Multiplicity of positive solutions of nonlinear Schr\"odinger \'equations concentrating at a potential well

Abstract

We consider singularly perturbed nonlinear Schr\"odinger equations \be \label{eq:0.1} - \varepsilon^2 \Delta u + V(x)u = f(u), \ \ u > 0, \ \ v \in H^1(\R^N) \ee where $V \in C(\R^N, \R)$ and $f$ is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain $\Omega \subset \R^N$ such that \[m_0 \equiv \inf_{x \in \Omega} V(x) < \inf_{x \in \partial \Omega} V(x) \] 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.