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A three critical points theorem revisited. (English) Zbl 1214.47079

Let X be a reflexive Banach space, I an interval; Φ:X a sequentially weakly lower semicontinuous C 1 functional, bounded on each bounded subset of X, whose derivative admits a continuous inverse on X * ; J:X a C 1 functional with compact derivative. Assume that lim x (Φ(x)+λJ(x))=+ for all λI, and there exists ρ such that sup λI inf xX (Φ(x)+λ(J(x)+ρ))<inf xX sup λI (Φ(x)+λ(J(x)+ρ)).

Then there exists a subset AI, A, and r>0 with the following property: for every λA 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)=0 has at least three solutions in X whose norms are less than r.

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