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Multiple critical points of invariant functionals and applications. (English) Zbl 0628.35037

Some multiplicity results for an abstract equation (1) \(Lu=\nabla F(u)\), in a closed subspace of \(L^ 2(\Omega,{\mathbb{R}}^ N)\) where \(F\in C^ 1({\mathbb{R}}^ N,{\mathbb{R}})\) is strictly convex function and L is unbounded selfadjoint operator are proved in this work. The solvability of (1) is treated by Lusternik-Schnirelman theory arguments and “dual action” principle of Clarke and Ekeland, provided (1) is equivariant with respect to some group action. Applications are made to nonlinear Dirichlet problems and periodic solutions for nonlinear wave equations.
Reviewer: S.Tersian

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35L70 Second-order nonlinear hyperbolic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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[1] Benci, V., A geometrical index for the group \(S^1\) and some applications to the study of periodic solutions of ordinary differential equations, Communs pure appl. Math., 34, 393-432 (1981) · Zbl 0447.34040
[2] Benci, V., On critical point theory for indefinite functionals in the presence of symmetries, Trans. Am. math. Soc., 274, 533-572 (1982) · Zbl 0504.58014
[3] Castro, A.; Lazer, A. C., Critical point theory and the number of solutions of a nonlinear Dirichlet problem, Annali Mat. pura appl., 120, 113-137 (1979) · Zbl 0426.35038
[4] Clark, D. C., A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J., 22, 65-74 (1972) · Zbl 0228.58006
[5] Clarke, F. H.; Ekeland, I., Hamiltonian trajectories with prescribed minimal period, Communs pure appl. Math., 33, 103-116 (1980) · Zbl 0403.70016
[6] Coron, J. M., Periodic solutions of a nonlinear wave equation withour assumption of monotonicity, Math. Annln, 262, 273-285 (1983) · Zbl 0489.35061
[7] Costa, D. G., An application of the Lusternik-Schnirelman theory, Proceedings of 15th Brazilian Seminar of Analysis, 211-223 (1982)
[8] Costa, D. G.; Willem, M., Multiple critical points of invariant functionals and applications, M.R.C. Technical Summary Report 2532 (1983) · Zbl 0628.35037
[9] Costa, D. G.; Willem, M., Point critiques multiples de fonctionnelles invariantes, C.r. hebd. Séanc. Acad. Sci. Paris, 298, 381-384 (1984) · Zbl 0581.58010
[10] Ekeland, I.; Lasry, J.-M., On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, Ann. Math., 112, 283-319 (1980) · Zbl 0449.70014
[11] Ekeland, I.; Lasry, J.-M., Duality in non-convex variational problems, (Aubin, J. P.; Bensoussan, A.; Ekeland, I., Advances in Hamiltonian Systems (1983), Birkhäuser: Birkhäuser Stuttgart) · Zbl 0541.58022
[12] Fadell, E. R.; Rabinowitz, P. H., Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math., 45, 139-174 (1978) · Zbl 0403.57001
[13] Fadell, E. R.; Husseini, S. Y.; Rabinowitz, P. H., Borsuk-Ulam theorems for arbitrary \(S^1\) actions and applications, Trans. Am. math. Soc., 274, 345-360 (1982) · Zbl 0506.58010
[15] Van Groesen, E. W.C., Applications of natural constraints in critical point theory to boundary value problems with rotation symmetry, M.R.C. Technical Summary Report 2594 (1983) · Zbl 0553.35017
[16] Watson, G. N., A Treatise on the Theory of Bessel Functions (1966), Cambridge University Press: Cambridge University Press London · Zbl 0174.36202
[17] Willem, M., Density of the range of potential operators, Proc. Am. math Soc., 83, 341-344 (1981) · Zbl 0478.49012
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