Global existence and blowup for a class of the focusing nonlinear Schrödinger equation with inverse-square potential

We consider a class of the focusing nonlinear Schrödinger equation with inverse-square potentiali∂tu+Δu−c|x|−2u=−|u|αu,u(0)=u0∈H1,(t,x)∈R×Rd, where d≥3, 4d≤α≤4d−2 and c≠0 satisfies c>−λ(d):=−(d−22)2. In the mass-critical case α=4d, we prove the global existence and blowup below ground states for the equation with d≥3 and c>−λ(d). In the mass and energy intercritical case 4d<α<4d−2, we prove the global existence and blowup below the ground state threshold for the equation. This extends similar results of [18] and [22] to any dimensions d≥3 and a full range c>−λ(d). We finally prove the blowup below ground states for the equation in the energy-critical case α=4d−2 with d≥3 and c>−d2+4d(d+2)2λ(d).