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Bound states for fractional Schrödinger-Poisson system with critical exponent
School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China |
$ \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}(-\Delta )^su+V(x)u+K(x)\phi u = |u|^{2_{s}^{*}-2}u, & \text{in}\ {\Bbb R}^3,\\ (-\Delta)^{t}\phi = K(x)u^2, & \text{in}\ {\Bbb R}^3, \end{array} \right. \end{equation*} $ |
$ s\in (\frac{3}{4}, 1) $ |
$ t\in(0, 1) $ |
$ \varepsilon $ |
$ 2_{s}^{*} = \frac{6}{3-2s} $ |
$ K(x)\in L^{\frac{6}{2t+4s-3}}({\Bbb R}^3) $ |
$ V(x)\in L^{\frac{3}{2s}}({\Bbb R}^3) $ |
$ V(x) $ |
$ {\Bbb R}^3 $ |
References:
[1] |
A. Ambrosetti and D. Ruiz,
Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.
doi: 10.1142/S021919970800282X. |
[2] |
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. |
[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. |
[4] |
V. Benci and G. Cerami,
Existence of positive solutions of the equation $-\Delta u+a(x)u = u^{(N+2)/(N-2)}$ in ${\Bbb R}^N$, J. Funct. Anal., 88 (1990), 90-117.
doi: 10.1016/0022-1236(90)90120-A. |
[5] |
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. |
[6] |
G. Cerami and R. Molle,
Positive bound state solutions for some Schrödinger-Poisson systems, Nonlinearity, 29 (2016), 3103-3119.
doi: 10.1088/0951-7715/29/10/3103. |
[7] |
J. Chabrowski and J. Yang,
Multiple semiclassical solutions of the Schrödinger equation involving a critical Sobolev exponent, Portugal. Math., 57 (2000), 273-284.
|
[8] |
W. Chen, C. Li and B. Ou,
Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[9] |
T. D'Aprile and D. Mugnai,
Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.
doi: 10.1515/ans-2004-0305. |
[10] |
T. D'Aprile and D. Mugnai,
Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906.
doi: 10.1017/S030821050000353X. |
[11] |
T. D'Aprile and J. Wei,
On bound states concentrating on spheres for the Maxwell-Schrödinger equation, SIAM J. Math. Anal., 37 (2005), 321-342.
doi: 10.1137/S0036141004442793. |
[12] |
E. Di 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. |
[13] |
R. L. Frank and E. H. Lieb,
Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Calc. Var. Partial Differential Equations, 39 (2010), 85-99.
doi: 10.1007/s00526-009-0302-x. |
[14] |
R. L. Frank and E. H. Lieb,
A new, rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality, Oper. Theory Adv. Appl., 219 (2012), 55-67.
doi: 10.1007/978-3-0348-0263-5_4. |
[15] |
L. Guo and Q. Li, Multiple bound state solutions for fractional Choquard equation with Hardy-Littlewood-Sobolev critical exponent, J. Math. Phys., 61 (2020), 121501, 20 pp.
doi: 10.1063/5.0013475. |
[16] |
I. Ianni and G. Vaira,
On concentration of positive bound states for the Schrödinger-Poisson problem with potentials, Adv. Nonlinear Stud., 8 (2008), 573-595.
doi: 10.1515/ans-2008-0305. |
[17] |
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. |
[18] |
E. H. Lieb,
Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118 (1983), 349-374.
doi: 10.2307/2007032. |
[19] |
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. |
[20] |
E. G. Murcia and G. Siciliano,
Positive semiclassical states for a fractional Schrödinger-Poisson system, Differential Integral Equations, 30 (2017), 231-258.
|
[21] |
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. |
[22] |
D. Ruiz and G. Vaira,
Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential, Rev. Mat. Iberoam., 27 (2011), 253-271.
doi: 10.4171/RMI/635. |
[23] |
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. |
[24] |
K. Teng and R. P. Agarwal,
Existence and concentration of positive ground state solutions for nonlinear fractional Schrödinger-Poisson system with critical growth, Math. Methods Appl. Sci., 41 (2018), 8258-8293.
doi: 10.1002/mma.5289. |
[25] |
M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[26] |
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), Paper No. 116, 25 pp.
doi: 10.1007/s00526-017-1199-4. |
[27] |
J. Zhang, J. M. 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. |
[28] |
H. Zhang, J. Xu and F. Zhang, Multiplicity of semiclassical states for Schrödinger-Poisson systems with critical frequency, Z. Angew. Math. Phys., 71 (2020), Paper No. 5, 15 pp.
