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

We consider a class of the focusing nonlinear Schrodinger equation with inverse square potential i partial derivative(t)u + Delta u - c vertical bar x vertical bar(-2)u = vertical bar u vertical bar(alpha)u, u(0) = u(0) is an element of H-1, (t, x) is an element of R x R-d, where d >= 3, 4/d <= alpha <= 4/d-2 and c not equal 0 satisfies c > -lambda(d) := - (d-2/2)(2). In the mass-critical case alpha = 4/d, we prove the global existence and blowup below ground states for the equation with d >= 3 and c > -lambda(d). In the mass and energy intercritical case 4/d < alpha < 4/d-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 > -lambda(d). We finally prove the blowup below ground states for the equation in the energy-critical case alpha = 4/d-2 with d >= 3 and c > -d(2)+4d/(d+2)(2) lambda(d). (C) 2018 Elsevier Inc. All rights reserved.