The Semiclassical Limit of the Nonlinear Schrödinger Equation in a Radial Potential

Abstract

In this paper, we are concerned with the following nonlinear Schrödinger equation: i h ̵ ∂ψ ∂t =− h ̵ 2 2m Δψ+V(x)ψ−γ h ̵ ∣ψ∣ p−2ψ, γ h ̵ >0, x∈ R 2, where ħ>0, 2< p<6, ψ : R 2→ C , and the potential V is radially symmetric. Our main purpose is to obtain positive solutions among the functions having the form ψ(r,θ,t)=exp(Im h ̵ θ/ h ̵ +iEt/ h ̵ )v(r) , being r, θ the polar coordinates in the plane. Since we assume M ħ >0, the functions in this special class have nontrivial angular momentum as it will be specified in the Introduction. Furthermore, our solutions exhibit a “spike-layer” pattern when the parameter ħ approaches zero; our object is to analyse the appearance of such type of concentration asymptotic behaviour in order to locate the asymptotic peaks.