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: iℏ ∂ψ/∂t = - ℏ2/2m Δψ + V(x)ψ - γℏ ψ p-2ψ, γℏ>0, xε ℝ2, where ℏ> 0, 2 < p < 6, ψ: ℝ2 → 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(iMhθ/ℏ + iEt/ℏ)ν(r), being r, θ the polar coordinates in the plane. Since we assume Mh > 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. © 2002 Elsevier Science (USA).