Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (1980)
- Volume: 7, Issue: 4, page 539-603
- ISSN: 0391-173X
Access Full Article
topHow to cite
topAmann, H., and Zehnder, E.. "Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 7.4 (1980): 539-603. <http://eudml.org/doc/83846>.
@article{Amann1980,
author = {Amann, H., Zehnder, E.},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {nonresonance problems; elliptic boundary value problems; periodic solutions of nonlinear wave equations; Hamiltonian systems},
language = {eng},
number = {4},
pages = {539-603},
publisher = {Scuola normale superiore},
title = {Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations},
url = {http://eudml.org/doc/83846},
volume = {7},
year = {1980},
}
TY - JOUR
AU - Amann, H.
AU - Zehnder, E.
TI - Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 1980
PB - Scuola normale superiore
VL - 7
IS - 4
SP - 539
EP - 603
LA - eng
KW - nonresonance problems; elliptic boundary value problems; periodic solutions of nonlinear wave equations; Hamiltonian systems
UR - http://eudml.org/doc/83846
ER -
References
top- [1] S. Ahmad - A.C. Lazer - J.L. Paul, Elementary critical point theory and perturbations of elliptic boundary value problems at resonance, Indiana Univ. Math. I., 25 (1976), pp. 933-944. Zbl0351.35036MR427825
- [2] H. Amann, Saddle points and multiple solutions of differential equations, Math. Z., 169 (1979), pp. 127-166. Zbl0414.47042MR550724
- [2a] H. Amann - E. Zehnder, Multiple periodic solutions for a class of nonlinear autonomous wave equations, to appear in Houston J. of Math. Zbl0481.35061MR638944
- [2b] H. Amann - E. Zehnder, Periodic solutions of asymptotically linear Hamiltonian equations, to appear.
- [3] A. Ambrosetti - G. Mancini, Theorems of existence and multiplicity for nonlinear elliptic problems with noninvertible linear part, Ann. Scuola Norm. Sup. Pisa, 5 (1978), pp. 15-28. Zbl0375.35024MR487001
- [4] A. Ambrosetti - G. Mancini, Existence and multiplicity results for nonlinear elliptic problems with linear part at resonance. The case of the single eigenvalue, J. Differential Equations, 28 (1978), pp. 220-245. Zbl0393.35032MR492839
- [5] A. Ambrosetti - G. Prodi, Analisi non Lineare, Scuola Norm. Sup. Pisa, 1973. Zbl0352.47001
- [6] A. Ambrosetti - P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), pp. 349-381. Zbl0273.49063MR370183
- [7] M. Berger, On a family of periodic solutions of Hamiltonian systems. J. Differential Equations, 10 (1971), pp. 17-26. Zbl0237.34067MR280802
- [8] M. Berger, On periodic solutions of second order Hamiltonian systems, J. Math. Anal. Appl., 29 (1970), pp. 512-522. Zbl0206.09904MR257470
- [9] M. Berger, Periodic solutions of second order dynamical systems and isoperimetric variational problems, Amer. J. Math., 93 (1971), pp. 1-10. Zbl0222.34042MR276848
- [10] D. Blackmore, On local normal forms for diffeomorphisms and flows, Notices Amer. Math. Soc., (1977), A-313.
- [11] M. Bottkol, Bifurcation of periodic orbits on manifolds, and Hamiltonian systems. Thesis N.Y.U. (1978). MR440615
- [12] N. Bourgoyne - R. Cushman, Normal forms for real linear Hamiltonian systems, in Lie Groups : History, Frontiers, and Applications, vol. VII, editors : C. Martin and R. Hermann, Math. Sci. Press, Brookline Mass., 1977, pp. 483-529. MR488135
- [13] H. Brezis - L. Nirenberg, Forced vibrations for a nonlinear wave equation. Comm. Pure Appl. Math., 31 (1978), pp. 1-30. Zbl0378.35040MR470377
- [14] H. Brezis - L. Nirenberg, Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Scuola Norm. Sup. Pisa, Ser. IV, 5 (1978), pp. 225-326. Zbl0386.47035MR513090
- [15] A. Castro - A.C. Lazer, Critical point theory and the number of solutions of a nonlinear Dirichlet problem, Ann. Mat. pura e appl., (IV) 120 (1979), pp.113-137. Zbl0426.35038MR551063
- [15a] D.C. Clark, Periodic solutions of variational systems of ordinary differential equations, J. Differential Equations, 28 (1978), pp. 354-368. Zbl0369.34019MR492562
- [16] F.H. Clarke, Periodic solutions to Hamiltonian inclusions, Preprint, Vancouver, 1978. MR614215
- [17] F.H. Clarke - I. Ekeland, Hamiltonian trajectories having prescribed minimal period, Cahiers de mathématiques de la Decision N. 7822, Université de Paris IX (1978).
- [18] C.C. Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Math., 38 (1978), AMS, Providence, R.I. Zbl0397.34056MR511133
- [19] C.C. Conley - R.W. Easton, Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc., 158 (1971), pp. 35-61. Zbl0223.58011MR279830
- [20] J.M. Coron, Résolution de l'équation Au + Bu = f où A est linéaire autoadjoint et B déduit d'un potential convexe, C. R. Acad. Sci. Paris Sér. A-B, 288 (1979), pp. A805-A808. Zbl0398.47039MR535640
- [21] I. Ekeland, Periodic solutions of Hamiltonian equations and a theorem of P. Rabinowitz, Cahiers de Mathématiques de la Decison N. 7827, Université de Paris IX (1978). MR555325
- [22] I. Ekeland - J.-M. Lasry, Nombre de solutions périodiques des équations de Hamilton, Preprint, Paris (1978). Zbl0397.34049MR525925
- [23] I. Ekeland - R. Temam, Analyse convexe et problèmes variationels, Dunod, Paris (1974). Zbl0281.49001MR463993
- [24] E.R. Fadell - P. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math., 45 (1978), pp. 139-174. Zbl0403.57001MR478189
- [25] A. Friedman, Partial Differential equations, Holt, Rinehart and Winston, Inc., New York, 1969. Zbl0224.35002MR445088
- [26] P. Hess, Solutions nontriviales d'un problème aux limites elliptique non linéaire, C.R. Acad. Sci. Paris, to appear.
