On the blowup for the L^2‐critical focusing nonlinear Schrödinger equation in higher dimensions below the energy class

We consider the focusing mass‐critical nonlinear Schrödinger equation and prove that blowup solutions to this equation with initial data in 𝐻𝑠⁡(\R𝑑), 𝑠 >𝑠0⁡(𝑑) and 𝑑 ≥3, concentrate at least the mass of the ground state at the blowup time. This extends recent work by Colliander et al. [Math. Res. Lett., 12 (2005), pp. 357–375], Hmidi and Keraani [Remarks on the Blowup for the 𝐿2‐Critical Nonlinear Schrödinger Equations, preprint], and Tzirakis [SIAM J. Math. Anal., 37 (2006), pp. 1923–1946] on the blowup of the two‐dimensional and one‐dimensional mass‐critical focusing nonlinear Schrödinger equation below the energy space to all dimensions 𝑑 ≥3.