Bifurcation results for critical points of families of functionals

Abstract

Recently, the first author studied in [17] the bifurcation of critical points of families of functionals on a Hilbert space, which are parametrized by a compact and orientable manifold having a non-vanishing first integral cohomology group. We improve this result in two directions: topologically and analytically. From the analytical point of view, we generalize it to a broader class of functionals. From the topological point of view, we allow the parameter space to be a metrizable Banach manifold. Our methods are, in particular, powerful if the parameter space is simply connected. As an application of our results, we consider families of geodesics in (semi-) Riemannian manifolds.