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Bound state solutions of Schrödinger-Poisson system with critical exponent
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China |
$\tag{P}\label{0.1} \begin{cases}- Δ u+V(x)u+K(x)φ u=|u|^{2^*-2}u, &x∈ \mathbb{R}^3,\\-Δ φ=K(x)u^2,&x∈ \mathbb{R}^3,\end{cases}$ |
$2^*=6 $ |
$\mathbb R^3$ |
$ K∈ L^{\frac{1}{2}}(\mathbb{R}^3)$ |
$V∈ L^{\frac{3}{2}}(\mathbb{R}^3)$ |
$|V|_{\frac{3}{2}}+|K|_{\frac{1}{2}}$ |
References:
[1] |
A. Ambrosetti,
On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.
doi: 10.1007/s00032-008-0094-z. |
[2] |
A. Ambrosetti and D. Ruiz,
Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.
doi: 10.1142/S021919970800282X. |
[3] |
A. Azzollini and A. Pomponio,
Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anual. Appl., 345 (2008), 90-108.
doi: 10.1016/j.jmaa.2008.03.057. |
[4] |
V. Benci and D. Fortunato,
Some compact embedding theorems for weighted Sobolev spaces, Boll. Un. Mat. Ital. B (5), 13 (1976), 832-843.
|
[5] |
V. Benci and C. Cerami,
Existence of positive solutions of the equation $-Δ u+a(x)u=u^{(N+2)/(N-2)}$ in $\mathbb R^3$, J. Funct. Anal., 88 (1990), 90-117.
doi: 10.1016/0022-1236(90)90120-A. |
[6] |
V. Benci and D. Fortunato,
An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.
|
[7] |
V. Benci and D. Fortunato,
Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.
doi: 10.1142/S0129055X02001168. |
[8] | |
[9] |
G. M. Coclite,
A multiplicity result for the Schrödinger-Maxwell equations with negative potential, Ann. Polon. Math., 79 (2002), 21-30.
|
[10] |
T. D'Aprile and D. Mugnai,
Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.
|
[11] |
T. D'Aprile and J. C. Wei,
On bound states concentrating on spheres for the Maxwell-Schrödinger equations, SIAM J. Math. Anal., 37 (2005), 321-342.
doi: 10.1137/S0036141004442793. |
[12] |
X. M. He and W. M. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth J. Math. Phys. 53 (2012), 023702, 19pp.
doi: 10.1063/1.3683156. |
[13] |
I. Ianni and G. Vaira,
Solutions of the Schrödinger-Poisson problem concentrating on spheres, Part I: Necessary condition, Math. Models Methods Appl. Sci., 19 (2009), 707-720.
doi: 10.1142/S0218202509003589. |
[14] |
I. Ianni,
Solutions of the Schrödinger-Poisson problem concentrating on spheres, Part II: Existence, Math. Models Methods Appl. Sci., 19 (2009), 877-910.
doi: 10.1142/S0218202509003656. |
[15] |
Y. S. Jiang and H. S. Zhou,
Schrödinger-Poisson system with steep potential well, J. Differential. Equations., 251 (2011), 582-608.
doi: 10.1016/j.jde.2011.05.006. |
[16] |
G. B. Li, S. J. Peng and C. H. Wang, Multi-bump solutions for the nonlinear Schrödinger-Poisson system J. Maht. phys. 52 (2011), 053505, 19pp.
doi: 10.1063/1.3585657. |
[17] |
P. L. Lions,
The concentration-compactness method in the calculus of variations. The locally compact case, parts 1 and 2, Ann. Inst. H. Poincaré Anual. Non Linéair, 1 (1984), 109-145,223-283.
|
[18] |
Z. S. Liu and S. J. Guo,
On ground state solutions for the Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 412 (2014), 435-448.
doi: 10.1016/j.jmaa.2013.10.066. |
[19] |
D. Ruiz,
Semiclassical states for coupled Schrödinger-Maxwell equations: Concentration around a sphere, Math. Models Methods Appl. Sci., 15 (2005), 141-164.
doi: 10.1142/S0218202505003939. |
[20] |
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. |
[21] |
A. Salvatore,
Multiple solitary waves for a non-homogeneous Schrödinger-Maxwell system in $\mathbb{R}^3$, Adv. Nonlinear strud., 6 (2006), 157-169.
doi: 10.1515/ans-2006-0203. |
[22] |
M. Struwe,
A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.
doi: 10.1007/BF01174186. |
[23] |
J. Wang, J. X. Xu, F. B. Zhang and X. M. Chen,
Existence of multi-bump solutions for a semilinear Schrödinger-Poisson system, Nonlinearity, 26 (2013), 1377-1399.
doi: 10.1088/0951-7715/26/5/1377. |
[24] | |
[25] |
M. Willem,
Minimax Theorems Birkhäuser, Boston, Basel, Berlin, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[26] |
Z. P. Wang and H. S. Zhou,
Positive solutions 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. |
[27] |
J. Zhang,
On the Schrödinger-Poisson equations with a general nonlinearity in the critical growth, Nonlinea Anal., 75 (2012), 6391-6401.
doi: 10.1016/j.na.2012.07.008. |
[28] |
J. Zhang,
On ground state and nodal solutions of Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 428 (2015), 387-404.
doi: 10.1016/j.jmaa.2015.03.032. |
[29] |
L. G. Zhao, H. Liu and F. K. Zhao,
Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Differential. Equations, 255 (2013), 1-23.
