The kinetic energy inequality, which is dual to the

fundamental Lieb-Thirring inequality, plays an important role in the

proof of the stability of matter, and they have a huge literature in

the mathematical physics. In this talk, we develop the profile

decomposition associated with the kinetic energy inequality and prove

existence of a maximizer for the Weinstein functional. From the

variational problem, we also derive the Euler-Lagrange equation, which

is a system of infinitely coupled nonlinear elliptic equations. This

is the joint work with Soonsik Kwon and Haewon Yoon at KAIST in Korea.