Virial identities for nonlinear Schrödinger equations with a critical coefficient inverse-square potential

Abstract

Virial identities for nonlinear Schrödinger equations with some strongly singular potential (a|x|−2 ) are established. Here if a = a(N) :=−(N−2)2/4 , then Pa(N) :=−Δ+a(N)|x|−2 is nonnegative selfadjoint in the sense of Friedrichs extension. But the energy class D((1 + Pa(N))) does not coincide with H1(RN ) . Thus justification of the virial identities has a lot of difficulties. The identities can be applicable for showing blow-up in finite time and for proving the existence of scattering states. Mathematics subject classification (2010): 35Q55, 35Q40, 81Q15.