July  2020, 19(7): 3735-3768. doi: 10.3934/cpaa.2020165

Positive bound states for fractional Schrödinger-Poisson system with critical exponent

Department of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China

*Corresponding author

Received  September 2019 Revised  January 2020 Published  April 2020

Fund Project: The work is supported by NSFC grant 11501403, by fund program for the Scientific Activities of Selected Returned Overseas Professionals in Shanxi Province (2018) and by the Natural Science Foundation of Shanxi Province (No. 201901D111085)

In this paper, we consider the following fractional critical Schrödinger-Poisson system without perturbation terms
$ \begin{equation*} \begin{cases} (-\Delta)^su+V(x)u+K(x)\phi u = |u|^{2_s^{\ast}-2}u & \hbox{in $\mathbb{R}^3$,} \\ (-\Delta)^t\phi = K(x)u^2& \hbox{in $\mathbb{R}^3$,} \end{cases} \end{equation*} $
where
$ s\in(\frac{3}{4},1) $
,
$ t\in(0,1) $
. Under some suitable assumptions on potentials
$ V(x) $
and
$ K(x) $
, we show that the ground state positive solutions do not exist, by using the topological argument, we establish the existence of positive bound state solutions in the range
$ (\frac{s}{3}\mathcal{S}_s^{\frac{3}{2s}},\frac{2s}{3}\mathcal{S}_s^{\frac{3}{2s}}) $
.
Citation: 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
References:
[1]

C. O. Alves and O. H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R}^N$ via penalization method, Calc. Var. Partial Differ. Equ., 55 (2016), 47: 19. doi: 10.1007/s00526-016-0983-x.

[2]

A. AzzolliniP. d'Avenia and A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term, Ann. Inst. Henri Poincare Anal. Non Lineaire, 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 $-\Delta u+a(x)u = u^{\frac{N+2}{N-2}}$ in $\mathbb{R}^N$, 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, Nonlinear Anal., 11 (1998), 283-293.  doi: 10.12775/TMNA.1998.019.

[6]

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.

[7] G. M. BisciV. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, 162 Cambridge, 2016.  doi: 10.1017/CBO9781316282397.
[8]

L. BrascoM. Squassina and Y. Yang, Global compactness results for nonlocal problems, Discrete Contin. Dyn. Syst. S, 11 (2018), 391-424.  doi: 10.3934/dcdss.2018022.

[9]

G. Cerami and R. Molle, Multiple positive bound states for critical Schrödinger-Poisson systems, preprint, arXiv: 1802.02539v1. doi: 10.1051/cocv/2018071.

[10]

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

[11]

R. Cont and P. Tankov, Financial Modeling with Jump Processes, Chapman Hall/CRC Financial Mathematics Series, Boca Raton, 2004.

[12]

J. N. Correia and G. M. Figueiredo, Existence of positive solution of the equation $(-\Delta)^su+a(x)u = |u|^{2_s^{\ast}-2}u$, Calc. Var. Partial Differ. Equ., 58 (2019), Art. 63. doi: 10.1007/s00526-019-1502-7.

[13]

S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb{R}^n$, Lecture Notes, Scuola Normale Superiore di Pisa (New Series), Vol. 15, Edizioni della Normale, Pisa, 2017. doi: 10.1007/978-88-7642-601-8.

[14]

S. DipierroG. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional laplacian, Matematiche LXVIII, 68 (2013), 201-216.  doi: 10.4418/2013.68.1.15.

[15]

P. FelmerA. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. R. Soc. Edinb. Sect. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.

[16]

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.

[17]

L. R. HuangE. Rocha and J. Q. Chen, Positive and sign-changing solutions of a Schrödinger-Poisson system involving a critical nonlinearity, J. Math. Anal. Appl., 408 (2013), 55-69.  doi: 10.1016/j.jmaa.2013.05.071.

[18]

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

[19]

H. Kikuchi, On the existence of a solution for elliptic system related to the Maxwell-Schrödinger equations, Nonlinear Anal., 67 (2007), 1445-1456.  doi: 10.1016/j.na.2006.07.029.

[20]

H. Kikuchi, Existence and stability of standing waves for Schrödinger-Poisson-Slater equation, Adv. Nonlinear Stud., 7 (2007), 407-437.  doi: 10.1515/ans-2007-0305.

