Existence of solutions for a perturbed problem with logarithmic potential in $\mathbb{R}^2$

Federico Bernini; Simone Secchi; Federico Bernini; Simone Secchi

doi:10.3934/mine.2020020

Mathematics in Engineering

2020, Volume 2, Issue 3: 438-458. doi: 10.3934/mine.2020020

Previous Article Next Article

Research article Special Issues

Existence of solutions for a perturbed problem with logarithmic potential in $\mathbb{R}^2$

Federico Bernini ,
Simone Secchi ^,

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

Received: 31 October 2019 Accepted: 20 February 2020 Published: 28 February 2020

We study a perturbed Schrödinger equation in the plane arising from the coupling of quantum physics with Newtonian gravitation. We obtain some existence results by means of a perturbation technique in Critical Point Theory.
- variational methods,
- perturbation methods,
- finite-dimensional reduction
Citation: Federico Bernini, Simone Secchi. Existence of solutions for a perturbed problem with logarithmic potential in $\mathbb{R}^2$[J]. Mathematics in Engineering, 2020, 2(3): 438-458. doi: 10.3934/mine.2020020

Related Papers:

Abstract

We study a perturbed Schrödinger equation in the plane arising from the coupling of quantum physics with Newtonian gravitation. We obtain some existence results by means of a perturbation technique in Critical Point Theory.

References

[1]	Ambrosetti A, Badiale M (1998) Homoclinics: Poincaré-Melnikov type results via a variational approach. Ann Inst H Poincaré Anal Non Linéaire 15: 233-252.
[2]	Ambrosetti A, Badiale M (1998) Variational perturbative methods and bifurcation of bound states from the essential spectrum. Proc Roy Soc Edinburgh Sect A 128: 1131-1161. doi: 10.1017/S0308210500027268
[3]	Ambrosetti A, Badiale M, Cingolani S (1997) Semiclassical states of nonlinear Schrödinger equations. Arch Ration Mech Anal 140: 285-300. doi: 10.1007/s002050050067
[4]	Ambrosetti A, Malchiodi A (2006) Perturbation methods and semilinear elliptic problems on $\mathbb{R}^n$, Basel: Birkhäuser Verlag.
[5]	Bernini F, Mugnai D (2020) On a logarithmic Hartree equation. Adv Nonlinear Anal 9: 850-865.
[6]	Bonheure D, Cingolani S, Van Schaftingen J (2017) The logarithmic Choquard equation: Sharp asymptotics and nondegeneracy of the groundstate. J Funct Anal 272: 5255-5281. doi: 10.1016/j.jfa.2017.02.026
[7]	Choquard P, Stubbe J, Vuffray M (2008). Stationary solutions of the Schrödinger-Newton model-an ODE approach. Differ Integral Equ 21: 665-679.
[8]	Cingolani S, Weth T (2016) On the planar Schrödinger-Poisson system. Ann Inst H Poincaré Anal Non Linéaire 33: 169-197.
[9]	Harrison R, Moroz I, Tod KP (2003) A numerical study of the Schrödinger-Newton equations. Nonlinearity 16: 101-122. doi: 10.1088/0951-7715/16/1/307
[10]	Lenzmann E (2009) Uniqueness of ground states for pseudorelativistic Hartree equations. Anal PDE 2: 1-27. doi: 10.2140/apde.2009.2.1
[11]	Lieb EH (1977) Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation. Stud in Appl Math 57: 93-105. doi: 10.1002/sapm197757293
[12]	Lieb EH, Loss M (2001) Analysis, 2 Eds., Vol 14 of Graduate Studies in Mathematics, Providence: American Mathematical Society.
[13]	Secchi S (2010) A note on Schrödinger-Newton systems with decaying electric potential. Nonlinear Anal 72: 3842-3856. doi: 10.1016/j.na.2010.01.021
[14]	Stubbe J (2008) Bound states of two-dimensional Schrödinger-Newton equations. arXiv:0807.4059.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)