Advertisement

Critical Point Theory and Applications to Differential Equations: A Survey

  • Paul H. Rabinowitz
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 15)

Abstract

The purpose of this paper is to survey developments in the field of critical point theory and its applications to differential equations that have occurred during the past 20–25 years. This is too broad a theme for a single survey and we will focus on three particular areas. First we will examine contributions to the minimax approach to critical point theory. In particular the Mountain Pass Theorem, the Saddle Point Theorem, and variants thereupon will be discussed in Part 1. Then in Part 2, applications of critical point theory to the existence of periodic solutions of Hamiltonian systems of differential equations will be surveyed. Many of the different questions that have been studied will be described and a variety of results will be presented. Lastly, Part 3 deals with connecting orbits of Hamiltonian systems, mainly homoclinic and heteroclinic orbits. This last set of material represents the least developed subject matter treated here and therefore it can be described more completely than the earlier topics.

Keywords

Periodic Solution Hamiltonian System Process Since Homoclinic Orbit Morse Index 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Al]
    Ambrosetti, A., Critical Points and Nonlinear Variational Problems, Mémoire 49, Soc. Math. de France, 1992.Google Scholar
  2. [A2]
    Ambrosetti, A., Elliptic equations with jumping nonlinearities, J. Math. Phys. Sci. (Madras), 1984, 1–10.Google Scholar
  3. [AB]
    Ambrosetti, A. and M. L. Bertotti, Homoclinics for second order conservative systems, Proc. Conf. in honor of L. Nirenberg, to appear.Google Scholar
  4. [ABL]
    Ambrosetti, A., V. Benci, and Y. Long, A note on the existence of multiple brake orbits, to appear in Nonlinear Analysis: TMA.Google Scholar
  5. [ACZ]
    Ambrosetti, A. and V. Coti Zelati, Multiple homoclinic orbits for a class of conservative systems, preprint.Google Scholar
  6. [AL]
    Alama, S. and Y. Y. Li, On “multibump” bound states for certain semilinear elliptic equations, preprint.Google Scholar
  7. [ALP]
    Ahmad, S. A. C. Lazer and J. L. Paul, Elementary critical point theory and perturbations of elliptic boundary value problems at resonance, Indiana Univ. Math. J. 25, 1976, 933–944.MathSciNetzbMATHGoogle Scholar
  8. [Am]
    Amann, H., Saddle points and multiple solutions of differential equations, Math. Z. 196, 1979, 127–166.MathSciNetGoogle Scholar
  9. [AM1]
    Ambrosetti, A. and G. Mancini, On a theorem by Ekeland and Lasry concerning the number of periodic Hamiltonian trajectories, J. Differential Equations 43, 1981, 1–6.MathSciNetGoogle Scholar
  10. [AM2]
    Ambrosetti, A. and G. Mancini, Solutions of minimal period for a class of convex Hamiltonian systems, Math. Ann. 255, 1981, 405–421.MathSciNetzbMATHGoogle Scholar
  11. [AP]
    Ambrosetti, A. and G. Prodi, A Primer of Nonlinear Analysis, Cambridge Studies in Advanced Math. 34, 1993.Google Scholar
  12. [AR]
    Ambrosetti, A. and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14, 1973, 349–381.MathSciNetzbMATHGoogle Scholar
  13. [AZ1]
    Amann, H. and E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7, 1980, 539–603.MathSciNetzbMATHGoogle Scholar
  14. [AZ2]
    Amann, H. and E. Zehnder, Periodic solutions of asymptotically linear Hamiltonian systems, Manus. Math. 32, 1980, 149–189.MathSciNetzbMATHGoogle Scholar
