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Extremal functions for a class of trace Trudinger-Moser inequalities on a compact Riemann surface with smooth boundary
Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems
1. | Dip. di Scienza e Alta Tecnologia, Università degli Studi dell'Insubria, and RISM – Riemann International School of Mathematics, Villa Toeplitz, Via G.B. Vico, 46 – 21100 Varese, Italy |
2. | College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, China |
$ \mathbb R^3 $ |
$ \begin{equation*} \begin{cases} - \varepsilon^2\Delta v+V(x)v+\phi(x)v = f(v)\\ - \Delta\phi = v^2 . \end{cases} \end{equation*} $ |
$ V $ |
$ \varepsilon \rightarrow 0 $ |
References:
[1] |
P. Bechouche, N. J. Mauser and F. Poupaud,
Semiclassical limit for the Schrödinger-Poisson equation in a crystal, Commun. Pure Appl. Math., 54 (2001), 851-890.
doi: 10.1002/cpa.3004. |
[2] |
D. Bonheure, J. Di Cosmo and C. Mercuri, Concentration on circles for nonlinear Schrödinger-Poisson systems with unbounded potentials vanishing at infinity, Commun. Contemp. Math., 14 (2012), 31 pp.
doi: 10.1142/S0219199712500095. |
[3] |
H. Brezis and T. Kato,
Remarks on the Schrödinger operator with singularly complex potentials, J. Math. Pures Appl., 58 (1979), 137-151.
|
[4] |
J. Byeon, J. J. Zhang and W. M. Zou,
Singularly perturbed nonlinear Dirichlet problems involving critical growth, Calc. Var. Partial Differ. Equ., 47 (2013), 65-85.
doi: 10.1007/s00526-012-0511-6. |
[5] |
V. Benci and D. Fortunato, Variational Methods in Nonlinear Field Equations, Springer 2014. |
[6] |
S. Chen, A. Fiscella, P. Pucci and X. Tang,
Semiclassical ground state solutions for critical Schrödinger-Poisson systems with lower perturbations, J. Differ. Equ., 268 (2020), 2672-2716.
doi: 10.1016/j.jde.2019.09.041. |
[7] |
P. L. Cunha, Existence and concentration of positive bound states for Schrödinger-Poisson systems with potential functions, Electron. J. Differ. Equ., (2015), 15 pp. |
[8] |
P. d'Avenia, A. Pomponio and D. Ruiz,
Semiclassical states for the nonlinear Schrödinger equation on saddle points of the potential via variational methods, J. Funct. Anal., 262 (2012), 4600-4633.
doi: 10.1016/j.jfa.2012.03.009. |
[9] |
M. J. Esteban and P. L. Lions,
Existence and nonexistence results for semilinear elliptic problems in unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A, 93 (1982/1983), 1-14.
doi: 10.1017/S0308210500031607. |
[10] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren 224, Springer, Berlin Heidelberg, New York and Tokyo, 1983.
doi: 10.1007/978-3-642-61798-0. |
[11] |
I. Ianni,
Solutions of the Schrödinger-Poisson problem concentrating on spheres. II. Existence, Math. Models Methods Appl. Sci., 19 (2009), 877-910.
doi: 10.1142/S0218202509003656. |
[12] |
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. |
[13] |
L. Jeanjean and K. Tanaka,
A remark on least energy solutions in $ \mathbb R^N$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408.
doi: 10.1090/S0002-9939-02-06821-1. |
[14] |
X. Liu and J. Zhao,
$p$-Laplacian Equations in $ \mathbb R^N$ with finite potential via the truncation method, Adv. Nonlinear Stud., 17 (2017), 595-610.
doi: 10.1515/ans-2015-5059. |
[15] |
A. Mao, L. Yang, A. Qian and S. Luan,
Existence and concentration of solutions of Schr${\rm{\ddot d}}$inger-Poisson system, Appl. Math. Lett., 68 (2017), 8-12.
doi: 10.1016/j.aml.2016.12.014. |
[16] |
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. |
[17] |
M. A. S. Souto,
On the location of the peaks of least-energy solutions to semilinear Dirichlet problems with critical growth, Abstr. Appl. Anal., 7 (2002), 547-561.
doi: 10.1155/S1085337502206028. |
[18] |
K. Teng and P. R. 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. |
[19] |
J. Wang, L. Tian, J. Xu and F. Zhang,
Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in $ \mathbb R^3$, Calc. Var. Partial Differ. Equ., 48 (2013), 243-273.
doi: 10.1007/s00526-012-0548-6. |
[20] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[21] |
M. Yang, Z. Shen and Y. Ding,
Multiple semiclassical solutions for the nonlinear Maxwell-Schrödinger system, Nonlinear Anal., 71 (2009), 730-739.
doi: 10.1016/j.na.2008.10.105. |
[22] |
Y. Yu, F. Zhao and L. Zhao, The concentration behavior of ground state solutions for a fractional Schrödinger-Poisson system, Calc. Var. Partial Differ. Equ., 56 (2017), 25 pp.
doi: 10.1007/s00526-017-1199-4. |
[23] |
H. Zhang, J. Xu and F. Zhang, Multiplicity of semiclassical states for Schrödinger-Poisson systems with critical frequency, Z. Angew. Math. Phys., 71 (2020), 15 pp.
