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We construct an extremizer for the kinetic energy inequality (except theendpoint cases) developing the concentration-compactness technique for operatorvalued inequality in the formulation of the profile decomposition. Moreover, weinvestigate the properties of the extremizer, such as the system ofEuler-Lagrange equations, regularity and summability. As an application, westudy a dynamical consequence of a system of nonlinear Schr\"odinger equationswith focusing cubic nonlinearities in three dimension when each wave functionis restricted to be orthogonal. Using the critical element of the kineticenergy inequality, we establish a global existence versus finite time blowupdichotomy. This result extends the single particle result of Holmer andRoudenko to infinitely many particles system.

Global existence versus finite time blowup dichotomy for the system of nonlinear Schr\"odinger equations