2013/05/16 by Silvia Cingolani, Louis Jeanjean, Cingolani, Silvia +3
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP
paper · pdf · doi:10.48550/arxiv.1305.3685
arxiv created 2013/05/16 · arxiv updated 2013/05/17
We consider singularly perturbed nonlinear Schrödinger equations \be - ε2 Δu + V(x)u = f(u), u > 0, v ∈ H1(\RN) \ee 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 Ω⊂ \RN such that m0 ≡ infx ∈ Ω V(x) < infx ∈ ∂ Ω V(x) and we set K = \x ∈ Ω | V(x) = m0\. For \e >0 small we prove the existence of at least \cuplength(K) + 1 solutions to (\refeq:0.1) concentrating, as \e → 0 around K. We remark that, under our assumptions of f, the search of solutions to (\refeq:0.1) cannot be reduced to the study of the critical points of a functional restricted to a Nehari manifold.