JournalsjemsVol. 8 , No. 2DOI 10.4171/jems/51

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

  • Patricio A. Felmer

    Universidad de Chile, Santiago, Chile
  • Salomé Martínez

    Universidad de Chile, Santiago, Chile
  • Kazunaga Tanaka

    Waseda University, Tokyo, Japan
On the number of positive solutions of singularly perturbed 1D NLS cover

Abstract

We study singularly perturbed 1D nonlinear Schr\"odinger equations (\ref{eq:1.1}). When V(x)V(x) has multiple critical points, (\ref{eq:1.1}) has a wide variety of positive solutions for small ε\varepsilon and the number of positive solutions increases to \infty as ε0\varepsilon\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)V(x). Envelope functions or equivalently adiabatic profiles of high frequency solutions play an important role in the proof.