Essential critical points in product manifolds
Let \(M\) be a Banach space and \(f : M \to \mathbb{R}\) a continuous function. A point \(u \in M\) is said to be essentially regular for \(f\), if, up to a local homeomorphism, \(f\) is a nonconstant affine function near \(u\). A point \(u \in M\) is said to be essentially critical, if it is not essentially regular. These notions can be easily extended to the case where \(M\) is a Banach manifold and has a topological nature. They were introduced by \textit{F. Wille} [Contemp. Math. 72, 253-268 (1988; Zbl 0659.58004)] where the existence of such essential critical points was established when \(M\) is finite-dimensional. The authors consider a \(C^2\)-Finsler manifold \(M\), a metrizable Banach manifold \(P\) and a continuous function \(f : M \times P \to \mathbb{R}\) which is locally Lipschitz with respect to the first variable. They impose a kind of Palais-Smale condition, which seems to be rather natural when \(P\) is compact. Then they extend some of Wille's results to this infinite- dimensional setting. Finally, they give an application to the existence of periodic solutions for an ordinary differential equation.
scientific article; zbMATH DE number 796915
scientific article