  15. [Ba]
    Bahri, A., Critical Points at Infinity in some Variational Problems, Wiley, 1989.zbMATHGoogle Scholar
  16. [B]
    Berestycki, H., Solutions periodiques de systemes hamiltoniens, Sem. Bourbaki 1982/83, expose 603, Astérisque, 1983, 105–128.Google Scholar
  17. [BaL1]
    Bahri, A. and P. L. Lions, Remarks on the variational theory of critical points and applications, C. R. Acad. Sc. Paris, Sci. I. Math. 301, 1985, 145–148.MathSciNetzbMATHGoogle Scholar
  18. [BaL2]
    Bahri, A. and P. L. Lions, Solutions of superlinear elliptic equations and their Morse indices, Comm. Pure Appl. Math. 45, 1992, 1205–1215.MathSciNetzbMATHGoogle Scholar
  19. [BaR]
    Bahri, A. and P. Rabinowitz, Periodic solutions of Hamiltonian systems of 3-body type, Ann. IHP-Analyse Nonlin. 8, 1991, 561–649.MathSciNetzbMATHGoogle Scholar
  20. [BB1]
    Bahri, A. and H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc. 267, 1981, 1–32.MathSciNetzbMATHGoogle Scholar
  21. [BB2]
    Bahri, A. and H. Berestycki, Forced vibrations of superquadratic Hamiltonian systems, Acta. Math. 152, 1984, 143–197.MathSciNetzbMATHGoogle Scholar
  22. [BB3]
    Bahri, A. and H. Berestycki, Existence of forced oscillations for some nonlinear differential equations, Comm. Pure Appl. Math. 37, 1984, 403–442.MathSciNetzbMATHGoogle Scholar
  23. [BC]
    Bahri, A. and J-M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math. 41, 1981, 253–294.MathSciNetGoogle Scholar
  24. [BCl]
    Bartsch, T. and M. Clapp, The compact category and multiple periodic solutions of Hamiltonian systems on symmetric starshaped energy surfaces, Math. Ann. 293, 1992, 523–542.MathSciNetzbMATHGoogle Scholar
  25. [BCN]
    Brezis, H., J. M. Coron, and L. Nirenbergy, Free vibrations of a semilinear wave equation and a theorem of Rabinowitz, Comm. Pure Appl. Math. 33, 1980, 667–689.MathSciNetzbMATHGoogle Scholar
  26. [Be]
    Benci, V., A geometrical index for the group S 1 and some applications to the research of periodic solutions of O.D.E.’s, Comm. Pure Appl. Math. 34, 1981, 393–432.MathSciNetzbMATHGoogle Scholar
  27. [BF]
    Benci, V. and D. Fortunato, ‘Birkhoff-Lewis’ type result for a class of Hamiltonian systems, Manus. Math. 59, 1987, 441–456.MathSciNetzbMATHGoogle Scholar
  28. [BFG]
    Benci, V., D. Fortunato, and F. Giannoni, On the existence of periodic trajectories in static Lorentz manifolds with nonsmooth boundary, Nonlinear Analysis: A tribute in honor of G. Prodi, Quad. del. Sc. Norm. Sup. Pisa, A. Ambrosetti and A. Marino ed., 1991.Google Scholar
  29. [Bg]
    Berger, M. S., Nonlinearity and functional analysis, Academic Press, New York, 1978.Google Scholar
  30. [BG]
    Benci, V. and F. Giannoni, Homoclinic orbits on compact manifolds, J. Math. Anal. and Appl. 157, 1991, 568–576.MathSciNetzbMATHGoogle Scholar
  31. [BHR]
    Benci, V., H. Hofer, and P. H. Rabinowitz, A remark on a priori bounds and existence for periodic solutions of Hamiltonian systems, Periodic Solutions of Hamiltonian Systems and Related Topics, P. H. Rabinowitz et. al. eds., D. Reidel, 1987, 85–88.Google Scholar
  32. [Bi]
    Birkhoff, G. D., Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc. 18, 1917, 199–300.MathSciNetzbMATHGoogle Scholar
  33. [BLMR]
    Berestycki, H., J. M. Lasry, G. Mancini and B. Ruf, Existence of multiple periodic orbits on star-shaped Hamiltonian surfaces, Comm. Pure Appl. Math. 38, 1985, 253–290.MathSciNetzbMATHGoogle Scholar
