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Multiple solutions to a magnetic nonlinear Choquard equation

  • Silvia Cingolani
  • Mónica Clapp
  • Simone Secchi
Article

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 < (2Nα)/(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.

Mathematics Subject Classification (2010)

35Q55 35Q40 35J20 35B06 

Keywords

Nonlinear Choquard equation Nonlocal nonlinearity Electromagnetic potential Multiple solutions Intertwining solutions 

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Silvia Cingolani
    • 1
  • Mónica Clapp
    • 2
  • Simone Secchi
    • 3
  1. 1.Dipartimento di MatematicaPolitecnico di BariBariItaly
  2. 2.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoMéxico D.F.Mexico
  3. 3.Dipartimento di Matematica ed ApplicazioniUniversità di Milano-BicoccaMilanoItaly

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