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

TL;DR

This paper proves the existence of multiple positive solutions to a nonlinear Schrödinger equation that concentrate around a potential well, with the number of solutions related to the topological complexity of the well.

Contribution

It establishes the existence of at least (K)+1 solutions concentrating near the potential minimum set, without relying on Nehari manifold reduction.

Findings

At least (K)+1 solutions exist.

Solutions concentrate around the set where the potential attains its minimum.

The number of solutions relates to the topological complexity of the minimum set.

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.