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

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

This contribution is part of the Special Issue: Qualitative Analysis and Spectral Theory for Partial Differential Equations
Guest Editor: Veronica Felli
Link: http://www.aimspress.com/newsinfo/1371.html

Special Issues: Qualitative Analysis and Spectral Theory for Partial Differential

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.
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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.    

3. Ambrosetti A, Badiale M, Cingolani S (1997) Semiclassical states of nonlinear Schrödinger equations. Arch Ration Mech Anal 140: 285-300.    

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.    

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.    

10. Lenzmann E (2009) Uniqueness of ground states for pseudorelativistic Hartree equations. Anal PDE 2: 1-27.    

11. Lieb EH (1977) Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation. Stud in Appl Math 57: 93-105.    

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.    

14. Stubbe J (2008) Bound states of two-dimensional Schrödinger-Newton equations. arXiv:0807.4059.

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

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