Nonlinear functional analysis and its applications, Proc. NATO Adv. Study Inst., Maratea/Italy 1985, NATO ASI Ser., Ser. C 173, 393-399 (1986).
[For the entire collection see Zbl 0583.00019.]
Let X be a real Banach space, : a convex, proper (i.e. and lower semicontinuous function, and . It is also supposed that the following compactness condition is satisfied: If is a sequence such that and where , then possesses a convergent subsequence. Here is the subdifferential of .
The author gives the ideas of proofs for some criteria for existence of critical points. For example: Theorem 3. Let and , are even. Assume also that (i) there exists a subspace of X of finite codimension, and numbers such that , (ii) there is a finite dimensional subspace of X, dim , such that I(u) as , . Then I has at least dim distinct pairs of nontrivial critical points.
Application to nonlinear boundary value problems are also given. Theorem 6. Let f(t) be an odd function such that , f is nondecreasing and f’(t) as . If , then the boundary value problem in , has at least k distinct pairs of nontrivial solutions u such that . Here is kth eigenvalue of in .