Mass-critical focusing inhomogeneous NLS with inverse square potential

This work investigates a mass-critical focusing inhomogeneous NLS with an inverse square potential. The paper proves the finite time blow-up of solutions for datum with negative energy without any radial or finite variance assumption. Moreover, the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{2}$\end{document} concentration of nonglobal mass-critical solutions is established. This note complements (arXiv:2306.15210v1 [math.AP]) to the mass-critical regime and (Nonlinearity 35(8), 2022) to the case with inverse square potential. The proof is based on a localized Virial type identity via the decaying factor |x|−b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$|x|^{-b}$\end{document}, which gives a control, away from the origin, for the terms arising from the nonlinearity. Moreover, Hardy estimate is used to get the norm equivalence ∥⋅∥H˙1(RN)∼∥−Δ+a|x|2⋅∥L2(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\|\cdot \|_{\dot{H}^{1}(\mathbb{R}^{N})}\sim \|\sqrt{-\Delta + \frac {a}{|x|^{2}}}\cdot \|_{L^{2}(\mathbb{R}^{N})}$\end{document}, which enables to handle the inverse square potential. This needs in particular the restrictions N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N\geq 3$\end{document} and a>−(N−2)24\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a>-\frac{(N-2)^{2}}{4}$\end{document}. Furthermore, the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{2}$\end{document} concentration is based on a compactness result in the spirit of (Int. Math. Res. Not. 46:2815-2828, (2005))