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Multiplicity results for variational problems and applications. (English) Zbl 0634.35002
Let M be a compact riemannian manifold or an open bounded set in $${\mathbb{R}}^ n,$$ H$$L$$ 2(M) a Hilbert space and $$f\in C$$ 1(H,$${\mathbb{R}})$$ a functional of the form $$f(u)=(1/2)<Lu,\quad u>-\psi (u),$$ where $$\psi$$ ’ is a Nemicki operator and L is linear and self-adjoint. First we give general conditions for the compactness of $$\psi$$ ’, which allows to generalize results of several authors on existence and multiplicity of critical points of F. Using other techniques we obtain also results in a weak sublinear context. Finally we give applications to the existence of non trivial solutions of elliptic equations and periodic, time dependent, solutions for the d’Alembert operator on spheres.

##### MSC:
 35A15 Variational methods applied to PDEs 49J20 Existence theories for optimal control problems involving partial differential equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces