On the GWP of focusing energy-ciritical inhomogeneous NLS

Abstract

We consider the focussing energy-critical inhomogeneous nonlinear Schr\"odinger equation: $$ iu_t + \Delta u + g|u|^2u = 0, u(0)= \varphi \in \dot{H}^1,\;\; 0 \le g_i \le |x|g \le g_s.$$ On the road map of Kenig-Merle \cite{km} we show the global well-posedness and scattering of radial solutions under energy condition $$E_g(\varphi) < E_g(Q),\;\;\mbox{and}\;\; g_s\|\varphi\|_{\dot{H}^1}^2 < \|Q\|_{\dot{H}^1}^2,$$ where $Q$ is the solution of $\Delta Q + |x|^{-1}Q^3 = 0$, together with scaling condition $|g(x)| + |x||\nabla g(x)| \lesssim |x|^{-1}$, variational condition $g_s(2-g_i) \le 1$, and rigidity condition $-g(x) \le x\cdot \nabla g(x)$. We also provide sharp finite time blowup results for nonradial and radial solutions. For this we utilize the localized virial identity.