Ljusternik-Schnirelmann theory on \(C^ 1\)-manifolds (Q1113146)

The author extends to \(C^ 1\) manifolds the Lyusternik-Schnirelman theory. More precisely, let M be a complete Finsler manifold of class \(C^ 1\). It is shown that, if M contains a compact subset of category k in M, then each function \(f\in C^ 1(M,{\mathbb{R}})\) which is bounded from below and satisfies the Palais-Smale condition must necessarily have k critical points. The proof depends on an interesting application of Ekeland's variational principle. An application is given to an eigenvalue problem for a quasilinear differential equation involving the p- Laplacian.