Palais–Smale condition

The Palais–Smale condition is a concept from mathematical analysis, particularly in calculus of variations and critical point theory. It essentially provides a framework for finding solutions to complex equations. More specifically, it states that when studying a functional (a mathematical expression that assigns a number to a function), if a sequence of inputs leads to a bounded output and the functional values converge, then there is a certain "compactness" that allows us to extract a solution. This condition helps in ensuring that the search for solutions remains manageable and fruitful in various mathematical contexts.