Multiple solutions to a magnetic nonlinear Choquard equation
type: Item
instance of: scholarly article
title: Multiple solutions to a magnetic nonlinear Choquard equation
zbMATH Open document ID: 1247.35141
author: Silvia Cingolani
author: Mónica Clapp
author: Simone Secchi
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 2012-05-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1109.1386
review text: The authors prove the existence of multiple complex valued solutions of the stationary magnetic nonlinear Choquard equation \[ \begin{multlined} (-i \nabla + A(x))^2 u + V(x)u = \left(\frac{1}{|x|^\alpha} \star |u|^p \right) |u|^{p-2} u, \\ u \in L^2( \mathbb{R}^N,\mathbb{C}), \enspace \nabla u + iA(x)u \in L^2(\mathbb{R}^N,\mathbb{C}^N),\end{multlined} \] where \(A: \mathbb{R}^N \rightarrow \mathbb{R}^N\) is a real-valued vector potential, \(V: \mathbb{R}^N \rightarrow \mathbb{R}\) is a real-valued scalar potential, \(N \geq 3\), \(\alpha \in (0,N)\), and \( 2 - (\alpha / N) < p < (2N - \alpha) / (N -2)\), for the case that both the vector and the scalar potential satisfy the symmetry conditions \(A(gx) = gA(x)\), \(V(gx) = V(x)\) for all \(g \in G, x \in \mathbb{R}^N\), where \(G\) is a closed subgroup of the group \(O(N)\) of linear isometries of \( \mathbb{R}^N\). Precisely, one finds solutions that satisfy the symmetry condition \(u(gx) = \tau(g) u(x)\), where \(\tau: G \rightarrow S^1\) is a known group holomorphism into the unit complex numbers \(S^1\). The main results are formulated in two theorems whose proofs are based on variational methods. For the lack of compactness problem they refer to \textit{P.-L. Lions} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 109--145 (1984; Zbl 0541.49009)]. This equation is a generalization of the stationary Choquard equation \[ -\Delta u + u = \left(\frac{1}{|x|} \star |u|^2 \right) u, \enspace u \in H^1(\mathbb{R}^3) \] that comes up in the Hartree-Fock theory of plasma. Existence and uniqueness results for this equation had been already established by \textit{E. H. Lieb} [Studies Appl. Math. 57, 93--105 (1977; Zbl 0369.35022)] and \textit{P.-L. Lions} [Nonlinear Anal., Theory Methods Appl. 4, 1063--1072 (1980; Zbl 0453.47042)].
reviewed by: Georg Hebermehl
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35B06
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 6035603
zbMATH Keywords: nonlinear Choquard equation
zbMATH Keywords: nonlocal nonlinearity
zbMATH Keywords: electromagnetic potential
zbMATH Keywords: existence of multiple solutions
zbMATH Keywords: intertwining solutions
MaRDI profile type: Publication
OpenAlex ID: W3098015761
arXiv ID: 1109.1386
cites work: On a periodic Schrödinger equation with nonlocal superlinear part
cites work: Dual variational methods in critical point theory and applications
cites work: Intertwining semiclassical bound states to a nonlinear magnetic Schrödinger equation
cites work: Intertwining semiclassical solutions to a Schrödinger-Newton system
cites work: Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities
cites work: Q3209889
cites work: Transformation groups
cites work: Borsuk-Ulam Theorems for Arbitrary S 1 Actions and Applications
cites work: Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation
cites work: The Choquard equation and related questions
cites work: Classification of positive solitary solutions of the nonlinear Choquard equation
cites work: Spherically-symmetric solutions of the Schrödinger-Newton equations
cites work: An analytical approach to the Schrödinger-Newton equations
cites work: Breathing modes for the Schrödinger-Poisson system with a multiple-well external potential
cites work: The principle of symmetric criticality
cites work: On gravity's role in quantum state reduction
cites work: Quantum computation, entanglement and state reduction
cites work: Q3557982
cites work: A note on Schrödinger-Newton systems with decaying electric potential
cites work: The ground state energy of the Schrödinger-Newton equation
cites work: Strongly interacting bumps for the Schrödinger–Newton equations
cites work: Minimax theorems
cites work: Existence of a nontrivial solution for Choquard's equation
DOI: 10.1007/S00033-011-0166-8
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description: scientific article; zbMATH DE number 6035603