A Nonlinear Schrödinger Equation Resonating at an Essential Spectrum

Abstract EN

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.