  34. [BN1]
    Brezis, H. and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36, 1982, 437–477.MathSciNetGoogle Scholar
  35. [BN2]
    Brezis, H. and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. 44, 1991, 939–963.MathSciNetzbMATHGoogle Scholar
  36. [Bol]
    Bolotin, S., Libration motions of natural dynamical systems, Vestnik Moskov. Univ. Ser. I Matem. Mekh. 6, 1978, 72–77.MathSciNetGoogle Scholar
  37. [Bo2]
    Bolotin, S., Existence of homoclinic motions, Vestnik Moskov. Univ. Ser I, Matem. Mekh 6, 1983, 98–103.MathSciNetGoogle Scholar
  38. [Bo3]
    Bolotin, S., Homoclinic orbits to invariant tori of symplectic diffeomorphisms and Hamiltonian systems, preprint.Google Scholar
  39. [Br]
    Brezis, H., Periodic solutions of nonlinear vibrating strings and duality principles, Bull. Amer. Math. Soc. 8, 1983, 409–426.MathSciNetzbMATHGoogle Scholar
  40. [BrC]
    Brezis, H. and J. M. Coron, Periodic solutions of nonlinear wave equations and Hamiltonian systems, Amer. J. Math. 103, 1981, 559–570.MathSciNetzbMATHGoogle Scholar
  41. [BR1]
    Benci, V. and P. H. Rabinowitz, Critical point theorems for indefinite Junctionals, Invent. Math. 52, 1979, 241–273.MathSciNetzbMATHGoogle Scholar
  42. [BR2]
    Benci, V. and P. H. Rabinowitz, A priori bounds for periodic solutions of Hamiltonian systems, Ergodic Th. and Dynam. Sys. 8, 1988, 27–31.MathSciNetGoogle Scholar
  43. [Bs]
    Bessi, H., preprint.Google Scholar
  44. [Bw1]
    Browder, F. E., Infinite dimensional manifolds and nonlinear eigenvalue problems, Ann. of Math. (2) 82, 1965, 459–477.MathSciNetzbMATHGoogle Scholar
  45. [Bw2]
    Browder, F. E., Nonlinear eigenvalues and group invariance, Functional Analysis and Related Fields (F. E. Browder, ed.), Springer-Verlag, Berlin and New York, 1970, 1–58.Google Scholar
  46. [C]
    Clark, D., On periodic solutions of autonomous Hamiltonian systems of ordinary differential equations, Proc. AMS 39, 1973, 579–584.zbMATHGoogle Scholar
  47. [CC]
    Conley, C. C., Isolated invariant sets and the Morse index, CBMS Regional Conf. Ser. in Math., no. 38, Amer. Math. Soc., Providence, R. I., 1978.zbMATHGoogle Scholar
  48. [Ce]
    Cerami, G., Un criterio di esistenza per i punti critici su varieta illimitate, Rend. Acad. Sci. Let. Ist. Lombardo 112, 1978, 332–336.MathSciNetGoogle Scholar
  49. [CE]
    Clarke, F. and I. Ekeland, Hamiltonian trajectories having prescribed minimal period, Comm. Pure Appl. Math. 33, 1980, 103–116.MathSciNetzbMATHGoogle Scholar
  50. [Ch1]
    Chang, K. C., Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhaüser, 1993.zbMATHGoogle Scholar
  51. [Ch2]
    Chang, K. C., Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80, 1981, 102–129.MathSciNetzbMATHGoogle Scholar
  52. [Ch3]
    Chang, K. C., On the periodic nonlinearity and the multiplicity of solutions, Nonlinear Analysis TMA 13, 1989, 527–538.zbMATHGoogle Scholar
  53. [Ch4]
    Chang, K. C., Applications of homology theory to some problems in differential equations and analysis, Nonlinear Analysis and Applications, F. E. Browder ed., Proc. Symp. Pure Math. V45 AMS, 1986, 253–262.Google Scholar
  54. [Cl1]
    Clarke, F., A classical variational principle for periodic Hamiltonian trajectories, Proc. A.M.S. 76, 1979, 186–189.zbMATHGoogle Scholar
  55. [Cl2]
    Clarke, F., Periodic solutions of Hamiltonian inclusions, J. Diff. Eq. 40, 1981, 1–6.zbMATHGoogle Scholar
  56. [CL]
