Invariant Sets of Descending Flow in Critical Point Theory with Applications to Nonlinear Differential Equations

Properties of invariant sets of descending flow defined by a pseudogradient vector field of a functional in a Banach space are studied. In this way, several critical points can be found by constructing different invariant sets on which the functional is bounded below. Under suitable conditions, the existence of at least four critical points of a functional is proved, each critical point being in a certain invariant set. The theoretical results are applied to nonlinear elliptic boundary value problems and nonlinear systems of ordinary differential equations. In variant cases, at least four solutions are obtained for these equations.