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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.

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