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Multiple critical points for a class of periodic lower semicontinuous functionals. (English) Zbl 1281.34024
The authors develope a Lusternik-Schnirelman-type theory for a class of variational problems motivated by the search for periodic solutions of systems of the type \[ (\phi(u'))' = \nabla_u F(t,u) + h(t), \] where \(\phi = \nabla \Phi : \overline B(a) \to \mathbb R^N\) belongs to some class of homeomorphisms, \(F\) is \(\omega_i\)-periodic with respect to \(u_i\) (\(1 \leq i \leq N\)) and \(\int_0^T h(t)\,dt = 0\). The difficulty of the problem lies in the fact that the corresponding action functional, which is invariant for a suitable group, is defined only for functions \(u\) such that \(\|u'\|_\infty \leq 1\). To overcome this difficulty, the authors make use of Szulkin’s minimax principle for lower semicontinuous functions and prove an adapted deformation lemma. The result is applied to a new proof of the existence of at least \(N +1\) geometrically distinct periodic solutions for the system above.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
47J30 Variational methods involving nonlinear operators
34C25 Periodic solutions to ordinary differential equations
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