Liu and S. J. Guo, On ground state solutions for the Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 412 (2014), 435-448.  doi: 10.1016/j.jmaa.2013.10.066.  Google Scholar

[19]

D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell equations: Concentration around a sphere, Math. Models Methods Appl. Sci., 15 (2005), 141-164.  doi: 10.1142/S0218202505003939.  Google Scholar

[20]

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.  Google Scholar

[21]

A. Salvatore, Multiple solitary waves for a non-homogeneous Schrödinger-Maxwell system in $\mathbb{R}^3$, Adv. Nonlinear strud., 6 (2006), 157-169.  doi: 10.1515/ans-2006-0203.  Google Scholar

[22]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.  doi: 10.1007/BF01174186.  Google Scholar

[23]

J. Wang, J. X. Xu, F. B. Zhang and X. M. Chen, Existence of multi-bump solutions for a semilinear Schrödinger-Poisson system, Nonlinearity, 26 (2013), 1377-1399.  doi: 10.1088/0951-7715/26/5/1377.  Google Scholar

[24]

M. Willem, Analyse Harmomque Réelle, Hermann, Parias, 1995.  Google Scholar

[25]

M. Willem, Minimax Theorems Birkhäuser, Boston, Basel, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[26]

Z. P. Wang and H. S. Zhou, Positive solutions for a nonlinear stationary Schrödinger-Poisson system in $\mathbb{R}^3 $, Discrete Contin. Dyn. Syst., 18 (2007), 809-816.  doi: 10.3934/dcds.2007.18.809.  Google Scholar

[27]

J. Zhang, On the Schrödinger-Poisson equations with a general nonlinearity in the critical growth, Nonlinea Anal., 75 (2012), 6391-6401.  doi: 10.1016/j.na.2012.07.008.  Google Scholar

[28]

J. Zhang, On ground state and nodal solutions of Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 428 (2015), 387-404.  doi: 10.1016/j.jmaa.2015.03.032.  Google Scholar

[29]

L. G. Zhao, H. Liu and F. K. Zhao, Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Differential. Equations, 255 (2013), 1-23.  doi: 10.1016/j.jde.2013.03.005.  Google Scholar

[30]

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

[31]

L. G. Zhao and F. K. Zhao, On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anual. Appl., 346 (2008), 155-169.  doi: 10.1016/j.jmaa.2008.04.053.  Google Scholar

[1]

Mengyao Chen, Qi Li, Shuangjie Peng. Bound states for fractional Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1819-1835. doi: 10.3934/dcdss.2021038
[2]

Xia Sun, Kaimin Teng. Positive bound states for fractional Schrödinger-Poisson system with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3735-3768. doi: 10.3934/cpaa.2020165
[3]

Xinsheng Du, Qi Li, Zengqin Zhao, Gen Li. Bound state solutions for fractional Schrödinger-Poisson systems. Mathematical Foundations of Computing, 2021  doi: 10.3934/mfc.2021023
[4]

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 & Applied Analysis, 2019, 18 (5) : 2299-2324. doi: 10.3934/cpaa.2019104
[5]

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 & Applied Analysis, 2017, 16 (5) : 1587-1602. doi: 10.3934/cpaa.2017076
[6]

Sitong Chen, Xianhua Tang. Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4685-4702. doi: 10.3934/dcdsb.2018329
[7]

Sitong Chen, Junping Shi, Xianhua Tang. Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5867-5889. doi: 10.3934/dcds.2019257
[8]

Xiaoping Chen, Chunlei Tang. Least energy sign-changing solutions for Schrödinger-Poisson system with critical growth. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2291-2312. doi: 10.3934/cpaa.2021077
[9]

Caixia Chen, Aixia Qian. Multiple positive solutions for the Schrödinger-Poisson equation with critical growth. Mathematical Foundations of Computing, 2021  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 & Applied Analysis, 2019, 18 (4) : 1663-1693. doi: 10.3934/cpaa.2019079
[11]

Chunhua Wang, Jing Yang. Positive solutions for a nonlinear Schrödinger-Poisson system. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5461-5504. doi: 10.3934/dcds.2018241
[12]

Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991
[13]

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
[14]

Kaimin Teng, Xian Wu. Concentration of bound states for fractional Schrödinger-Poisson system via penalization methods. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2022014
[15]

Yao Du, Jiabao Su, Cong Wang. On the critical Schrödinger-Poisson system with $ p $-Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2022020
[16]

Xianhua Tang, Sitong Chen. Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4973-5002. doi: 10.3934/dcds.2017214
[17]

Sitong Chen, Wennian Huang, Xianhua Tang. Existence criteria of ground state solutions for Schrödinger-Poisson systems with a vanishing potential. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3055-3066. doi: 10.3934/dcdss.2020339
[18]

Qian Shen, Na Wei. Stability of ground state for the Schrödinger-Poisson equation. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2805-2816. doi: 10.3934/jimo.2020095
[19]

Mingzheng Sun, Jiabao Su, Leiga Zhao. Infinitely many solutions for a Schrödinger-Poisson system with concave and convex nonlinearities. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 427-440. doi: 10.3934/dcds.2015.35.427
[20]

Claudianor O. Alves, Minbo Yang. Existence of positive multi-bump solutions for a Schrödinger-Poisson system in $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 5881-5910. doi: 10.3934/dcds.2016058

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (331)
  • HTML views (72)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy