Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations

H. Amann; E. Zehnder

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (1980)

  • Volume: 7, Issue: 4, page 539-603
  • ISSN: 0391-173X

How to cite

top

Amann, 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. [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. [2] H. Amann, Saddle points and multiple solutions of differential equations, Math. Z., 169 (1979), pp. 127-166. Zbl0414.47042MR550724
  3. [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
  4. [2b] H. Amann - E. Zehnder, Periodic solutions of asymptotically linear Hamiltonian equations, to appear. 
  5. [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
  6. [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
  7. [5] A. Ambrosetti - G. Prodi, Analisi non Lineare, Scuola Norm. Sup. Pisa, 1973. Zbl0352.47001
  8. [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
  9. [7] M. Berger, On a family of periodic solutions of Hamiltonian systems. J. Differential Equations, 10 (1971), pp. 17-26. Zbl0237.34067MR280802
  10. [8] M. Berger, On periodic solutions of second order Hamiltonian systems, J. Math. Anal. Appl., 29 (1970), pp. 512-522. Zbl0206.09904MR257470
  11. [9] M. Berger, Periodic solutions of second order dynamical systems and isoperimetric variational problems, Amer. J. Math., 93 (1971), pp. 1-10. Zbl0222.34042MR276848
  12. [10] D. Blackmore, On local normal forms for diffeomorphisms and flows, Notices Amer. Math. Soc., (1977), A-313. 
  13. [11] M. Bottkol, Bifurcation of periodic orbits on manifolds, and Hamiltonian systems. Thesis N.Y.U. (1978). MR440615
  14. [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
  15. [13] H. Brezis - L. Nirenberg, Forced vibrations for a nonlinear wave equation. Comm. Pure Appl. Math., 31 (1978), pp. 1-30. Zbl0378.35040MR470377
  16. [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
  17. [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
  18. [15a] D.C. Clark, Periodic solutions of variational systems of ordinary differential equations, J. Differential Equations, 28 (1978), pp. 354-368. Zbl0369.34019MR492562
  19. [16] F.H. Clarke, Periodic solutions to Hamiltonian inclusions, Preprint, Vancouver, 1978. MR614215
  20. [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). 
  21. [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
  22. [19] C.C. Conley - R.W. Easton, Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc., 158 (1971), pp. 35-61. Zbl0223.58011MR279830
  23. [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
  24. [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
  25. [22] I. Ekeland - J.-M. Lasry, Nombre de solutions périodiques des équations de Hamilton, Preprint, Paris (1978). Zbl0397.34049MR525925
  26. [23] I. Ekeland - R. Temam, Analyse convexe et problèmes variationels, Dunod, Paris (1974). Zbl0281.49001MR463993
  27. [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
  28. [25] A. Friedman, Partial Differential equations, Holt, Rinehart and Winston, Inc., New York, 1969. Zbl0224.35002MR445088
  29. [26] P. Hess, Solutions nontriviales d'un problème aux limites elliptique non linéaire, C.R. Acad. Sci. Paris, to appear. 
  30. [27] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966. Zbl0148.12601MR203473
  31. [28] A. Liapunoff, Problème générale de la stabilité du mouvement, Ann. Fac. Sci. Toulouse (2) (1907), pp. 203-474. Zbl38.0738.07MR21186JFM38.0738.07
  32. [29] J.L. Lions - E. Magenes, Non-Homogeneous Boundary Value Problems and Applications - I, Springer-Verlag, Berlin-Heidelberg -New York, 1972. Zbl0223.35039
  33. [30] G. Mancini, Periodic solutions of some semilinear autonomious wave equations, Boll. Un. Mat. Ital., (5), 15-B (1978), pp. 649-672. Zbl0393.35005MR518496
  34. [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
  35. [31a] J. Moser, New aspects in the theory of stability of Hamiltonian systems, Comm. Pure Appl. Math., 11 (1958), pp. 81-114. Zbl0082.40801MR96872
  36. [32] K.J. Palmer, Linearization near an integral manifold, J. Math. Anal. Appl., 51 (1975), pp. 243-255. Zbl0311.34056MR374564
  37. [33] P. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations, Comm. Pure Appl. Math., 20 (1967), pp. 145-205. Zbl0152.10003MR206507
  38. [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
  39. [35] P. Rabinowitz, Free vibrations for a semilinear wave equation, Comm. Pure Appl. Math., 31 (1978), pp. 31-68. Zbl0341.35051MR470378
  40. [36] P. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), pp. 157-184. Zbl0358.70014MR467823
  41. [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
  42. [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
  43. [39] C.L. Siegel - J. Moser, Lectures on Celestial Mechanics, Springer-Verlag, New York, 1971. Zbl0312.70017MR502448
  44. [40] E.H. Spanier, Algebraic Topology, McGraw-Hill Book Co., Inc., New York, 1966. Zbl0145.43303MR210112
  45. [41] K. Thews, A reduction method for some nonlinear Dirichlet problems, J. Nonlinear Analysis. Theory, Methods, Appl., 3 (1979), pp. 795-813. Zbl0419.35027MR548953
  46. [42] K. Thews, Nontrivial solutions of elliptic equations at resonance, Proc. Roy. Soc. Edinburgh, 85A (1980), pp. 119-129. Zbl0431.35040MR566069
  47. [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
  48. [44] A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math., 108 (1978), pp. 507-518. Zbl0403.58001MR512430
  49. [45] A. Weinstein, Bifurcations and Hamilton's principle, Math. Z., 159 (1978), pp. 235-248. Zbl0366.58003MR501163
  50. [46] A. Weinstein, Normal modes for nonlinear Hamiltonian systems, Invent. Math., 20 (1973), pp. 47-57. Zbl0264.70020MR328222
  51. [47] A. Weinstein, Lagrangian submanifolds and Hamiltonian systems, Ann. of Math., 98 (1973), pp. 377-410. Zbl0271.58008MR331428
  52. [48] G.W. Whitehead, Elements of Homotopy Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1978. Zbl0406.55001MR516508
  53. [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
  1. Ivar Ekeland, Une théorie de Morse pour les systèmes hamiltoniens convexes
  2. Paul H. Rabinowitz, On nontrivial solutions of a semilinear wave equation
  3. Jaroslav Jaroš, On the unique solvability of semi-linear elliptic systems
  4. K. C. Chang, J. Q. Liu, M. J. Liu, Nontrivial periodic solutions for strong resonance hamiltonian systems
  5. A. Salvatore, Periodic solutions of asymptotically linear systems without symmetry
  6. Yiming Long, The minimal period problem of classical hamiltonian systems with even potentials
  7. Antonio Marino, Claudio Saccon, Some variational theorems of mixed type and elliptic problems with jumping nonlinearities
  8. Henri Berestycki, Solutions périodiques de systèmes hamiltoniens
  9. Antonio Ambrosetti, Critical points and nonlinear variational problems

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.