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Multiple solutions to a magnetic nonlinear Choquard equation

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Abstract

We consider the stationary nonlinear magnetic Choquard equation

$$(- {\rm i}\nabla+ A(x))^{2}u + V (x)u = \left(\frac{1}{|x|^{\alpha}}\ast |u|^{p}\right) |u|^{p-2}u,\quad x\in\mathbb{R}^{N}$$

where A is a real-valued vector potential, V is a real-valued scalar potential, N ≥ 3, \({\alpha \in (0, N)}\) and 2 − (α/N) < p < (2N − α)/(N−2). We assume that both A and V are compatible with the action of some group G of linear isometries of \({\mathbb{R}^{N}}\) . We establish the existence of multiple complex valued solutions to this equation which satisfy the symmetry condition

$$u(gx) = \tau(g)u(x)\quad{\rm for\, all }\ g \in G,\;x \in \mathbb{R}^{N},$$

where \({\tau : G \rightarrow \mathbb{S}^{1}}\) is a given group homomorphism into the unit complex numbers.

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Authors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Bari, via Orabona 4, 70125, Bari, Italy

    Silvia Cingolani

  2. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., 04510, México D.F., Mexico

    Mónica Clapp

  3. Dipartimento di Matematica ed Applicazioni, Università di Milano-Bicocca, via Cozzi 53, 20125, Milano, Italy

    Simone Secchi

Authors
  1. Silvia Cingolani
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  2. Mónica Clapp
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  3. Simone Secchi
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Corresponding author

Correspondence to Mónica Clapp.

Additional information

S. Cingolani is supported by the MIUR proyect Variational and topological methods in the study of nonlinear phenomena (PRIN 2007).

M. Clapp is supported by CONACYT grant 129847 and PAPIIT grant IN101209 (Mexico).

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Cingolani, S., Clapp, M. & Secchi, S. Multiple solutions to a magnetic nonlinear Choquard equation. Z. Angew. Math. Phys. 63, 233–248 (2012). https://doi.org/10.1007/s00033-011-0166-8

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