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 & Applied Analysis, 2020, 19 (7) : 3735-3768. doi: 10.3934/cpaa.2020165
References:
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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.  Google Scholar

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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.  Google Scholar

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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

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[9]

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

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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