A Schrödinger equation with time-oscillating critical nonlinearity

In this paper, we study the time-oscillating critical nonlinear Schrödinger equation iut+Δu+θ(ωt)|u|4n−2u=0 in Rn(n≥3), where θθ is a periodic function. We show that, for a given initial condition u(0)=φu(0)=φ in H1H1, the solution uωuω converges as |ω|→∞|ω|→∞ to the solution UU of the limiting equation iUt+ΔU+I(θ)|U|4n−2U=0 with the same initial condition, where I(θ)I(θ) is the average of θθ. We also show that if the solution UU is global and has a certain decay property, then uωuω is also global if |ω||ω| is sufficiently large. Similar results for the subcritical problem are given by Cazenave and Scialom (2010) [9].