We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations 0.1LA,Vħu=f(|u|2)uinRN\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} L^{\hbar }_{A,V} u = f(|u|^2)u \quad \hbox {in}\quad \mathbb {R}^N \end{aligned}$$\end{document}where N≥3\documentclass[12pt]{minimal}
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\begin{document}$$N \ge 3$$\end{document}, LA,Vħ\documentclass[12pt]{minimal}
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\begin{document}$$L^{\hbar }_{A,V}$$\end{document} is the Schrödinger operator with a magnetic field having source in a C1\documentclass[12pt]{minimal}
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\begin{document}$$C^1$$\end{document} vector potential A and a scalar continuous (electric) potential V defined by 0.2LA,Vħ=-ħ2Δ-2ħiA·∇+|A|2-ħidivA+V(x).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} L^{\hbar }_{A,V}= -\hbar ^2 \Delta -\frac{2\hbar }{i} A \cdot \nabla + |A|^2- \frac{\hbar }{i}\mathrm{div}A + V(x). \end{aligned}$$\end{document}Here, f is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain Ω⊂RN\documentclass[12pt]{minimal}
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\begin{document}$$\Omega \subset \mathbb {R}^N$$\end{document} such that m0≡infx∈ΩV(x)<infx∈∂ΩV(x)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} m_0 \equiv \inf _{x \in \Omega } V(x) < \inf _{x \in \partial \Omega } V(x) \end{aligned}$$\end{document}and we set K={x∈Ω|V(x)=m0}\documentclass[12pt]{minimal}
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\begin{document}$$K = \{ x \in \Omega \ | \ V(x) = m_0\}$$\end{document}. For ħ>0\documentclass[12pt]{minimal}
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\begin{document}$$\hbar >0$$\end{document} small we prove the existence of at least cupl(K)+1\documentclass[12pt]{minimal}
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\begin{document}$${\mathrm{cupl}}(K) + 1$$\end{document} geometrically distinct, complex-valued solutions to (0.1) whose moduli concentrate around K as ħ→0\documentclass[12pt]{minimal}
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\begin{document}$$\hbar \rightarrow 0$$\end{document}.