doi: 10.1007/s00033-019-1226-8. |
[24] |
J. Zhang, The existence and concentration of positive solutions for a nonlinear Schrödinger-Poisson system with critical growth, J. Math. Phys., 55 (2014), 14 pp.
doi: 10.1063/1.4868617. |
[25] |
H. Zhu, Semi-classical states for Schrödinger-Poisson systems on $ \mathbb R^3$, Electron. J. Differ. Equ., (2016), 15 pp. |
show all references
References:
[1] |
P. Bechouche, N. J. Mauser and F. Poupaud,
Semiclassical limit for the Schrödinger-Poisson equation in a crystal, Commun. Pure Appl. Math., 54 (2001), 851-890.
doi: 10.1002/cpa.3004. |
[2] |
D. Bonheure, J. Di Cosmo and C. Mercuri, Concentration on circles for nonlinear Schrödinger-Poisson systems with unbounded potentials vanishing at infinity, Commun. Contemp. Math., 14 (2012), 31 pp.
doi: 10.1142/S0219199712500095. |
[3] |
H. Brezis and T. Kato,
Remarks on the Schrödinger operator with singularly complex potentials, J. Math. Pures Appl., 58 (1979), 137-151.
|
[4] |
J. Byeon, J. J. Zhang and W. M. Zou,
Singularly perturbed nonlinear Dirichlet problems involving critical growth, Calc. Var. Partial Differ. Equ., 47 (2013), 65-85.
doi: 10.1007/s00526-012-0511-6. |
[5] |
V. Benci and D. Fortunato, Variational Methods in Nonlinear Field Equations, Springer 2014. |
[6] |
S. Chen, A. Fiscella, P. Pucci and X. Tang,
Semiclassical ground state solutions for critical Schrödinger-Poisson systems with lower perturbations, J. Differ. Equ., 268 (2020), 2672-2716.
doi: 10.1016/j.jde.2019.09.041. |
[7] |
P. L. Cunha, Existence and concentration of positive bound states for Schrödinger-Poisson systems with potential functions, Electron. J. Differ. Equ., (2015), 15 pp. |
[8] |
P. d'Avenia, A. Pomponio and D. Ruiz,
Semiclassical states for the nonlinear Schrödinger equation on saddle points of the potential via variational methods, J. Funct. Anal., 262 (2012), 4600-4633.
doi: 10.1016/j.jfa.2012.03.009. |
[9] |
M. J. Esteban and P. L. Lions,
Existence and nonexistence results for semilinear elliptic problems in unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A, 93 (1982/1983), 1-14.
doi: 10.1017/S0308210500031607. |
[10] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren 224, Springer, Berlin Heidelberg, New York and Tokyo, 1983.
doi: 10.1007/978-3-642-61798-0. |
[11] |
I. Ianni,
Solutions of the Schrödinger-Poisson problem concentrating on spheres. II. Existence, Math. Models Methods Appl. Sci., 19 (2009), 877-910.
doi: 10.1142/S0218202509003656. |
[12] |
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. |
[13] |
L. Jeanjean and K. Tanaka,
A remark on least energy solutions in $ \mathbb R^N$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408.
doi: 10.1090/S0002-9939-02-06821-1. |
[14] |
X. Liu and J. Zhao,
$p$-Laplacian Equations in $ \mathbb R^N$ with finite potential via the truncation method, Adv. Nonlinear Stud., 17 (2017), 595-610.
doi: 10.1515/ans-2015-5059. |
[15] |
A. Mao, L. Yang, A. Qian and S. Luan,
Existence and concentration of solutions of Schr${\rm{\ddot d}}$inger-Poisson system, Appl. Math. Lett., 68 (2017), 8-12.
doi: 10.1016/j.aml.2016.12.014. |
[16] |
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. |
[17] |
M. A. S. Souto,
On the location of the peaks of least-energy solutions to semilinear Dirichlet problems with critical growth, Abstr. Appl. Anal., 7 (2002), 547-561.
doi: 10.1155/S1085337502206028. |
[18] |
K. Teng and P. R. 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. |
[19] |
J. Wang, L. Tian, J. Xu and F. Zhang,
Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in $ \mathbb R^3$, Calc. Var. Partial Differ. Equ., 48 (2013), 243-273.
doi: 10.1007/s00526-012-0548-6. |
[20] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[21] |
M. Yang, Z. Shen and Y. Ding,
Multiple semiclassical solutions for the nonlinear Maxwell-Schrödinger system, Nonlinear Anal., 71 (2009), 730-739.
doi: 10.1016/j.na.2008.10.105. |
[22] |
Y. Yu, F. Zhao and L. Zhao, The concentration behavior of ground state solutions for a fractional Schrödinger-Poisson system, Calc. Var. Partial Differ. Equ., 56 (2017), 25 pp.
doi: 10.1007/s00526-017-1199-4. |
[23] |
H. Zhang, J. Xu and F. Zhang, Multiplicity of semiclassical states for Schrödinger-Poisson systems with critical frequency, Z. Angew. Math. Phys., 71 (2020), 15 pp.
doi: 10.1007/s00033-019-1226-8. |
[24] |
J. Zhang, The existence and concentration of positive solutions for a nonlinear Schrödinger-Poisson system with critical growth, J. Math. Phys., 55 (2014), 14 pp.
doi: 10.1063/1.4868617. |
[25] |
H. Zhu, Semi-classical states for Schrödinger-Poisson systems on $ \mathbb R^3$, Electron. J. Differ. Equ., (2016), 15 pp. |
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