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Minimax principles for a class of lower semicontinuous functions and applications to nonlinear boundary value problems. (English) Zbl 0631.58002
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, ψ : X(-,] a convex, proper (i.e. ψ¬) and lower semicontinuous function, and ϕC 1 (X,). It is also supposed that the following compactness condition is satisfied: If (u n ) is a sequence such that I(u n ) (=ϕ(u n )+ψ(u n ))c and ϕ ' (u n )+ψ(u n )z n where z n 0, then (u n ) possesses a convergent subsequence. Here ψ is the subdifferential of ψ.

The author gives the ideas of proofs for some criteria for existence of critical points. For example: Theorem 3. Let I(0)=0 and ϕ, ψ are even. Assume also that (i) there exists a subspace X 1 of X of finite codimension, and numbers α,ρ>0 such that I| B ρ X 1 α, (ii) there is a finite dimensional subspace X 2 of X, dim X 2 >codimX 1 , such that I(u) as u, uX 2 . Then I has at least dim X 2 -codimX 1 distinct pairs of nontrivial critical points.

Application to nonlinear boundary value problems are also given. Theorem 6. Let f(t) be an odd C 1 function such that f(0)=f ' (0)=0, f is nondecreasing and f’(t) as |t|. If λ>λ k , then the boundary value problem (-Δ) m u+f(u)=λ(u) in Ω, uH 0 m has at least k distinct pairs of nontrivial solutions u such that uf(u)L 1 . Here λ k is kth eigenvalue of (-Δ) m in H 0 m .

Reviewer: P.Kucment
MSC:
58E05Abstract critical point theory
35J65Nonlinear boundary value problems for linear elliptic equations
35J20Second order elliptic equations, variational methods