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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.

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