Palais-Smale theorem

The Palais-Smale theorem is a concept in mathematical analysis that helps identify special points, called critical points, where a function's behavior changes, such as peaks or valleys. Specifically, it states that if a sequence of points has energies approaching a certain value, and the slopes (or derivatives) near these points get closer to zero, then there is a converging subsequence that reaches a critical point. This is useful in solving complex equations in physics and engineering, as it guarantees the existence of these critical points under certain conditions, facilitating the analysis of solutions to variational problems.