Dynamics of the non-radial energy-critical inhomogeneous NLS

Abstract

We consider the focusing inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_t u + \Delta u + |x|^{-b}|u|^\alpha u = 0\qtq{on}\R\times\R^N, \] with $\alpha=\tfrac{4-2b}{N-2}$, $N=\{3,4,5\}$ and $0<b\leq \min\Big\{\tfrac{6-N}{2},\tfrac{4}{N}$\Big\}. This paper establishes global well-posedness and scattering for the non-radial energy-critical case in $\dot{H}^1(\R^N)$. It extends the previous research by Murphy and the first author \cite{GM}, which focused on the case $(N,\alpha,b)=(3,2,1)$. The novelty here, beyond considering higher dimensions, lies in our assumption of the condition $\sup_{t\in I}\|\nabla u(t)\|_{L^2}<\|\nabla Q\|_{L^2}$, which is weaker than the condition stated in \cite{Guzman}. Consequently, if a solution has energy and kinetic energy less than the ground state $Q$ at some point, then the solution is global and scatters. Moreover, we show scattering for the defocusing case. On the other hand, in this work, we also investigate the blow-up issue with nonradial data for $N\geq 3$ in $H^1(\mathbb{R}^N)$. This implies that our result holds without classical assumptions such as spherically symmetric data or $|x|u_0 \in L^2(\mathbb{R}^N)$. \ \noindent Mathematics Subject Classification. 35A01, 35QA55, 35P25.