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 $L^{2}$ 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 $|x|^{-b}$ , 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 $\|\cdot \|_{\dot{H}^{1}(\mathbb{R}^{N})}\sim \|\sqrt{-\Delta + \frac {a}{|x|^{2}}}\cdot \|_{L^{2}(\mathbb{R}^{N})}$ , which enables to handle the inverse square potential. This needs in particular the restrictions $N\geq 3$ and $a>-\frac{(N-2)^{2}}{4}$ . Furthermore, the $L^{2}$ concentration is based on a compactness result in the spirit of (Int. Math. Res. Not. 46:2815-2828, (2005))