Concentration phenomena for the Schrödinger-Poisson system in <inline-formula><tex-math id="M1">\begin{document}$  \mathbb{R}^2 $\end{document}</tex-math></inline-formula>

Denis Bonheure; Silvia Cingolani; Simone Secchi

doi:10.3934/dcdss.2020447

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Preface: Applications of mathematical analysis to problems in theoretical physics

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

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

Denis Bonheure ^1, , Silvia Cingolani ^2,, and Simone Secchi ^3,

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 May 2021 Early access 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) $

$ \ge \rightarrow 0 $

, leaning on a nongeneracy result in [3].

Keywords: Schrödinger-Poisson system, nonlocal nonlinearity, logarithmic potential, semiclassical solutions.

Mathematics Subject Classification: Primary: 35J20, 35Q55, 35J61; Secondary: 35Q40, 35B06.

Citation: Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447

References:

[1]	A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285-300. doi: 10.1007/s002050050067.
[2]	A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbb{R}^n$, Birkhäuser Verlag, 2006.
[3]	D. Bonheure, S. 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.
[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.
[5]	P. Choquard, J. Stubbe and M. Vuffray, Stationary solutions of the Schrödinger-Newton model — an ODE approach, Differ. Integral Equ., 21 (2008), 665-679.
[6]	S. Cingolani, M. 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.
[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.
[8]	S. Cingolani, S. 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.
[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.
[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.
[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.
[12]	R. Harrison, I. 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.
[13]	E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27. doi: 10.2140/apde.2009.2.1.
[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.
[15]	P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. doi: 10.1016/0362-546X(80)90016-4.
[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.
[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.
[18]	I. M. Moroz, R. 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.
[19]	R. Penrose, On gravity's role in quantum state reduction, Gen. Rel. Grav., 28 (1996), 581-600. doi: 10.1007/BF02105068.
[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.
[21]	R. Penrose, The Road to Reality. A Complete Guide to the Laws of the Universe, Alfred A. Knopf Inc., New York, 2005.
[22]	J. Stubbe, Bound states of two-dimensional Schrödinger-Newton equations, preprint, arXiv: 0807.4059v1, 2008.
[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.
[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.

show all references

References:

[1]	A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285-300. doi: 10.1007/s002050050067.
[2]	A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbb{R}^n$, Birkhäuser Verlag, 2006.
[3]	D. Bonheure, S. 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.
[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.
[5]	P. Choquard, J. Stubbe and M. Vuffray, Stationary solutions of the Schrödinger-Newton model — an ODE approach, Differ. Integral Equ., 21 (2008), 665-679.
[6]	S. Cingolani, M. 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.
[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.
[8]	S. Cingolani, S. 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.
[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.
[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.
[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.
[12]	R. Harrison, I. 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.
[13]	E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27. doi: 10.2140/apde.2009.2.1.
[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.
[15]	P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. doi: 10.1016/0362-546X(80)90016-4.
[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.
[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.
[18]	I. M. Moroz, R. 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.
[19]	R. Penrose, On gravity's role in quantum state reduction, Gen. Rel. Grav., 28 (1996), 581-600. doi: 10.1007/BF02105068.
[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.
[21]	R. Penrose, The Road to Reality. A Complete Guide to the Laws of the Universe, Alfred A. Knopf Inc., New York, 2005.
[22]	J. Stubbe, Bound states of two-dimensional Schrödinger-Newton equations, preprint, arXiv: 0807.4059v1, 2008.
[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.
[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.

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American Institute of Mathematical Sciences

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

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

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