[21]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[22]

N. Laskin, Fractional Schrödinger equation, Phys. Rev., 66 (2002), 56-108.  doi: 10.1103/PhysRevE.66.056108.

[23]

Y. Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. 

[24]

F. Y. Li, Y. H. Li and J. P. Shi, Existence of positive solutions to Schrödinger-Poisson type systems with critical exponent, Commun. Contemp. Math., 16 (2014), Art. 1450036. doi: 10.1142/S0219199714500369.

[25]

P. L. Lions, The concentration-compactness principle in the calculus of variation. The locally compact case. Part I, Ann. Inst. Henri Poincare (C) Non Linear Anal., 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.

[26]

P. L. Lions, The concentration-compactness principle in the calculus of variation. The locally compact case. Part II, Ann. Inst. Henri Poincare (C) Non Linear Anal., 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.

[27]

Z. S. Liu and J. 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.

[28]

R. Metzler and J. Klafter, The random walks guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.

[29]

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

[30]

E. D. NezzaG. 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.

[31]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbb{R}^N$, J. Math. Phys., 54 (2013), Art. 031501. doi: 10.1063/1.4793990.

[32]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.

[33]

X. D. Shang and J. H. Zhang, Ground states for fractional Schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207.  doi: 10.1088/0951-7715/27/2/187.

[34]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.

[35]

K. M. Teng, Multiple solutions for a class of fractional Schrödinger equation in $\mathbb{R}^N$, Nonlinear Anal. Real World Appl., 21 (2015), 76-86.  doi: 10.1016/j.nonrwa.2014.06.008.

[36]

K. M. Teng, Ground state solutions for the nonlinear fractional Schrödinger-Poisson system, Appl. Anal., 98 (2019), 1959-1996.  doi: 10.1080/00036811.2018.1441998.

[37]

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

[38]

K. M. Teng, Concerntration phenomenon for fractional nonlinear Schrödinger-Poisson system with critical nonlinearity, preprint, arXiv: 1906.11563.

[39]

K. M. Teng and X. M. He, Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent, Commun. Pure Appl. Anal., 15 (2016), 991-1008.  doi: 10.3934/cpaa.2016.15.991.

[40]

J. WangL. X. TianJ. 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]

J. ZhangM. 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.

[42]

X. ZhangS. Ma and Q. Xie, Bound state solutions of Schrödinger-Poisson system with critical exponent, Discrete Contin. Dyn. Syst., 37 (2017), 605-625.  doi: 10.3934/dcds.2017025.

[43]

L. ZhaoH. Liu and F. Zhao, Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Differ. Equ., 255 (2013), 1-23.  doi: 10.1016/j.jde.2013.03.005.

[44]

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]

C. O. Alves and O. H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R}^N$ via penalization method, Calc. Var. Partial Differ. Equ., 55 (2016), 47: 19. doi: 10.1007/s00526-016-0983-x.

[2]

A. AzzolliniP. d'Avenia and A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term, Ann. Inst. Henri Poincare Anal. Non Lineaire, 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 $-\Delta u+a(x)u = u^{\frac{N+2}{N-2}}$ in $\mathbb{R}^N$, 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, Nonlinear Anal., 11 (1998), 283-293.  doi: 10.12775/TMNA.1998.019.

[6]

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.

[7] G. M. BisciV. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, 162 Cambridge, 2016.  doi: 10.1017/CBO9781316282397.
[8]

L. BrascoM. Squassina and Y. Yang, Global compactness results for nonlocal problems, Discrete Contin. Dyn. Syst. S, 11 (2018), 391-424.  doi: 10.3934/dcdss.2018022.

[9]

G. Cerami and R. Molle, Multiple positive bound states for critical Schrödinger-Poisson systems, preprint, arXiv: 1802.02539v1. doi: 10.1051/cocv/2018071.

[10]

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

[11]

R. Cont and P. Tankov, Financial Modeling with Jump Processes, Chapman Hall/CRC Financial Mathematics Series, Boca Raton, 2004.

[12]

J. N. Correia and G. M. Figueiredo, Existence of positive solution of the equation $(-\Delta)^su+a(x)u = |u|^{2_s^{\ast}-2}u$, Calc. Var. Partial Differ. Equ., 58 (2019), Art. 63. doi: 10.1007/s00526-019-1502-7.

