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} $

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.
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 and 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.

[2]

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

[3]

A. AzzolliniP. 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.

[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.

[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.

[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.

[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.

[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.

[9]

W. ChoiS. 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.

[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.

[11]

J. D'avilaM. 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.

[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.

[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.

[14]

S. DipierroG. 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.

[15]

M. FallF. 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.

[16]

P. FelmerA. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[24]

G. LiS. 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.

[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.

[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.

[27]

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[36]

J. WangL. TianJ. 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.

[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.

[38]

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

[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.

[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.

[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.

[42]

J. ZhangM. 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.

[43]

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.

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.

[2]

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

[3]

A. AzzolliniP. 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.

[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.

[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.

[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.

[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.

[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.

[9]

W. ChoiS. 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.

[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.

[11]

J. D'avilaM. 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.

[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.

[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.

[14]

S. DipierroG. 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.

[15]

M. FallF. 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.

[16]

P. FelmerA. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[24]

G. LiS. 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.

[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.

[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.

[27]

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[36]

J. WangL. TianJ. 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.

[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.

[38]

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

[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.

[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.

[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.

[42]

J. ZhangM. 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.

[43]

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.

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