May  2021, 14(5): 1631-1648. doi: 10.3934/dcdss.2020447

Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $

1. 

Département de Mathématiques, Université Libre de Bruxelles, CP 214, Boulevard du Triomphe, B-1050 Bruxelles, Belgium

2. 

Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Via Orabona 4, 70125 Bari, Italy

3. 

Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Via Roberto Cozzi 55, 20125 Milano, Italy

* Corresponding author: Silvia Cingolani

Received  April 2020 Revised  July 2020 Published  November 2020

We perform a semiclassical analysis for the planar Schrödinger-Poisson system
$ \begin{gather} \begin{cases} -\varepsilon^{2} \Delta\psi+V(x)\psi = E(x) \psi \quad \text{in $ \mathbb{R}^2$}, \\ -\Delta E = |\psi|^{2} \quad \text{in $ \mathbb{R}^2$}, \end{cases} \end{gather}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (S{P_\varepsilon }) $
where
$ \varepsilon $
is a positive parameter corresponding to the Planck constant and
$ V $
is a bounded external potential. We detect solution pairs
$ (u_\varepsilon, E_\varepsilon) $
of the system
$ (SP_\varepsilon) $
as
$ \ge \rightarrow 0 $
, leaning on a nongeneracy result in [3].
Citation: Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447
References:
[1]

A. AmbrosettiM. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285-300.  doi: 10.1007/s002050050067.  Google Scholar

[2]

A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbb{R}^n$, Birkhäuser Verlag, 2006.  Google Scholar

[3]

D. BonheureS. Cingolani and J. Van Schaftingen, The logarithmic Choquard equation: Sharp asymptotics and nondegeneracy of the groundstate, J. Funct. Anal., 272 (2017), 5255-5281.  doi: 10.1016/j.jfa.2017.02.026.  Google Scholar

[4]

P. Choquard and J. Stubbe, The one-dimensional Schrödinger-Newton equations, Lett. Math. Phys., 81 (2007), 177-184.  doi: 10.1007/s11005-007-0174-y.  Google Scholar

[5]

P. ChoquardJ. Stubbe and M. Vuffray, Stationary solutions of the Schrödinger-Newton model — an ODE approach, Differ. Integral Equ., 21 (2008), 665-679.   Google Scholar

[6]

S. CingolaniM. Clapp and S. Secchi, Intertwining semiclassical solutions to a Schrödinger-Newton system, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 891-908.  doi: 10.3934/dcdss.2013.6.891.  Google Scholar

[7]

S. Cingolani and L. Jeanjean, Stationary waves with prescribed $L^2$-norm for the Schrödinger-Poisson system, SIAM J. Math. Anal., 51 (2019), 3533-3568.  doi: 10.1137/19M1243907.  Google Scholar

[8]

S. CingolaniS. Secchi and M. Squassina, Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 973-1009.  doi: 10.1017/S0308210509000584.  Google Scholar

[9]

S. Cingolani and K. Tanaka, Semi-classical states for the nonlinear Choquard equations: Existence, multiplicity and concentration at a potential well, Rev. Mat. Iberoam., 35 (2019), 1885-1924.  doi: 10.4171/rmi/1105.  Google Scholar

[10]

S. Cingolani and T. Weth, On the planar Schrödinger-Poisson systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 169-197.  doi: 10.1016/j.anihpc.2014.09.008.  Google Scholar

[11]

M. Du and T. Weth, Ground states and high energy solutions of the planar Schrödinger-Poisson system, Nonlinearity, 30 (2017), 3492-3515.  doi: 10.1088/1361-6544/aa7eac.  Google Scholar

[12]

R. HarrisonI. Moroz and K. P. Tod, A numerical study of the Schrödinger–Newton equation, Nonlinearity, 16 (2003), 101-122.  doi: 10.1088/0951-7715/16/1/307.  Google Scholar

[13]

E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27.  doi: 10.2140/apde.2009.2.1.  Google Scholar

[14]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1977), 93-105.  doi: 10.1002/sapm197757293.  Google Scholar

[15]

P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4.  Google Scholar

[16]

S. Masaki, Local existence and WKB approximation of solutions to Schrödinger-Poisson system in the two-dimensional whole space, Comm. Partial Differential Equations, 35 (2010), 2253-2278.  doi: 10.1080/03605301003717142.  Google Scholar

[17]

V. Moroz and J. Van Schaftingen, Semi-classical states for the Choquard equations, Calc. Var. Partial Differential Equations, 52 (2015), 199-235.  doi: 10.1007/s00526-014-0709-x.  Google Scholar

[18]

I. M. MorozR. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.  doi: 10.1088/0264-9381/15/9/019.  Google Scholar

[19]

R. Penrose, On gravity's role in quantum state reduction, Gen. Rel. Grav., 28 (1996), 581-600.  doi: 10.1007/BF02105068.  Google Scholar

[20]

R. Penrose, Quantum computation, entanglement and state reduction, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927-1939.  doi: 10.1098/rsta.1998.0256.  Google Scholar

[21]

