Critical groups of critical points produced by local linking with applications. (English) Zbl 0972.58008

Let \(F:X\to \mathbb R\) be a \(C^1\) function on a real Banach space \(X\). For each isolated critical \(u_0\) of \(F\) with \(F(u_0)=c\), the \(q\)-th critical group is defined to be the relative integral homology group \(C_q(F,u_0)=H_q(F_c\cap U, (F_c\cap U)-\{u_0\})\) where \(F_c=\{u\in X\mid F(u)\leq c\}\). Suppose \(F\) satisfies the Palais-Smale condition and has isolated critical values each of which corresponds to a finite number of critical points.
Under the assumption of local linking near 0, a notion introduced by S. Li and J. Liu [Kexue Tongbao 29, 1025-1027 (1984)], the author proves an existence result of a non-zero critical point, in terms of the critical groups. A similar result is also obtained using a weaker local linking condition in the case where \(X\) is a Hilbert space and \(dF\) is assumed to be Lipschitz in a neighborhood of 0. Applications to elliptic boundary values problems are given.


58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
Full Text: DOI EuDML Link