- [27] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966. Zbl0148.12601MR203473
- [28] A. Liapunoff, Problème générale de la stabilité du mouvement, Ann. Fac. Sci. Toulouse (2) (1907), pp. 203-474. Zbl38.0738.07MR21186JFM38.0738.07
- [29] J.L. Lions - E. Magenes, Non-Homogeneous Boundary Value Problems and Applications - I, Springer-Verlag, Berlin-Heidelberg -New York, 1972. Zbl0223.35039
- [30] G. Mancini, Periodic solutions of some semilinear autonomious wave equations, Boll. Un. Mat. Ital., (5), 15-B (1978), pp. 649-672. Zbl0393.35005MR518496
- [31] J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Comm. Pure Appl. Math., 29 (1976), pp. 727-747. Zbl0346.34024MR426052
- [31a] J. Moser, New aspects in the theory of stability of Hamiltonian systems, Comm. Pure Appl. Math., 11 (1958), pp. 81-114. Zbl0082.40801MR96872
- [32] K.J. Palmer, Linearization near an integral manifold, J. Math. Anal. Appl., 51 (1975), pp. 243-255. Zbl0311.34056MR374564
- [33] P. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations, Comm. Pure Appl. Math., 20 (1967), pp. 145-205. Zbl0152.10003MR206507
- [34] P. Rabinowitz, Some minimax theorems and applications to nonlinear partial differential equations, Nonlinear Analysis, A Collection of Papers in Honor of Erich H. Rothe, pp. 161-177, Academic Press, 1978. Zbl0466.58015MR501092
- [35] P. Rabinowitz, Free vibrations for a semilinear wave equation, Comm. Pure Appl. Math., 31 (1978), pp. 31-68. Zbl0341.35051MR470378
- [36] P. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), pp. 157-184. Zbl0358.70014MR467823
- [37] R.T. Rockafellar, Monotone operators associated with saddle-functions and minimax theorems, in Nonlinear Functional Analysis, Part I, Proc. Symp. Pure Math., 18 (1970), pp. 241-250. Zbl0237.47030MR285942
- [38] A.N. Shoshitaishvili, Bifurcations of topological type at singular points of parametrized vector fields, Functional Anal. Appl., 6 (1972), pp. 169-170. Zbl0274.34028
- [39] C.L. Siegel - J. Moser, Lectures on Celestial Mechanics, Springer-Verlag, New York, 1971. Zbl0312.70017MR502448
- [40] E.H. Spanier, Algebraic Topology, McGraw-Hill Book Co., Inc., New York, 1966. Zbl0145.43303MR210112
- [41] K. Thews, A reduction method for some nonlinear Dirichlet problems, J. Nonlinear Analysis. Theory, Methods, Appl., 3 (1979), pp. 795-813. Zbl0419.35027MR548953
- [42] K. Thews, Nontrivial solutions of elliptic equations at resonance, Proc. Roy. Soc. Edinburgh, 85A (1980), pp. 119-129. Zbl0431.35040MR566069
- [43] O. Vejvoda, Periodic solutions of nonlinear partial differential equations of evolution, Proc. Symp. Diff. Eqs. Appl. at Bratislava, 1966, Acta Fac. Rerum Natur. Univ. Comenian. Math., 17 (1967), pp. 293-300. Zbl0183.10401MR249793
- [44] A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math., 108 (1978), pp. 507-518. Zbl0403.58001MR512430
- [45] A. Weinstein, Bifurcations and Hamilton's principle, Math. Z., 159 (1978), pp. 235-248. Zbl0366.58003MR501163
- [46] A. Weinstein, Normal modes for nonlinear Hamiltonian systems, Invent. Math., 20 (1973), pp. 47-57. Zbl0264.70020MR328222
- [47] A. Weinstein, Lagrangian submanifolds and Hamiltonian systems, Ann. of Math., 98 (1973), pp. 377-410. Zbl0271.58008MR331428
- [48] G.W. Whitehead, Elements of Homotopy Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1978. Zbl0406.55001MR516508
- [49] J. Williams, On the algebraic problem concerning the normal form of a linear dynamical system, Amer. J. Math., 58 (1936), pp. 141-163. MR1507138JFM62.1795.10
Citations in EuDML Documents
top- Ivar Ekeland, Une théorie de Morse pour les systèmes hamiltoniens convexes
- Paul H. Rabinowitz, On nontrivial solutions of a semilinear wave equation
- Jaroslav Jaroš, On the unique solvability of semi-linear elliptic systems
- K. C. Chang, J. Q. Liu, M. J. Liu, Nontrivial periodic solutions for strong resonance hamiltonian systems
- A. Salvatore, Periodic solutions of asymptotically linear systems without symmetry
- Yiming Long, The minimal period problem of classical hamiltonian systems with even potentials
- Antonio Marino, Claudio Saccon, Some variational theorems of mixed type and elliptic problems with jumping nonlinearities
- Henri Berestycki, Solutions périodiques de systèmes hamiltoniens
- Antonio Ambrosetti, Critical points and nonlinear variational problems
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.