A Critical Point Theorem Suggested by an Elliptic Problem with Asymmetric Nonlinearities

It is shown that an isolated critical point given by the Ambrosetti-Rabinowitz mountain-pass theorem,"limit" of minimaximizing paths with fixed end-points, cannot be the "limit" of a sequence of minimaximizing homotopical deformations of spheres, for a large class of functionals. This yields a multiplicity result which is applied to the study of elliptic problems with asymmetric nonlinearities.