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A further three critical points theorem. (English) Zbl 1187.47057

This paper establishes a new three critical points theorem for the equation

Φ ' (x)=λJ ' (x)+μΨ ' (x)

under specific hypotheses. If X is real Banach space, denote by 𝒲 X the class of functionals Φ:X possessing the following property: if {u n } is a sequence in X converging weakly to uX and lim inf n Φ(u n )Φ(u), then {u n } has a subsequence converging strongly to u.

The main result of the paper is as follows.

Theorem 1. Let X be a separable and reflexive real Banach space; I an interval; Φ:X a sequentially weakly lower semicontinuous C 1 functional from 𝒲 X , bounded on each bounded subset of X and whose derivative admits a continuous inverse on X * ; J:X a C 1 functional with compact derivative. Assume that, for each λI, the functional Φ-λJ is coercive and has a strict local, not global minimum, say x ^ λ .

Then, for each compact interval [a,b]I for which sup λ[a,b] (Φ(x ^ λ )-λJ(x ^ λ ))<+, there exists r>0 with the following property: for every λ[a,b] and every C 1 functional Ψ:X with compact derivative, there exists δ>0 such that, for each μ[0,δ], the equation

Φ ' (x)=λJ ' (x)+μΨ ' (x)

has at least three solutions whose norms are less than r.

Some applications of this result are also given.


MSC:
47J30Variational methods (nonlinear operator equations)
58E05Abstract critical point theory
49J35Minimax problems (existence)
35J60Nonlinear elliptic equations