    Castro, A. and A. Lazer, Applications of a maximum principle, Rev. Columbiana, Mat. 10, 1976, 141–149.MathSciNetzbMATHGoogle Scholar
  57. [CM]
    Costa, D. G. and C. A. Magalhães, A unified approach to a class of strongly indefinite functionals, preliminary version.Google Scholar
  58. [Co1]
    Coffman, C. V., A minimum-maximum principle for a class of nonlinear integral equations, J. Analyse Math. 22, 1969, 391–419.MathSciNetzbMATHGoogle Scholar
  59. [CW]
    Costa, D. and M. Willem, Multiple critical points of invariant functional and applications, J. Nonlin. Analysis 9, 1986, 843–852.MathSciNetGoogle Scholar
  60. [CZ]
    Conley, C. and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V. Arnold, Inv. Math. 73, 1983, 33–49.MathSciNetzbMATHGoogle Scholar
  61. [CZEL]
    Coti Zelati, V., I. Ekeland, and P. L. Lions, to appear.Google Scholar
  62. [CZES]
    Coti Zelati, V., I. Ekeland, and E. Séré, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann. 288, 1990, 133–160.MathSciNetzbMATHGoogle Scholar
  63. [CZR1]
    Coti Zelati, V. and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc. 4, 1992, 693–727.MathSciNetGoogle Scholar
  64. [CZR2]
    Coti Zelati, V. and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE onn, Comm. Pure Appl. Math., 45, 1992, 1217–1269.MathSciNetzbMATHGoogle Scholar
  65. [De]
    Deimling, K., Nonlinear Functional Analysis, Springer, 1985.zbMATHGoogle Scholar
  66. [Df]
    De Figueiredo, D. G., Lectures on the Ekeland Variational Principle with Applications and Detours, Tata Inst. of Fund. Research, 1989.zbMATHGoogle Scholar
  67. [E1]
    Ekeland, I., Convexity Methods in Hamiltonian Mechanics, Springer, 1990.zbMATHGoogle Scholar
  68. [EH1]
    Ekeland, I. and H. Hofer, Periodic solutions with prescribed minimal period for convex autonomous Hamiltonian systems, Invent. Math. 81, 1985, 155–188.MathSciNetzbMATHGoogle Scholar
  69. [EH2]
    Ekeland, I. and H. Hofer, Subharmonics for convex nonautonomous Hamiltonian systems, Comm. Pure Appl. Math. 40, 1987, 1–36.MathSciNetzbMATHGoogle Scholar
  70. [EH3]
    Ekeland, I. and Hofer, Symplectic topology and Hamiltonian dynamics, Math. Z. 200, 1989, 355–378.MathSciNetzbMATHGoogle Scholar
  71. [EH4]
    Ekeland, I. and H. Hofer, Convex Hamiltonian energy surfaces and their closed trajectories, Comm. Math. Phys. 113, 1987, 419–467.MathSciNetzbMATHGoogle Scholar
  72. [EL]
    Ekeland, I. and J.-M. Lasry, On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, Ann. of Math. (2) 112, 1980, 283–319.MathSciNetzbMATHGoogle Scholar
  73. [ELs]
    Ekeland, I. and L. Lassoued, Multiplicité des trajectories fermées d’un système Hamiltonian sur une hypersurface d’énergies convexe, Ann. IHP - Analyse nonlin. 4, 1987, 1–29.MathSciNetGoogle Scholar
  74. [Fl]
    Felmer, P. L., Subharmonic near an equilibrium point for Hamiltonian systems, Manu. Math. 66, 1990, 359–396.MathSciNetzbMATHGoogle Scholar
  75. [F2]
    Felmer, P. L., Periodic solutions of ‘superquadratic’ Hamiltonian systems, (to appear) J. Diff. Eq.Google Scholar
  76. [F3]
    Felmer, P. L., Periodic solutions of spatially periodic Hamiltonian systems, J. Diff. Eq. 98, 1992, 143–168.MathSciNetzbMATHGoogle Scholar
  77. [F4]
    Felmer, P. L., Multiple periodic solutions for Lagrangian systems in Tn, Nonlinear Analysis, TMA 15, 1990, 815–831.MathSciNetzbMATHGoogle Scholar
  78. [F5]
    Felmer, P. L., Heteroclinic orbits for spatially periodic Hamiltonian systems, Analyse nonlin. IHP 8, 1991, 477–497.MathSciNetzbMATHGoogle Scholar
