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Nonlinear problems with strong resonance at infinity: an abstract theorem and applications*

Published online by Cambridge University Press:  14 November 2011

A. Capozzi  and
A. Salvatore
A. Capozzi
Affiliation:
Dipartimento di Matematic, Università degli Studi di Bari, Bari, Italy
A. Salvatore
Affiliation:
S.I.S.S.A., Trieste, Italy
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Synopsis

In this paper, we consider the equation

where A is a linear operator, N = ψ′ with ψ ∈ C1(E, R), and E is an Hilbert space. We suppose that N has a derivative at infinity N′(∞) and that 0 belongs to the spectrum of A–N′(∞). We prove an abstract theorem for multiplicity of solutions for the above equation. We then apply this theorem to the study of periodic solutions of Hamiltonian systems and of semilinear wave equations when the period is prescribed.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 99 , Issue 3-4 , 1985 , pp. 333 - 345
Copyright
Copyright © Royal Society of Edinburgh 1985

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References

1Aman, H. and Zehnder, E.. Periodic solutions of asymptotically linear Hamiltonian systems. Manuscripta Math. 32 (1980), 149–189.CrossRefGoogle Scholar
2Bartolo, P., Benci, V. and Fortunato, D.Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity. J. Nonlinear Anal. T.M.A. 7 (1983), 981–1012.CrossRefGoogle Scholar
3Basile, N. and Mininni, M.. Multiple periodic solutions for a semilinear wave equation with nonmonotone nonlinearity. J. Nonlinear Anal. T.M.A., to appear.Google Scholar
4Benci, V.. A geometrical index for the group S1 and some applications to the study of periodic solutions of ordinary differential equations. Comm. Pure Appl. Math. 34 (1981), 393–432.CrossRefGoogle Scholar
5Benci, V.. On the critical point theory for indefinite functionals in the presence of symmetries. Trans. Amer. Math. Soc. 274 (1982), 533–572.Google Scholar
6Benci, V., Capozzi, A. and Fortunato, D.. Periodic solutions of Hamiltonian systems of prescribed period. Math. Research Center, Technical Summary Report n. 2508 (Univ. of Wisconsin-Madison, 1983).Google Scholar
7Benci, V., Capozzi, A. and Fortunato, D.. On asymptotically quadratic Hamiltonians systems. J. Nonlinear Anal. T.M.A. 7 (1983), 929–931.CrossRefGoogle Scholar
8Benci, V. and Fortunato, D.. The dual method in critical point theory. Multiplicity results for indefinite functionals. Ann. Mat. Pura Appl. 32 (1982), 215–242.Google Scholar
9Brezis, H.. Periodic solutions of nonlinear vibrating strings and duality principles. Proc. AMS symposium on the mathematical heritage of H. Poincaré (Bloomington, April 1980), and Bull. Amer. Math. Soc. 8 (1983), 409–426.Google Scholar
10Brezis, H., Coron, J. M. and Nirenberg, L.. Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz. Comm. Pure Appl. Math. 33 (1980), 667–689.CrossRefGoogle Scholar
11Capozzi, A. and Salvatore, A.. Periodic solutions for nonlinear problems with strong resonance atinfinity. Comm. Math. Univ. Carolin. 23, 3 (1982), 415–425.Google Scholar
12Capozzi, A. and Salvatore, A.. A note on a class of autonomous Hamiltonian systems with strong resonance at infinity. Lecture Notes in Mathematics 1017, 132–139 (Berlin: Springer, 1983).Google Scholar
13Coron, J. M.. Periodic solutions of a nonlinear wave equation without assumptions of monotonicity. Math. Ann. 252 (1983), 273–285.CrossRefGoogle Scholar
14Rabinowitz, P. H.. Free vibrations for a semilinear wave equation. Comm. Pure Appl. Math. 31 (1978), 31–68.CrossRefGoogle Scholar
15Rabinowitz, P. H.. Periodic solutions of Hamiltonian systems. Comm. Pure Appl. Math. 31 (1978), 157–184.Google Scholar
16Rabinowitz, P. H.. Periodic solutions of Hamiltonian systems: a survey. SIAM J. Math. Anal. 13 (1982), 343–352.CrossRefGoogle Scholar
17Rabinowitz, P. H.. Large amplitude time periodic solutions of a semilinear wave equation. Math. Research Center Technical Summary Report 2458 (University of Wisconsin-Madison, 1982).Google Scholar
18Thews, K.. Nontrivial solutions of elliptic equations at resonance. Proc. Roy. Soc. Edinburgh Sect. A 85 (1980), 119–129.Google Scholar
19Kato, T.. Perturbation theory for linear operators (Berlin: Springer, 1976).Google Scholar