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Multiplicity of the solutions of some problems of variational type and applications. (Italian. English summary) Zbl 0607.58014
Differential problems and theory of critical points, Meet. Bari/Italy 1984, 65-76 (1984).
[For the entire collection see Zbl 0594.00007.]
Let M be a compact Riemannian manifold or an open bounded set in \({\mathbb{R}}^ n\), \(H\hookrightarrow \hookrightarrow L^ 2(M)\) a Hilbert space and \(f\in C^ 1(H,{\mathbb{R}})\) a functional of the form \(f(u)=<Lu,u>- \psi (u)\), 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 nontrivial solutions of elliptic equations and periodic, time dependent, solutions for the d’Alembert operator on spheres.
MSC:
58E30 Variational principles in infinite-dimensional spaces
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
47J05 Equations involving nonlinear operators (general)
58J99 Partial differential equations on manifolds; differential operators