American Institute of Mathematical Sciences

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July  2019, 18(4): 1663-1693. doi: 10.3934/cpaa.2019079

Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $ \mathbb{R} ^{3} $

Lun Guo 1, , Wentao Huang 2,, and  Huifang Jia 3,

1. 

College of Science, Huazhong Agricultural University, Wuhan, 430070, China
2. 

School of Science, East China JiaoTong University, Nanchang, 330013, China
3. 

School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

* Corresponding author

Received  May 2018 Revised  October 2018 Published  January 2019

We consider the existence of positive solutions for the following fractional Schrödinger-Poisson system
$ \begin{equation*} \begin{cases} \varepsilon^{2s}(-\Delta)^{s}u+V(x)u+\phi(x)u = K(x)f(u)+|u|^{2_{s}^{*}-2}u, \ \ & x\in \mathbb{R} ^3, \\ \varepsilon^{2s}(-\Delta)^{s}\phi = u^{2}, \ \ & x \in \mathbb{R} ^3, \end{cases} \end{equation*} $
where
$ s \in (\frac{3}{4}, 1) $
,
$ \varepsilon $
 is a small and positive parameter,
$ V $
 and
$ K $
 are nonnegative potential functions.
$ 2_{s}^{*} $
 is the critical exponent with respect to fractional Sobolev embedding theorem. Under some suitable conditions on the nonlinearity
$ f $
 and potential functions
$ V $
 and
$ K $
, we prove that for
$ \varepsilon $
 small, the system has a positive ground state solution concentrating around a concrete set related to
$ V $
 and
$ K $
. This result generalizes the result for fractional Schrödinger-Poisson system with subcritical exponent by Yu et al. [39] to critical exponent. Moreover, when
$ V $
 attains its minimum and
$ K $
 attains its maximum, we also obtain multiple solutions by Ljusternik-Schnirelmann theory.
Keywords: Fractional Schrödinger-Poisson system, ground state solution, concentration behavior, multiple solutions.
Mathematics Subject Classification: Primary: 35J62; Secondary: 35J20, 35J60.
Citation: 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
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 Differential Equations, 55 (2016), 19pp. doi: 10.1007/s00526-016-0983-x.  Google Scholar

[2]

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

[3]

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.  Google Scholar

[4]

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

[5]

V. Benci and G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations, 2 (1994), 29-48.  doi: 10.1007/BF01234314.  Google Scholar

[6]

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.  Google Scholar

[7]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

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K. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar

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W. Choi, S. Kim and K.-A. Lee, Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian, J. Funct. Anal., 266 (2014), 6531-6598.  doi: 10.1016/j.jfa.2014.02.029.  Google Scholar

[10]

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.  Google Scholar

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J. D'avila, M. Del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892.  doi: 10.1016/j.jde.2013.10.006.  Google Scholar

[12]

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.  Google Scholar

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S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critial growth in the whole of $ \mathbb R ^N$, Edizioni della Normale Pisa, 15 (2017), viii+152. doi: 10.1007/978-88-7642-601-8.  Google Scholar

[14]

S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Mathematics, 68 (2013), 201-216.  doi: 10.4418/2013.68.1.15.  Google Scholar

[15]

M. Fall, F. Mahmoudi and E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961.  doi: 10.1088/0951-7715/28/6/1937.  Google Scholar

[16]

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

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X. He, Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations, Z. Angew. Math. Phys., 62 (2011), 869-889.  doi: 10.1007/s00033-011-0120-9.  Google Scholar

[18]

X. He and W. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 53 (2012), 023702, 19 pp. doi: 10.1063/1.3683156.  Google Scholar

[19]

X. He and W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differential Equations, 55 (2016), 39 pp. doi: 10.1007/s00526-016-1045-0.  Google Scholar

[20]

Y. He and G. 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

[21]

I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson system with potentials, Adv. Nonlinear Stud., 8 (2008), 573-595.  doi: 10.1515/ans-2008-0305.  Google Scholar

[22]

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

[23]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E (3), 66 (2002), 56-108. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[24]

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.  Google Scholar

[25]

E. H. Lieb and M. Loss, Analysis, 2nd edition, Graduate Studies in Mathematics, American Mathematical Society, Providence, Rhoad Island, 2001. doi: 10.1002/zamm.200490006.  Google Scholar

[26]

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.  Google Scholar

[27]

E. D. 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.  Google Scholar

[28]

G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829.  doi: 10.1007/s00526-013-0656-y.  Google Scholar

[29]

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.  Google Scholar

[30]

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

[31]

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

[32]

X. Shang and J. 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

[33]

X. Shang and J. Zhang, Existence and concentration of positive solutions for fractional nonlinear Schrödinger equation with critical growth, J. Math. Phys., 58 (2017), 081502, 18 pp. doi: 10.1063/1.4996578.  Google Scholar

[34]

L. Silvestre, Regularity of the obstable 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. 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.  Google Scholar

