Variational method

A variational method seeks to construct or control solutions to an equation by interpreting that equation as the Euler-Lagrange equation for a functional, and then constructing critical points or extremizers to that functional. One way to do this is to start with a minimizing sequence for the functional (which always exists, if the functional is bounded from below), and exploit some sort of compactness to extract a convergent subsequence, which then should hopefully converge to a minimizer. The presence of this compactness is known as the Palais-Smale condition. In many situations this condition breaks down due to the presence of non-compact symmetries (or approximate symmetries) in the equation and in the functional, however this can sometimes be counteracted using concentration compactness as a substitute for compactness.