Scattering for the radial 3D cubic focusing inhomogeneous nonlinear Schrödinger equation

Original · EN

The purpose of this work is to study the 3D focusing inhomogeneous nonlinear Schrödinger equation i uₜ +Δu+|x|⁻ᵇ|u|² u = 0, where 0<b<1/2. Let Q be the ground state solution of -Q+ΔQ+ |x|⁻ᵇ|Q|²Q=0 and sc=(1+b)/2. We show that if the radial initial data u₀ belongs to H¹(R³) and satisfies E(u₀)ˢᶜM(u₀)¹⁻ˢᶜ<E(Q)ˢᶜM(Q)¹⁻ˢᶜ and ∇ u₀ ₗ₂ˢᶜ u₀ₗ₂¹⁻ˢᶜ<∇ Q ₗ₂ˢᶜ Qₗ₂¹⁻ˢᶜ, then the corresponding solution is global and scatters in H¹(R³). Our proof is based in the ideas introduced by Kenig-Merle KENIG in their study of the energy-critical NLS and Holmer-Roudenko HOLROU for the radial 3D cubic NLS.