doi: 10.1016/j.jde.2013.03.005. |
[30] |
L. G. Zhao and F. K. Zhao,
Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164.
doi: 10.1016/j.na.2008.02.116. |
[31] |
L. G. Zhao and F. K. Zhao,
On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anual. Appl., 346 (2008), 155-169.
doi: 10.1016/j.jmaa.2008.04.053. |
show all references
References:
[1] |
A. Ambrosetti,
On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.
doi: 10.1007/s00032-008-0094-z. |
[2] |
A. Ambrosetti and D. Ruiz,
Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.
doi: 10.1142/S021919970800282X. |
[3] |
A. Azzollini and A. Pomponio,
Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anual. Appl., 345 (2008), 90-108.
doi: 10.1016/j.jmaa.2008.03.057. |
[4] |
V. Benci and D. Fortunato,
Some compact embedding theorems for weighted Sobolev spaces, Boll. Un. Mat. Ital. B (5), 13 (1976), 832-843.
|
[5] |
V. Benci and C. Cerami,
Existence of positive solutions of the equation $-Δ u+a(x)u=u^{(N+2)/(N-2)}$ in $\mathbb R^3$, J. Funct. Anal., 88 (1990), 90-117.
doi: 10.1016/0022-1236(90)90120-A. |
[6] |
V. Benci and D. Fortunato,
An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.
|
[7] |
V. Benci and D. Fortunato,
Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.
doi: 10.1142/S0129055X02001168. |
[8] | |
[9] |
G. M. Coclite,
A multiplicity result for the Schrödinger-Maxwell equations with negative potential, Ann. Polon. Math., 79 (2002), 21-30.
|
[10] |
T. D'Aprile and D. Mugnai,
Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.
|
[11] |
T. D'Aprile and J. C. Wei,
On bound states concentrating on spheres for the Maxwell-Schrödinger equations, SIAM J. Math. Anal., 37 (2005), 321-342.
doi: 10.1137/S0036141004442793. |
[12] |
X. M. He and W. M. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth J. Math. Phys. 53 (2012), 023702, 19pp.
doi: 10.1063/1.3683156. |
[13] |
I. Ianni and G. Vaira,
Solutions of the Schrödinger-Poisson problem concentrating on spheres, Part I: Necessary condition, Math. Models Methods Appl. Sci., 19 (2009), 707-720.
doi: 10.1142/S0218202509003589. |
[14] |
I. Ianni,
Solutions of the Schrödinger-Poisson problem concentrating on spheres, Part II: Existence, Math. Models Methods Appl. Sci., 19 (2009), 877-910.
doi: 10.1142/S0218202509003656. |
[15] |
Y. S. Jiang and H. S. Zhou,
Schrödinger-Poisson system with steep potential well, J. Differential. Equations., 251 (2011), 582-608.
doi: 10.1016/j.jde.2011.05.006. |
[16] |
G. B. Li, S. J. Peng and C. H. Wang, Multi-bump solutions for the nonlinear Schrödinger-Poisson system J. Maht. phys. 52 (2011), 053505, 19pp.
doi: 10.1063/1.3585657. |
[17] |
P. L. Lions,
The concentration-compactness method in the calculus of variations. The locally compact case, parts 1 and 2, Ann. Inst. H. Poincaré Anual. Non Linéair, 1 (1984), 109-145,223-283.
|
[18] |
Z. S. Liu and S. J. Guo,
On ground state solutions for the Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 412 (2014), 435-448.
doi: 10.1016/j.jmaa.2013.10.066. |
[19] |
D. Ruiz,
Semiclassical states for coupled Schrödinger-Maxwell equations: Concentration around a sphere, Math. Models Methods Appl. Sci., 15 (2005), 141-164.
doi: 10.1142/S0218202505003939. |
[20] |
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. |
[21] |
A. Salvatore,
Multiple solitary waves for a non-homogeneous Schrödinger-Maxwell system in $\mathbb{R}^3$, Adv. Nonlinear strud., 6 (2006), 157-169.
doi: 10.1515/ans-2006-0203. |
[22] |
M. Struwe,
A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.
doi: 10.1007/BF01174186. |
[23] |
J. Wang, J. X. Xu, F. B. Zhang and X. M. Chen,
Existence of multi-bump solutions for a semilinear Schrödinger-Poisson system, Nonlinearity, 26 (2013), 1377-1399.
doi: 10.1088/0951-7715/26/5/1377. |
[24] | |
[25] |
M. Willem,
Minimax Theorems Birkhäuser, Boston, Basel, Berlin, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[26] |
Z. P. Wang and H. S. Zhou,
Positive solutions 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. |
[27] |
J. Zhang,
On the Schrödinger-Poisson equations with a general nonlinearity in the critical growth, Nonlinea Anal., 75 (2012), 6391-6401.
doi: 10.1016/j.na.2012.07.008. |
[28] |
J. Zhang,
On ground state and nodal solutions of Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 428 (2015), 387-404.
doi: 10.1016/j.jmaa.2015.03.032. |
[29] |
L. G. Zhao, H. Liu and F. K. Zhao,
Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Differential. Equations, 255 (2013), 1-23.
doi: 10.1016/j.jde.2013.03.005. |
[30] |
L. G. Zhao and F. K. Zhao,
Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164.
doi: 10.1016/j.na.2008.02.116. |
[31] |
L. G. Zhao and F. K. Zhao,
On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anual. Appl., 346 (2008), 155-169.
doi: 10.1016/j.jmaa.2008.04.053. |
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