We construct an extremizer for the Lieb-Thirring energy inequality (except the endpoint cases) developing the concentration-compactness technique for operator valued inequality in the formulation of the profile decomposition. Moreover, we investigate the properties of the extremizer, such as the system of Euler-Lagrange equations, regularity and summability. As an application, we study a dynamical consequence of a system of nonlinear Schrodinger equations with focusing cubic nonlinearities in three dimension when each wave function is restricted to be orthogonal. Using the critical element of the Lieb-Thirring inequality, we establish a global existence versus finite time blowup dichotomy. This result extends the single particle result of Holmer-Roudenko  to infinitely many particles system. (C) 2018 Elsevier Masson SAS. All rights reserved.