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April  2021, 20(4): 1737-1754. doi: 10.3934/cpaa.2021039

Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems

Daniele Cassani 1,, , Luca Vilasi 1, and  Jianjun Zhang 2,

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

* Corresponding author

Received  July 2020 Revised  January 2021 Published  April 2021 Early access  March 2021

Fund Project: The third named author was supported by NSFC(No.11871123)

We consider in
$ \mathbb R^3 $
 the singularly perturbed Schrödinger-Poisson system
$ \begin{equation*} \begin{cases} - \varepsilon^2\Delta v+V(x)v+\phi(x)v = f(v)\\ - \Delta\phi = v^2 . \end{cases} \end{equation*} $
Using variational techniques, we construct solutions which concentrate around the saddle points of the external potential
$ V $
, as
$ \varepsilon \rightarrow 0 $
.
Keywords: Nonlinear Schrödinger equations, singular perturbation, semiclassical states, variational methods.
Mathematics Subject Classification: Primary: 35J47, Secondary: 35J50, 35J61.
Citation: Daniele Cassani, Luca Vilasi, Jianjun Zhang. Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1737-1754. doi: 10.3934/cpaa.2021039
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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