Dynamics of the Non-radial Energy-critical Inhomogeneous NLS

We consider the focusing inhomogeneous nonlinear Schr & ouml;dinger equation i partial derivative tu+Delta u+|x|-b|u|alpha u=0onRxRN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} i\partial _t u + \Delta u + |x|<^>{-b}|u|<^>\alpha u = 0\quad \text {on}\quad \mathbb {R}\times \mathbb {R}<^>N, \end{aligned}$$\end{document}with alpha=4-2bN-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =\frac{4-2b}{N-2}$$\end{document}, N={3,4,5}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=\{3,4,5\}$$\end{document} and 01(\mathbb {R}<^>N)$$\end{document}. It extends the previous research by Murphy and the first author Guzm & aacute;n and Murphy (J. Diff. Equ. 295, 187-210, 2021), which focused on the case (N,alpha,b)=(3,2,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(N,\alpha ,b)=(3,2,1)$$\end{document}. The novelty here, beyond considering higher dimensions, lies in our assumption of the condition supt is an element of I & Vert;del u(t)& Vert;L2<& Vert;del Q & Vert;L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sup _{t\in I}\Vert \nabla u(t)\Vert _{L<^>2}<\Vert \nabla Q\Vert _{L<^>2}$$\end{document}, which is weaker than the condition stated in Guzm & aacute;n (Nonlinear Anal. Real World Appl. 37, 249-286, 2017). Consequently, if a solution has energy and kinetic energy less than the ground state Q at some point, then the solution is global and scatters. Moreover, we show scattering for the defocusing case. On the other hand, in this work, we also investigate the blow-up issue with nonradial data for N >= 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 3$$\end{document} in H1(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>1(\mathbb {R}<^>N)$$\end{document}. This implies that our result holds without classical assumptions such as spherically symmetric data or |x|u0 is an element of L2(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|x|u_0 \in L<^>2(\mathbb {R}<^>N)$$\end{document}.