On the number of positive solutions of singularly perturbed 1D NLS

Abstract

We study singularly perturbed 1D nonlinear Schrödinger equations (1.1). When $V(x)$ has multiple critical points, (1.1) has a wide variety of positive solutions for small $\epsilon$ and the number of positive solutions increases to $\infty$ as $\epsilon \to 0$. We give an estimate of the number of positive solutions whose growth order depends on the number of local maxima of $V(x)$. Envelope functions or equivalently adiabatic profiles of high frequency solutions play an important role in the proof.