On the number of positive solutions of singularly perturbed ID nonlinear Schrödinger equations

We study singularly perturbed ID nonlinear Schrödinger equations (1.1). When V(x) has multiple critical points, (1.1) has a wide variety of positive solutions for small ε and the number of positive solutions increases to ∞ as ε → 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.