Ljusternik-Schnirelmann theory on $C^1$-manifolds
Ljusternik-Schnirelmann theory on $C^1$-manifolds
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DOI：
10.1016/s0294-1449(16)30348-1
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发表时间：
1988-03
影响因子：
1.9
通讯作者：
A. Szulkin
中科院分区：
文献类型：
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作者：
A. Szulkin
设M是C1类完备Finsler流形.证明了：若M包含范畴k（in M）的紧子集，则满足Palais-Smale条件的下有界函数f∈ C1（M，n）必有k个临界点.这应该与已知的结果相比，f至少有猫（M）的临界点提供M是类C2。本文给出了一个应用于含有p-Laplacian− div的拟线性微分方程的特征值问题（|u| p− 2 u），1< p<∞。
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∈ C 1 (M, ℝ) which is bounded below and satisfies the Palais-Smale condition must necessarily have k critical points. This should be compared with the known result that f has at least cat (M) critical points provided M is of class C 2. An application is given to an eigenvalue problem for a quasilinear differential equation involving the p-Laplacian− div (|∇ u| p− 2∇ u), 1< p<∞.