Multiple complex-valued solutions for nonlinear magnetic Schrodinger equations

We study, in the semiclassical limit, the singularly perturbed nonlinear Schr"odinger equations $$L_{A,V}^{\mathrm{\hslash }}u=f(\mathrm{\mid }u{\mathrm{\mid }}^2)uin{R}^N$$ where $N\ge 3$, $L_{A,V}^{\mathrm{\hslash }}$ is the Schr"odinger operator with a magnetic field having source in a $C^1$ vector potential $A$ and a scalar continuous (electric) potential $V$ defined by \begin{equation} L^{\hbar}_{A,V}= -\hbar^2 \Delta-\frac{2\hbar}{i} A \cdot \nabla + |A|^2- \frac{\hbar}{i}\operatorname{div}A + V(x). \end{equation} Here $f$ is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain $\mathrm{\Omega }\subset R^N$ such that $$m_0\equiv \underset{x\in \mathrm{\Omega }}{\inf }V(x)<\underset{x\in \mathrm{\partial }\mathrm{\Omega }}{\inf }V(x)$$ and we set $K=\{x\in \mathrm{\Omega }\mathrm{\mid }V(x)=m_0\}$. For $\mathrm{\hslash }>0$ small we prove the existence of at least $cuplenght(K)+1$ geometrically distinct, complex-valued solutions whose modula concentrate around $K$ as $\mathrm{\hslash }\to 0$.