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