Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities

By Silvia Cingolani, Simone Secchi, Marco Squassina. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2009

144 citations recorded by the source corpus.

Abstract The semi-classical regime of standing wave solutions of a Schrödinger equation in the presence of non-constant electric and magnetic potentials is studied in the case of non-local nonlinearities of Hartree type. It is shown that there exists a family of solutions having multiple concentration regions which are located around the minimum points of the electric potential.

Subfields: Atomic and Molecular Physics, and Optics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/deefb432398d179a13247c8fd47edd0a90d03dcb

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In Atomic and Molecular Physics, and Optics

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Citations in the corpus

Positive and sign changing solutions to a nonlinear Choquard equation (2012, background): “Further results for related problems may be found in [1, 7, 8, 18, 22, 25, 26] and the references therein.”
Existence of groundstates for a class of nonlinear Choquard equations (2012, background): “The proof of Pohoˇzaev identity (1.5) is a generalization of the argument for f ( s ) = s p [31] (see also particular cases [29], [13, lemma 2.1]).”
Semi-classical states for the Choquard equation (2013, methodology): “This penalization scheme should not be confused with the global penalization scheme of J. Byeon and Z.-Q. Wang which penalizes globally the L 2 norm outside a concentration [15,17,19].”
Singularly perturbed critical Choquard equations (2016, methodology): “[11] applied the penalization arguments due to Byeon and Jeanjean [8] and showed that there exists a family of solutions having multiple concentration regions which are located around the minimum points of the potential.”
Mass minimizers and concentration for nonlinear Choquard equations in $\R^N$ (2015, background): “The concentration phenomena have also been studied in [21] and [4] by considering semiclassical limit of the Choquard equation −ε(2)∆u+ V u = ε(Iα ∗ |u|)|u|u in R .”