[36]

J. Wang, L. Tian, J. Xu and F. Zhao, Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in $ \mathbb R ^3$, Calc. Var. Partial Differential Equations, 48 (2013), 243-273.  doi: 10.1007/s00526-012-0548-6.  Google Scholar

[37]

Z. Wang and H. Zhou, Positive solution 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

[38]

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

[39]

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), 25pp. doi: 10.1007/s00526-017-1199-4.  Google Scholar

[40]

J. Zhang, The existence and concentration of positive solutions for a nonlinear Schrödinger-Poisson system with critical growth, J. Math. Phys., 55 (2014), 031507. doi: 10.1063/1.4868617.  Google Scholar

[41]

J. Zhang, Ground state and multiple solutions for Schrödinger-Poisson equations with critical nonlinearity, J. Math. Anal. Appl., 440 (2016), 466-482.  doi: 10.1016/j.jmaa.2016.03.062.  Google Scholar

[42]

J. Zhang, M. do Ó João 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

[43]

X. Zhang, S. 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

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 Differential Equations, 55 (2016), 19pp. doi: 10.1007/s00526-016-0983-x.  Google Scholar

[2]

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

[3]

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.  Google Scholar

[4]

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

[5]

V. Benci and G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations, 2 (1994), 29-48.  doi: 10.1007/BF01234314.  Google Scholar

[6]

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.  Google Scholar

[7]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[8]

K. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar

[9]

W. Choi, S. Kim and K.-A. Lee, Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian, J. Funct. Anal., 266 (2014), 6531-6598.  doi: 10.1016/j.jfa.2014.02.029.  Google Scholar

[10]

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.  Google Scholar

[11]

J. D'avila, M. Del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892.  doi: 10.1016/j.jde.2013.10.006.  Google Scholar

[12]

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.  Google Scholar

[13]

S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critial growth in the whole of $ \mathbb R ^N$, Edizioni della Normale Pisa, 15 (2017), viii+152. doi: 10.1007/978-88-7642-601-8.  Google Scholar

[14]

S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Mathematics, 68 (2013), 201-216.  doi: 10.4418/2013.68.1.15.  Google Scholar

[15]

M. Fall, F. Mahmoudi and E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961.  doi: 10.1088/0951-7715/28/6/1937.  Google Scholar

[16]

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

[17]

X. He, Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations, Z. Angew. Math. Phys., 62 (2011), 869-889.  doi: 10.1007/s00033-011-0120-9.  Google Scholar

[18]

X. He and W. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 53 (2012), 023702, 19 pp. doi: 10.1063/1.3683156.  Google Scholar

[19]

X. He and W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differential Equations, 55 (2016), 39 pp. doi: 10.1007/s00526-016-1045-0.  Google Scholar

[20]

Y. He and G. 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

[21]

I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson system with potentials, Adv. Nonlinear Stud., 8 (2008), 573-595.  doi: 10.1515/ans-2008-0305.  Google Scholar

[22]

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

[23]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E (3), 66 (2002), 56-108. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[24]

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.  Google Scholar

[25]

E. H. Lieb and M. Loss, Analysis, 2nd edition, Graduate Studies in Mathematics, American Mathematical Society, Providence, Rhoad Island, 2001. doi: 10.1002/zamm.200490006.  Google Scholar

[26]

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.  Google Scholar

[27]

E. D. 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.  Google Scholar

[28]

G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829.  doi: 10.1007/s00526-013-0656-y.  Google Scholar

[29]

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.  Google Scholar

[30]

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

[31]

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

[32]

X. Shang and J. 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

[33]

X. Shang and J. Zhang, Existence and concentration of positive solutions for fractional nonlinear Schrödinger equation with critical growth, J. Math. Phys., 58 (2017), 081502, 18 pp. doi: 10.1063/1.4996578.  Google Scholar

[34]

L. Silvestre, Regularity of the obstable 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. 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.  Google Scholar

[36]

J. Wang, L. Tian, J. Xu and F. Zhao, Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in $ \mathbb R ^3$, Calc. Var. Partial Differential Equations, 48 (2013), 243-273.  doi: 10.1007/s00526-012-0548-6.  Google Scholar

[37]

Z. Wang and H. Zhou, Positive solution 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

[38]

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

[39]

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), 25pp. doi: 10.1007/s00526-017-1199-4.  Google Scholar

[40]

J. Zhang, The existence and concentration of positive solutions for a nonlinear Schrödinger-Poisson system with critical growth, J. Math. Phys., 55 (2014), 031507. doi: 10.1063/1.4868617.  Google Scholar

[41]

J. Zhang, Ground state and multiple solutions for Schrödinger-Poisson equations with critical nonlinearity, J. Math. Anal. Appl., 440 (2016), 466-482.  doi: 10.1016/j.jmaa.2016.03.062.  Google Scholar

[42]

J. Zhang, M. do Ó João 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

[43]

X. Zhang, S. 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

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