Dynamics of the Non-radial Energy-critical Inhomogeneous NLS

We consider the focusing inhomogeneous nonlinear Schrödinger equation \(\begin{aligned} i\partial _t u + \Delta u + |x|^{-b}|u|^\alpha u = 0\quad \text {on}\quad \mathbb {R}\times \mathbb {R}^N, \end{aligned}\) $$\begin{array}{c}\hfill i{\partial }_{t}u+\mathrm{\Delta }u+{|x|}^{-b}{|u|}^{\alpha }u=0\text{on}\mathbb{R}\times {\mathbb{R}}^N,\end{array}$$ with \(\alpha =\frac{4-2b}{N-2}\) $\alpha =\frac{4-2b}{N-2}$ , \(N=\{3,4,5\}\) $N=\{3,4,5\}$ and \(0<b\le \min \Big \{\frac{6-N}{2},\frac{4}{N}\Big \}\) $0<b\le min\{\frac{6-N}{2},\frac{4}{N}\}$ . This paper establishes global well-posedness and scattering for the non-radial energy-critical case in \(\dot{H}^1(\mathbb {R}^N)\) ${\dot{H}}^1({\mathbb{R}}^N)$ . It extends the previous research by Murphy and the first author Guzmán and Murphy (J. Diff. Equ. 295, 187–210, 2021), which focused on the case \((N,\alpha ,b)=(3,2,1)\) $(N,\alpha ,b)=(3,2,1)$ . The novelty here, beyond considering higher dimensions, lies in our assumption of the condition \(\sup _{t\in I}\Vert \nabla u(t)\Vert _{L^2}<\Vert \nabla Q\Vert _{L^2}\) ${sup}_{t\in I}{\Vert \mathrm{\nabla }u(t)\Vert }_{L^2}<{\Vert \mathrm{\nabla }Q\Vert }_{L^2}$ , which is weaker than the condition stated in Guzmán (Nonlinear Anal. Real World Appl. 37, 249–286, 2017). 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 \(N\ge 3\) $N\ge 3$ in \(H^1(\mathbb {R}^N)\) $H^1({\mathbb{R}}^N)$ . This implies that our result holds without classical assumptions such as spherically symmetric data or \(|x|u_0 \in L^2(\mathbb {R}^N)\) $|x|u_0\in L^2({\mathbb{R}}^N)$ .