# American Institute of Mathematical Sciences

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 & 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 [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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##### 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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