  79. [FM]
    Fonda, A. and J. Mawhin, Multiple periodic solutions of conservative systems with periodic nonlinearity, preprint.Google Scholar
  80. [FS]
    Felmer, P. L. and E. Silva, Subharmonics near an equilibrium for some second order Hamiltonian systems, (to appear) Proc. Royal Soc. Edinburgh.Google Scholar
  81. [FW]
    Fournier, G. and M. Willem, Multiple solutions of the forced double pendulum problem, preprint.Google Scholar
  82. [Gh]
    Ghoussoub, N., Duality and Perturbation Methods in Critical Point Theory, preprint.Google Scholar
  83. [Gil]
    Giannoni, F., Periodic solutions of dynamical systems by a Saddle Point Theorem of Rabinowitz, Nonlinear Analysis TMA 13, 1989, 707–719.MathSciNetzbMATHGoogle Scholar
  84. [Gi2]
    Giannoni, F., On the existence of homoclinic orbits on Riemannian manifolds, preprint.Google Scholar
  85. [GM1]
    Girardi, M. and M. Matseu, Some results on solutions of minimal period to superquadratic Hamiltonian equations, Nonlinear Analysis TMA 7, 1983, 475–482.zbMATHGoogle Scholar
  86. [GM2]
    Girardi, M. and M. Matseu, Solutions of minimal period for a class of nonconvex Hamiltonian systems and applications to the fixed energy problem, Nonlinear Analysis TMA 10, 1986, 371–382.zbMATHGoogle Scholar
  87. [GM3]
    Girardi, M. and M. Matseu, Periodic solutions of convex Hamiltonian systems with a quadratic growth at the origin and superquadratic at infinity, Ann. Mat. Pura. Appl. 147, 1987, 21–72.MathSciNetzbMATHGoogle Scholar
  88. [GP]
    Ghoussoub, N. and D. Preiss, A general mountain pass principle for locating and classifying critical points, Ann. IHP - Analyse nonlin. 6, 1989, 321–330.MathSciNetzbMATHGoogle Scholar
  89. [Gr]
    Girardi, M., Multiple orbits for Hamiltonian systems on starshaped surfaces with symmetries, Ann. IHP - Analyse nonlin. 1, 1984, 285–294.MathSciNetzbMATHGoogle Scholar
  90. [GR]
    Giannoni, F. and P. H. Rabinowitz, On the multiplicity of homoclinic orbits on Riemannian manifolds for a class of second order Hamiltonian systems, (to appear) Nonlin. Diff. Eq. and Appl.Google Scholar
  91. [Ha]
    Han, Z., Periodic solutions of a class of dynamical systems of second order, J. Diff. Eq. 90, 1991, 408–417.zbMATHGoogle Scholar
  92. [H1]
    Hofer, H., A geometric description of the neighborhood of a critical point given by the Mountain Pass Theorem, J. London Math. Soc. (2) 31, 1985, 566–570.MathSciNetzbMATHGoogle Scholar
  93. [H2]
    Hofer, H., A note on the topological degree at a critical point of mountain pass type, Proc. AMS 49, 1984, 309–315.MathSciNetGoogle Scholar
  94. [H3]
    Hofer, H., Variational and topological methods in partially ordered spaces, Math. Ann. 261, 1982, 493–514.MathSciNetzbMATHGoogle Scholar
  95. [H4]
    Hofer, H., On strongly indefinite functionals with applications, Trans. Amer. Math. Soc. 275, 1983, 185–214.MathSciNetzbMATHGoogle Scholar
  96. [HV]
    Hofer, H. and C. Viterbo, The Weinstein conjecture in the presence of holomorphic spheres, preprint.Google Scholar
  97. [HW]
    Hofer, H. and K. Wysocki, First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems, preprint.Google Scholar
  98. [HZ1]
    Hofer, H. and E. Zehnder, A new capacity for symplectic manifolds, Analysis, et cetera (P. Rabinowitz and E. Zehnder eds.) Academic Press, 1990, 405–428.Google Scholar
  99. [HZ2]
    Hofer, H. and E. Zehnder, Periodic solutions on hypersurfaces and a result by C. Viterbo, Inv. Math. 90, 1987, 1–9.MathSciNetzbMATHGoogle Scholar
