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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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