Multiplicity of positive solutions of nonlinear Schrödinger équations concentrating at a potential well

Abstract

We consider singularly perturbed nonlinear Schrödinger equations \be \label{eq:0.1} - \varepsilon^2 Δu + V(x)u = f(u), \ \ u > 0, \ \ v \in H^1(\R^N) \ee where $V\in C({\mathbb{R}}^N,\mathbb{R})$ and $f$ is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain $\Omega \subset {\mathbb{R}}^N$ such that \[m_0 \equiv \inf_{x \in Ω} V(x) < \inf_{x \in \partial Ω} V(x) \] and we set $K=\{x\in \Omega |V(x)=m_0\}$. For $\e >0$ small we prove the existence of at least ${\cuplength}(K) + 1$ solutions to (\ref{eq:0.1}) concentrating, as $\e \to 0$ around $K$. We remark that, under our assumptions of $f$, the search of solutions to (\ref{eq:0.1}) cannot be reduced to the study of the critical points of a functional restricted to a Nehari manifold.