  100. [K]
    Krasnoselski, M. A., Topological methods in the theory of nonlinear integral equations, Macmillan, New York, 1964.Google Scholar
  101. [KS]
    Kirchgraber, U. and D. Stoffer, Chaotic behavior in simple dynamical systems, SIAM Review 32, 1990, 424–452.MathSciNetzbMATHGoogle Scholar
  102. [Kz]
    Kozlov, V. V., Calculus of variations in the large and classical mechanics, Russ. Math. Surv. 40, 1985, 37–71.zbMATHGoogle Scholar
  103. [La]
    Lassoued, L., Periodic solutions of a second order superquadratic system with a change in sign in the potential, J. Diff. Eq. 93, 1991, 1–18.MathSciNetzbMATHGoogle Scholar
  104. [LaS]
    Lazer, A. and S. Solimini, Nontrivial solutions of operator equations and Morse indices of critical points of Minimax type, Nonlinear Analysis TMA 12, 1988, 761–775.MathSciNetzbMATHGoogle Scholar
  105. [Li]
    Liu, J. Q., A generalized saddle point theorem, J. Diff. Eq., 1989.Google Scholar
  106. [LL]
    Li, S. and J. Liu, Some existence theorems on multiple critical points and their applications, Kexue Tonghao 17, 1984.Google Scholar
  107. [Lo1]
    Long, Y., The minimal period problem of periodic solutions for autonomous superquadratic second order Hamiltonian systems, (to appear) J. Diff. Eq.Google Scholar
  108. [Lo2]
    Long, Y., The minimal period problem for even autonomous super-quadratic second order Hamiltonian systems, preprint, 1992.Google Scholar
  109. [Lo3]
    Long, Y., Periodic solutions of perturbed superquadratic Hamiltonian systems, Ann. Sc. Norm. Sup. Pisa, Ser 4, 17, 1990, 35–77.zbMATHGoogle Scholar
  110. [Lo4]
    Long, Y., Periodic solutions of superquadratic Hamiltonian systems with bounded forcing terms, Math. Z. 203, 1990, 453–467.MathSciNetzbMATHGoogle Scholar
  111. [Lo5]
    Long, Y., Multiple solutions of perturbed superquadratic second order Hamiltonian systems, Trans. AMS 311, 1989, 749–780.zbMATHGoogle Scholar
  112. [Lo6]
    Long, Y., A Maslov-type index theory and asymptotically linear Hamiltonian systems, Dynamical Systems and Related Topics, K. Shirawa, ed., World Scientific, 1991, 333–341.Google Scholar
  113. [LPL]
    Lions, P. L., The concentration-compactness principle in the calculus of variations, Ann. IHP - Analyse nonlin. 1, 1984, 101–145.Google Scholar
  114. [LS]
    Ljusternik, L. and L. Schnirelmann, Methodes topologique dans les problémes variationnels, Hermann and Cie, Paris, 1934.Google Scholar
  115. [LW]
    Li, S. and M. Willem, Applications of local linking to critical point theory, preprint.Google Scholar
  116. [LY]
    Li, Y. Y., On -Δu = K(x)u 5 in3, (to appear) Comm. Pure Appl. Math.Google Scholar
  117. [LZ]
    Long, Y. and E. Zehnder, Morse Theory for forced oscillations of asymptotically linear Hamiltonian systems, Stochastic processes, Physics, and Geometry, S. Albevario et al. ed., World Scientific, 1990, 528–563.Google Scholar
  118. [M]
    Mather, J., Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z. 207, 1991, 169–207.MathSciNetzbMATHGoogle Scholar
  119. [Ma]
    Mawhin, J., Periodic solutions of ordinary differential equations: the Poincaré’ heritage, Differential Topology-Geometry and Related Fields, (Russian ed.), Teubner, 1985, 287–307.Google Scholar
  120. [MW]
    Mawhin, J. and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, 1989.zbMATHGoogle Scholar
  121. [N]
    Nirenberg, L., Variational and topological methods in nonlinear problems, Bull. Amer. Math. Soc. (N.S.) 4, 1981, 267–302.MathSciNetzbMATHGoogle Scholar
