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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 |
$ \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*} $ |
$ s \in (\frac{3}{4}, 1) $ |
$ \varepsilon $ |
$ V $ |
$ K $ |
$ 2_{s}^{*} $ |
$ f $ |
$ V $ |
$ K $ |
$ \varepsilon $ |
$ V $ |
$ K $ |
$ V $ |
$ K $ |
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. 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. |
[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. 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. |
[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'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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
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. 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. |
[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. 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. |
[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'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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
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