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On a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems. (English) Zbl 1014.49004

The authors prove a general theorem for the existence of at least three critical points for the functional being the sum of locally Lipschitz and convex, proper and lower semicontinuous functions on a separable, reflexive Banach space, depending on a real parameter λ and satisfying some additional continuity, compactness and growth conditions. The paper generalizes the result of [B. Ricceri, “On a three critical points theorem”, Arch. Math. 75, No. 3, 220-226 (2000; Zbl 0979.35040)].

Finally two applications of the above result are shown: one to a variational-hemivariational inequality and the other to an elliptic inequality problem with highly discontinuous nonlinearities.

49J40Variational methods including variational inequalities
58E05Abstract critical point theory
35J20Second order elliptic equations, variational methods
49J35Minimax problems (existence)