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Ljusternik-Schnirelmann theory on \(C^ 1\)-manifolds. (English) Zbl 0661.58009
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.
Reviewer: M.Willem

MSC:
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35J65 Nonlinear boundary value problems for linear elliptic equations
58J32 Boundary value problems on manifolds
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
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