On the global well-posedness of focusing energy-critical inhomogeneous NLS

We consider the focusing energy-critical inhomogeneous nonlinear Schrödinger equation: iut+Δu+g|u|p-1u=0,u(0)=φ∈H˙1,where 0 ≤ g_i≤ | x| ^bg≤ g_s, 0<b<43, and p= 5 - 2 b. On the road map of Kenig and Merle (Invent Math 166:645–675, 2006), we show the global well-posedness and scattering of radial solutions under energy condition Eg(φ)<Eg(Qb),andgs1p0‖φ‖H˙12<‖Qb‖H˙12,where Q_b is the solution of ΔQb+|x|-bQbp=0, together with scaling condition | x| | ∇ g(x) | ≲ | x| ^-b, variational condition gs2p-1(p+12-gi)≤p-12, and rigidity condition - bg(x) ≤ x· ∇ g(x). We also provide sharp finite time blow-up results for non-radial and radial solutions. For this, we utilize the localized virial identity.