Partial differential equations
Logarithmic estimates for mean-field models in dimension two and the Schrödinger–Poisson system
Comptes Rendus. Mathématique, Volume 359 (2021) no. 10, pp. 1279-1293.

In dimension two, we investigate a free energy and the ground state energy of the Schrödinger–Poisson system coupled with a logarithmic nonlinearity in terms of underlying functional inequalities which take into account the scaling invariances of the problem. Such a system can be considered as a nonlinear Schrödinger equation with a cubic but nonlocal Poisson nonlinearity, and a local logarithmic nonlinearity. Both cases of repulsive and attractive forces are considered. We also assume that there is an external potential with minimal growth at infinity, which turns out to have a logarithmic growth. Our estimates rely on new logarithmic interpolation inequalities which combine logarithmic Hardy–Littlewood–Sobolev and logarithmic Sobolev inequalities. The two-dimensional model appears as a limit case of more classical problems in higher dimensions.

Accepted:
Published online:
DOI: 10.5802/crmath.272
Classification: 35J50,  35Q55,  35J47
Jean Dolbeault 1; Rupert L. Frank 2; Louis Jeanjean 3

1 CEREMADE (CNRS UMR 7534), PSL university, Université Paris-Dauphine Place de Lattre de Tassigny, 75775 Paris 16, France
2 R.L. Frank: Mathematisches Institut, Ludwig-Maximilans Universität München, Theresienstr. 39, 80333 München, Germany, and Munich Center for Quantum Science and Technology, Schellingstr. 4, 80799 München, Germany, and Mathematics 253-37, Caltech, Pasadena, CA 91125, USA
3 Laboratoire de Mathématiques (CNRS UMR 6623), Université of Bourgogne Franche-Comté, 25030 Besançon Cedex, France
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Jean Dolbeault; Rupert L. Frank; Louis Jeanjean. Logarithmic estimates for mean-field models in dimension two and the Schrödinger–Poisson system. Comptes Rendus. Mathématique, Volume 359 (2021) no. 10, pp. 1279-1293. doi : 10.5802/crmath.272. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.272/

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