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We study the focusing nonlinear Schrödinger equation with inverse-square potential i ∂ t u + Δ u − a | x | 2 u + | u | p − 1 u = 0 with a > − 1 4 and 3 4 < p − 1 < 4 in three dimensions. In this paper we obtain blow-up and global existence criteria for the solution to ( N L S a ) by the method from Duyckaerts-Roudenko [1], which extend a recent work of Killip-Murphy-Visan-Zheng [2] to above the threshold. [ABSTRACT FROM AUTHOR]

We study the focusing inhomogeneous nonlinear Schrödinger equation with inverse‐square potential i∂tu+Δu−a|x|2u+|x|−b|u|2u=0,$$ i{\partial}_tu&amp;#x0002B;\Delta u-\frac{a}{{\left&amp;#x0007C;x\right&amp;#x0007C;}&amp;#x0005E;2}u&amp;#x0002B;{\left&amp;#x0007C;x\right&amp;#x0007C;}&amp;#x0005E;{-b}{\left&amp;#x0007C;u\right&amp;#x0007C;}&amp;#x0005E;2u&amp;#x0003D;0, $$where a>−14$$ a&gt;-\frac{1}{4} $$ and 0

In this paper, we consider the scattering theory of the radial solution to focusing energy‐subcritical Hartree equation with inverse‐square potential in the energy space H1(Rd) using the method from [4]. The main difficulties are due to the fact that the equation is not space‐translation invariant and that the nonlinearity is non‐local. Using the radial Sobolev embedding and a virial‐Morawetz type estimate we can exclude the concentration of mass near the origin. Besides, we can overcome the weak dispersive estimate when a<0, using the dispersive estimate established by [23]. [ABSTRACT FROM AUTHOR]