doi: 10.1007/s00033-019-1226-8. |
[29] |
H. Zhang and F. Zhang, Multiplicity of semiclassical states for fractional Schrödinger equations with critical frequency, Nonlinear Anal., 190 (2020), 111599, 15pp.
doi: 10.1016/j.na.2019.111599. |
[30] |
L. Zhao and F. Zhao,
On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl., 346 (2008), 155-169.
doi: 10.1016/j.jmaa.2008.04.053. |
show all references
References:
[1] |
A. Ambrosetti and D. Ruiz,
Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.
doi: 10.1142/S021919970800282X. |
[2] |
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. |
[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. |
[4] |
V. Benci and G. Cerami,
Existence of positive solutions of the equation $-\Delta u+a(x)u = u^{(N+2)/(N-2)}$ in ${\Bbb R}^N$, J. Funct. Anal., 88 (1990), 90-117.
doi: 10.1016/0022-1236(90)90120-A. |
[5] |
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. |
[6] |
G. Cerami and R. Molle,
Positive bound state solutions for some Schrödinger-Poisson systems, Nonlinearity, 29 (2016), 3103-3119.
doi: 10.1088/0951-7715/29/10/3103. |
[7] |
J. Chabrowski and J. Yang,
Multiple semiclassical solutions of the Schrödinger equation involving a critical Sobolev exponent, Portugal. Math., 57 (2000), 273-284.
|
[8] |
W. Chen, C. Li and B. Ou,
Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[9] |
T. D'Aprile and D. Mugnai,
Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.
doi: 10.1515/ans-2004-0305. |
[10] |
T. D'Aprile and D. Mugnai,
Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906.
doi: 10.1017/S030821050000353X. |
[11] |
T. D'Aprile and J. Wei,
On bound states concentrating on spheres for the Maxwell-Schrödinger equation, SIAM J. Math. Anal., 37 (2005), 321-342.
doi: 10.1137/S0036141004442793. |
[12] |
E. Di 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. |
[13] |
R. L. Frank and E. H. Lieb,
Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Calc. Var. Partial Differential Equations, 39 (2010), 85-99.
doi: 10.1007/s00526-009-0302-x. |
[14] |
R. L. Frank and E. H. Lieb,
A new, rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality, Oper. Theory Adv. Appl., 219 (2012), 55-67.
doi: 10.1007/978-3-0348-0263-5_4. |
[15] |
L. Guo and Q. Li, Multiple bound state solutions for fractional Choquard equation with Hardy-Littlewood-Sobolev critical exponent, J. Math. Phys., 61 (2020), 121501, 20 pp.
doi: 10.1063/5.0013475. |
[16] |
I. Ianni and G. Vaira,
On concentration of positive bound states for the Schrödinger-Poisson problem with potentials, Adv. Nonlinear Stud., 8 (2008), 573-595.
doi: 10.1515/ans-2008-0305. |
[17] |
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. |
[18] |
E. H. Lieb,
Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118 (1983), 349-374.
doi: 10.2307/2007032. |
[19] |
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. |
[20] |
E. G. Murcia and G. Siciliano,
Positive semiclassical states for a fractional Schrödinger-Poisson system, Differential Integral Equations, 30 (2017), 231-258.
|
[21] |
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. |
[22] |
D. Ruiz and G. Vaira,
Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential, Rev. Mat. Iberoam., 27 (2011), 253-271.
doi: 10.4171/RMI/635. |
[23] |
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. |
[24] |
K. Teng and R. P. Agarwal,
Existence and concentration of positive ground state solutions for nonlinear fractional Schrödinger-Poisson system with critical growth, Math. Methods Appl. Sci., 41 (2018), 8258-8293.
doi: 10.1002/mma.5289. |
[25] |
M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[26] |
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), Paper No. 116, 25 pp.
doi: 10.1007/s00526-017-1199-4. |
[27] |
J. Zhang, J. M. 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. |
[28] |
H. Zhang, J. Xu and F. Zhang, Multiplicity of semiclassical states for Schrödinger-Poisson systems with critical frequency, Z. Angew. Math. Phys., 71 (2020), Paper No. 5, 15 pp.
doi: 10.1007/s00033-019-1226-8. |
[29] |
H. Zhang and F. Zhang, Multiplicity of semiclassical states for fractional Schrödinger equations with critical frequency, Nonlinear Anal., 190 (2020), 111599, 15pp.
doi: 10.1016/j.na.2019.111599. |
[30] |
L. Zhao and F. Zhao,
On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl., 346 (2008), 155-169.
doi: 10.1016/j.jmaa.2008.04.053. |
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