Giovany Figueiredo, Marcelo Montenegro, Localized complex solutions for the Choquard equation with magnetic field

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DOI: 10.23952/jnva.10.2026.1.08

Volume 10, Issue 1, 1 April 2026, Pages 181-197

Abstract. We study the Choquard equation $( \frac{\varepsilon}{i}\nabla - A(x))^2 u + W(x)u = (|x|^{-\mu}*F(|u|^{2}))f(|u|^2)u$ in presence of a magnetic field $A$ , $i=\sqrt{-1}$ and $0\textless \mu \textless 2$ . We prescribe the lowest number of complex solutions $u \in H^1(\mathbb{R}^N,\mathbb{C})$ . The quantity is at least equals to the number of global minima of the potential $W$ , when $\varepsilon > 0$ is sufficiently small. We prove a projection lemma of the Nehari manifold corresponding to the energy functional, and then we use it to estimate the energy of each solution and to localize them around each minima of the potential $W$ . We analyze the energy levels of the solutions to distinguish them from each other. We do not use topological arguments, and no symmetric function spaces are used.

How to Cite this Article:
G. Figueiredo, M. Montenegro, Localized complex solutions for the Choquard equation with magnetic field, J. Nonlinear Var. Anal. 10 (2026), 181-197.