R. Penrose, The Road to Reality. A Complete Guide to the Laws of the Universe, Alfred A. Knopf Inc., New York, 2005.  Google Scholar

[22]

J. Stubbe, Bound states of two-dimensional Schrödinger-Newton equations, preprint, arXiv: 0807.4059v1, 2008. Google Scholar

[23]

P. Tod and I. M. Moroz, An analytical approach to the Schrödinger-Newton equations, Nonlinearity, 12 (1999), 201-216.  doi: 10.1088/0951-7715/12/2/002.  Google Scholar

[24]

J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger–Newton equation, J. Math. Phys., 50 (2009), 012905, 22 pp. doi: 10.1063/1.3060169.  Google Scholar

show all references

References:
[1]

A. AmbrosettiM. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285-300.  doi: 10.1007/s002050050067.  Google Scholar

[2]

A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbb{R}^n$, Birkhäuser Verlag, 2006.  Google Scholar

[3]

D. BonheureS. Cingolani and J. Van Schaftingen, The logarithmic Choquard equation: Sharp asymptotics and nondegeneracy of the groundstate, J. Funct. Anal., 272 (2017), 5255-5281.  doi: 10.1016/j.jfa.2017.02.026.  Google Scholar

[4]

P. Choquard and J. Stubbe, The one-dimensional Schrödinger-Newton equations, Lett. Math. Phys., 81 (2007), 177-184.  doi: 10.1007/s11005-007-0174-y.  Google Scholar

[5]

P. ChoquardJ. Stubbe and M. Vuffray, Stationary solutions of the Schrödinger-Newton model — an ODE approach, Differ. Integral Equ., 21 (2008), 665-679.   Google Scholar

[6]

S. CingolaniM. Clapp and S. Secchi, Intertwining semiclassical solutions to a Schrödinger-Newton system, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 891-908.  doi: 10.3934/dcdss.2013.6.891.  Google Scholar

[7]

S. Cingolani and L. Jeanjean, Stationary waves with prescribed $L^2$-norm for the Schrödinger-Poisson system, SIAM J. Math. Anal., 51 (2019), 3533-3568.  doi: 10.1137/19M1243907.  Google Scholar

[8]

S. CingolaniS. Secchi and M. Squassina, Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 973-1009.  doi: 10.1017/S0308210509000584.  Google Scholar

[9]

S. Cingolani and K. Tanaka, Semi-classical states for the nonlinear Choquard equations: Existence, multiplicity and concentration at a potential well, Rev. Mat. Iberoam., 35 (2019), 1885-1924.  doi: 10.4171/rmi/1105.  Google Scholar

[10]

S. Cingolani and T. Weth, On the planar Schrödinger-Poisson systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 169-197.  doi: 10.1016/j.anihpc.2014.09.008.  Google Scholar

[11]

M. Du and T. Weth, Ground states and high energy solutions of the planar Schrödinger-Poisson system, Nonlinearity, 30 (2017), 3492-3515.  doi: 10.1088/1361-6544/aa7eac.  Google Scholar

[12]

R. HarrisonI. Moroz and K. P. Tod, A numerical study of the Schrödinger–Newton equation, Nonlinearity, 16 (2003), 101-122.  doi: 10.1088/0951-7715/16/1/307.  Google Scholar

[13]

E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27.  doi: 10.2140/apde.2009.2.1.  Google Scholar

[14]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1977), 93-105.  doi: 10.1002/sapm197757293.  Google Scholar

[15]

P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4.  Google Scholar

[16]

S. Masaki, Local existence and WKB approximation of solutions to Schrödinger-Poisson system in the two-dimensional whole space, Comm. Partial Differential Equations, 35 (2010), 2253-2278.  doi: 10.1080/03605301003717142.  Google Scholar

[17]

V. Moroz and J. Van Schaftingen, Semi-classical states for the Choquard equations, Calc. Var. Partial Differential Equations, 52 (2015), 199-235.  doi: 10.1007/s00526-014-0709-x.  Google Scholar

[18]

I. M. MorozR. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.  doi: 10.1088/0264-9381/15/9/019.  Google Scholar

[19]

R. Penrose, On gravity's role in quantum state reduction, Gen. Rel. Grav., 28 (1996), 581-600.  doi: 10.1007/BF02105068.  Google Scholar

[20]

R. Penrose, Quantum computation, entanglement and state reduction, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927-1939.  doi: 10.1098/rsta.1998.0256.  Google Scholar

[21]

R. Penrose, The Road to Reality. A Complete Guide to the Laws of the Universe, Alfred A. Knopf Inc., New York, 2005.  Google Scholar

[22]

J. Stubbe, Bound states of two-dimensional Schrödinger-Newton equations, preprint, arXiv: 0807.4059v1, 2008. Google Scholar

[23]

P. Tod and I. M. Moroz, An analytical approach to the Schrödinger-Newton equations, Nonlinearity, 12 (1999), 201-216.  doi: 10.1088/0951-7715/12/2/002.  Google Scholar

[24]

J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger–Newton equation, J. Math. Phys., 50 (2009), 012905, 22 pp. doi: 10.1063/1.3060169.  Google Scholar

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