In this talk, we construct an extremizer for the Lieb-Thirring energy inequality using concentration-compactness principle. 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 Schrödinger 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 is the joint work with Younghun Hong (Chung-Ang Univ.) and Soonsik Kwon (KAIST).