Multiple complex-valued solutions for nonlinear magnetic Schrodinger equations

We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations LℏA,Vu=f(|u|2)uinRN (0.1) where N≥3, LℏA,V is the Schrödinger operator with a magnetic field having source in a C1 vector potential A and a scalar continuous (electric) potential V defined by LℏA,V=−ℏ2Δ−2ℏiA⋅∇+|A|2−ℏidivA+V(x). (0.2) Here, f is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain Ω⊂RN such that m0≡infx∈ΩV(x)<infx∈∂ΩV(x) and we set K={x∈Ω | V(x)=m0}. For ℏ>0 small we prove the existence of at least cupl(K)+1 geometrically distinct, complex-valued solutions to (0.1) whose moduli concentrate around K as ℏ→0.