  122. [Ni]
    Ni, W. M., Some minimax principles and their applications in nonlinear elliptic equations, J. Analyse Math. 37, 1980, 248–275.MathSciNetzbMATHGoogle Scholar
  123. [O]
    Offin, D., A class of periodic orbits in classical mechanics, J. Diff. Eq. 66, 1987, 90–117.MathSciNetzbMATHGoogle Scholar
  124. [PI]
    Palais, R. S., Critical point theory and the minimax principle, Proc. Sympos. Pure Math., vol. 15, Amer. Math. Soc., Providence, R. I., 1970, 185–212.Google Scholar
  125. [P2]
    Palais, R. S., Lusternik-Schnirelmann theory on Banach manifolds, Topology 5, 1966, 115–132.MathSciNetzbMATHGoogle Scholar
  126. [PS]
    Palais, R. S. and S. Smale, A generalized Morse Theory, Bull. AMS 70, 1964, 165–171.MathSciNetzbMATHGoogle Scholar
  127. [PSe1]
    Pucci, P. and J. Serrin, Extensions of the mountain pass theorem, J. Funct. Analysis 59, 1984, 185–210.MathSciNetzbMATHGoogle Scholar
  128. [PSe2]
    Pucci, P. and J. Serrin, The structure of the critical set in the mountain pass theorem, Trans. Amer. Math. Soc. 299, 1987, 115–132.MathSciNetzbMATHGoogle Scholar
  129. [R1]
    Rabinowitz, P. H., Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. in Math. 65, 1986.Google Scholar
  130. [R2]
    Rabinowitz, P. H., Periodic solutions of Hamiltonian systems: A survey, SIAM J. Math. Anal 13, 1982, 343–352.MathSciNetzbMATHGoogle Scholar
  131. [R3]
    Rabinowitz, P. H., Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31, 1978, 157–184.MathSciNetGoogle Scholar
  132. [R4]
    Rabinowitz, P. H., On a theorem of Hofer and Zehnder, Periodic Solutions of Hamiltonian systems and Related Topics, P. H. Rabinowitz et. al. eds., D. Reidel, 1987, 245–253.Google Scholar
  133. [R5]
    Rabinowitz, P. H., Periodic solutions of large norm of Hamiltonian systems, J. Differential Equations 50, 1983, 33–48.MathSciNetzbMATHGoogle Scholar
  134. [R6]
    Rabinowitz, P. H., On a class of functionals invariant under an action, Trans. AMS 310, 1988, 303–311.MathSciNetzbMATHGoogle Scholar
  135. [R7]
    Rabinowitz, P. H., Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edin. 114A, 1990, 33–38.MathSciNetGoogle Scholar
  136. [R8]
    Rabinowitz, P. H., Periodic and heteroclinic orbits for a periodic Hamiltonian system, Ann. IHP-Analyse nonlin. 6, 1989, 331–346.MathSciNetzbMATHGoogle Scholar
  137. [R9]
    Rabinowitz, P. H., Some recent results on heteroclinic and other connecting orbits of Hamiltonian systems, (to appear) Proc. Conf. on Variational Methods in Hamiltonian Systems and Elliptic Equations.Google Scholar
  138. [R10]
    Rabinowitz, P. H., A variational approach to heteroclinic orbits for a class of Hamiltonian systems, Frontiers in Pure and Applied Mathematics, R. Dautray ed., 1991, 267–278.Google Scholar
  139. [R11]
    Rabinowitz, P. H., Homoclinic and heteroclinic orbits for a class of Hamiltonian systems, Calculus of Var. 1, 1993, 1–36.MathSciNetzbMATHGoogle Scholar
  140. [R12]
    Rabinowitz, P. H., Heteroclinics for a reversible Hamiltonian system, preprint, 1993.Google Scholar
  141. [RT]
    Rabinowitz, P. H. and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z. 206, 1991, 473–499.MathSciNetzbMATHGoogle Scholar
  142. [S1]
    Schwartz, J. T., Generalizing the Lusternik-Schnirelmann theory of critical points, Comm. Pure Appl. Math. 17, 1964, 307–315.MathSciNetzbMATHGoogle Scholar
  143. [S2]
