A Nonlinear Schrödinger Equation Resonating at an Essential Spectrum

Abstract

We consider the nonlinear Schrödinger equation -Δu+f(u)=V(x)u in RN. The potential function V satisfies that the essential spectrum of the Schrödinger operator -Δ-V is [0,+∞) and this Schrödinger operator has infinitely many negative eigenvalues accumulating at zero. The nonlinearity f satisfies the resonance type condition limt→∞f(t)/t=0. Under some additional conditions on V and f, we prove that this equation has infinitely many solutions.