[13]

S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb{R}^n$, Lecture Notes, Scuola Normale Superiore di Pisa (New Series), Vol. 15, Edizioni della Normale, Pisa, 2017. doi: 10.1007/978-88-7642-601-8.

[14]

S. DipierroG. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional laplacian, Matematiche LXVIII, 68 (2013), 201-216.  doi: 10.4418/2013.68.1.15.

[15]

P. FelmerA. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. R. Soc. Edinb. Sect. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.

[16]

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.

[17]

L. R. HuangE. Rocha and J. Q. Chen, Positive and sign-changing solutions of a Schrödinger-Poisson system involving a critical nonlinearity, J. Math. Anal. Appl., 408 (2013), 55-69.  doi: 10.1016/j.jmaa.2013.05.071.

[18]

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

[19]

H. Kikuchi, On the existence of a solution for elliptic system related to the Maxwell-Schrödinger equations, Nonlinear Anal., 67 (2007), 1445-1456.  doi: 10.1016/j.na.2006.07.029.

[20]

H. Kikuchi, Existence and stability of standing waves for Schrödinger-Poisson-Slater equation, Adv. Nonlinear Stud., 7 (2007), 407-437.  doi: 10.1515/ans-2007-0305.

[21]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[22]

N. Laskin, Fractional Schrödinger equation, Phys. Rev., 66 (2002), 56-108.  doi: 10.1103/PhysRevE.66.056108.

[23]

Y. Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. 

[24]

F. Y. Li, Y. H. Li and J. P. Shi, Existence of positive solutions to Schrödinger-Poisson type systems with critical exponent, Commun. Contemp. Math., 16 (2014), Art. 1450036. doi: 10.1142/S0219199714500369.

[25]

P. L. Lions, The concentration-compactness principle in the calculus of variation. The locally compact case. Part I, Ann. Inst. Henri Poincare (C) Non Linear Anal., 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.

[26]

P. L. Lions, The concentration-compactness principle in the calculus of variation. The locally compact case. Part II, Ann. Inst. Henri Poincare (C) Non Linear Anal., 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.

[27]

Z. S. Liu and J. 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.

[28]

R. Metzler and J. Klafter, The random walks guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.

[29]

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

[30]

E. D. NezzaG. 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.

[31]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbb{R}^N$, J. Math. Phys., 54 (2013), Art. 031501. doi: 10.1063/1.4793990.

[32]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.

[33]

X. D. Shang and J. H. Zhang, Ground states for fractional Schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207.  doi: 10.1088/0951-7715/27/2/187.

[34]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.

[35]

K. M. Teng, Multiple solutions for a class of fractional Schrödinger equation in $\mathbb{R}^N$, Nonlinear Anal. Real World Appl., 21 (2015), 76-86.  doi: 10.1016/j.nonrwa.2014.06.008.

[36]

K. M. Teng, Ground state solutions for the nonlinear fractional Schrödinger-Poisson system, Appl. Anal., 98 (2019), 1959-1996.  doi: 10.1080/00036811.2018.1441998.

[37]

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

[38]

K. M. Teng, Concerntration phenomenon for fractional nonlinear Schrödinger-Poisson system with critical nonlinearity, preprint, arXiv: 1906.11563.

[39]

K. M. Teng and X. M. He, Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent, Commun. Pure Appl. Anal., 15 (2016), 991-1008.  doi: 10.3934/cpaa.2016.15.991.

[40]

J. WangL. X. TianJ. 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]

J. ZhangM. 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.

[42]

X. ZhangS. Ma and Q. Xie, Bound state solutions of Schrödinger-Poisson system with critical exponent, Discrete Contin. Dyn. Syst., 37 (2017), 605-625.  doi: 10.3934/dcds.2017025.

[43]

L. ZhaoH. Liu and F. Zhao, Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Differ. Equ., 255 (2013), 1-23.  doi: 10.1016/j.jde.2013.03.005.

[44]

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]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[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]

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

[14]

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

[15]

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

[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]

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

[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: 1.916

Metrics

  • PDF downloads (285)
  • HTML views (111)
  • Cited by (1)

Other articles
by authors

[Back to Top]