    Schwartz, J. T., Nonlinear functional analysis, Gordon & Breach, New York, 1969.zbMATHGoogle Scholar
  144. [Sc]
    Schechter, M., A bounded mountain pass lemma without the (PS) condition and applications, Trans. Amer. Math. Soc. 331, 1992, 681–704.MathSciNetzbMATHGoogle Scholar
  145. [Se1]
    Séré, E., Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z. 209, 1992, 27–42.MathSciNetzbMATHGoogle Scholar
  146. [Se2]
    Séré, E., Looking for the Bernoulli shift, preprint.Google Scholar
  147. [Se3]
    Séré, E., Homoclinic orbits on compact hypersurfaces inn of restricted contact type, preprint.Google Scholar
  148. [Sf]
    Seifert, H., Periodische Bewegungen mechanischer systeme, Math. Z. 51, 1948, 197–216.MathSciNetzbMATHGoogle Scholar
  149. [Si1]
    Silva, E., Critical point theorems and applications to differential equations, thesis, University of Wisconsin-Madison, 1988.Google Scholar
  150. [Si2]
    Silva, E., Linking theorems and applications to nonlinear elliptic equations at resonance, Nonlinear Analysis: TMA 16, 1991, 455–477.zbMATHGoogle Scholar
  151. [St1]
    Struwe, M., Variational Methods and their Applications to Partial Differential Equations and Hamiltonian Systems, Springer, 1990.Google Scholar
  152. [St2]
    Struwe, M., Existence of periodic solutions of Hamiltonian systems on almost every energy surface, preprint.Google Scholar
  153. [ST]
    Schechter, M. and C. Tintarov, Pairs of critical points produced by linking subsets with applications to semilinear elliptic problems, Bull. Soc. Math. Belg. 44, 1992, 249–261.zbMATHGoogle Scholar
  154. [SU]
    Sacks, P. and K. Uhlenbeck, On the existence of minimal immersions of 2-spheres, Ann. of Math. 113, 1981, 1–24.MathSciNetzbMATHGoogle Scholar
  155. [Sz1]
    Szulkin, A., Morse theory and existence of periodic solutions of convex Hamiltonian systems, Bull. SMF, 1988.Google Scholar
  156. [T1]
    Tanaka, K., Morse indices at critical points related to the symmetric mountain pass theorem and applications, Comm. PDE 14, 1989, 99–128.zbMATHGoogle Scholar
  157. [T2]
    Tanaka, K., Homoclinic orbits in a first order Hamiltonian system: Convergence of subharmonic orbits, Ann. IHP - Analyse nonlin. (to appear).Google Scholar
  158. [V]
    Viterbo, C., A proof of the Weinstein conjecture in2n, Ann. IHP - Analyse nonlin. 4, 1987, 337–356.MathSciNetzbMATHGoogle Scholar
  159. [VG1]
    van Groesen, E. W. C., Existence of multiple normal mode trajectories on convex energy surfaces of even classical Hamiltonian systems, J. Differential Equations 57, 1985, 70–89.MathSciNetzbMATHGoogle Scholar
  160. [W1]
    Weinstein, A., On the hypotheses of Rabinowitz’ periodic orbit theorems, J. Diff. Eq. 33, 1979, 353–358.zbMATHGoogle Scholar
  161. [W2]
    Weinstein, A., Periodic orbits for convex Hamiltonian systems, Ann. of Math. (2) 108, 1978, 507–518.MathSciNetzbMATHGoogle Scholar
  162. [W3]
    Weinstein, A., Normal modes for nonlinear Hamiltonian systems, Inv. Math. 20, 1973, 45–57.Google Scholar
  163. [Wi]
    Willem, M., Subharmonic oscillations of convex Hamiltonian systems, J. Nonlinear Anal. 9, 1985, 1303–1311.MathSciNetzbMATHGoogle Scholar
  164. [Z]
    Zehnder, E., Periodische Losungen von Hamiltonishe Systemen, Jahresber. Deutsch. Math. - Verein 89, N1, 1987, 33–59.MathSciNetzbMATHGoogle Scholar

Copyright information

© Birkhäuser Boston 1995

Authors and Affiliations

  • Paul H. Rabinowitz
    • 1
  1. 1.Mathematics DepartmentUniversity of WisconsinMadisonUSA

Personalised recommendations