Multiplicity of positive solutions of nonlinear Schrodinger equations concentrating at a potential well

We consider singularly perturbed nonlinear Schrödinger equations − ε2u + V(x)u = f (u), u > 0, v ∈ H1(RN ) (0.1) where V ∈ C(RN ,R) and f is a nonlinear term which satisfies the so-called Berestycki–Lions conditions. We assume that there exists a bounded domain Omega ⊂ RN such that m0 ≡ inf inf{V(x) |x∈ Omega} < inf{V(x) |x∈∂ Omega } and we set K = {x ∈ Omega | V(x) = m0}. For ε > 0 small we prove the existence of at least cupl(K) + 1 solutions to (0.1) concentrating, as ε → 0 around K. We remark that, under our assumptions of f , the search of solutions to (0.1) cannot be reduced to the study of the critical points of a functional restricted to a Nehari manifold.