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Vortex-type solutions to a magnetic nonlinear Choquard equation

  • Dora Salazar
Article

Abstract

We consider the stationary nonlinear magnetic Choquard equation
$$(-{\rm i}\nabla + A(x))^{2} u + W(x)u = \left(\frac{1}{|x|^{\alpha}} \ast |u|^{p}\right) |u|^{p - 2}u, \quad x \in \mathbb{R}^N,$$
where \({N \geq 3, \alpha \in (0, N), p \in \bigl[2, \frac{2N - \alpha}{N - 2}\bigr), A : \mathbb{R}^N \rightarrow \mathbb{R}^N}\) is a magnetic potential and \({W : \mathbb{R}^N \rightarrow \mathbb{R}}\) is a bounded electric potential. For a given group \({\Gamma}\) of linear isometries of \({\mathbb{R}^N}\), we assume that A(gx) = gA(x) and W(gx) = W(x) for all \({g \in \Gamma, x \in \mathbb{R}^N}\). Under some assumptions on the decay of A and W at infinity, we establish the existence of solutions to this problem which satisfy
$$u(\gamma{x}) = \phi(\gamma)u(x) \quad {\rm for\, all} \gamma \in \Gamma,\, x \in \mathbb{R}^N,$$
where \({\phi : \Gamma \rightarrow \mathbb{S}^1}\) is a given continuous group homomorphism into the unit complex numbers.

Mathematics Subject Classification (2010)

Primary 35Q55 Secondary 35Q40 35A01 35B06 35J20 

Keywords

Nonlinear Choquard equation Nonlocal nonlinearity Electromagnetic potential Vortex-type solutions 

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Centro de Modelamiento Matemático, UMI 2807 CNRS-UChileUniversidad de ChileSantiagoChile

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