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Variational Methods and Nonlinear Problems: Classical Results and Recent Advances

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Book cover Topological Nonlinear Analysis

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 15))

Abstract

Around the end of the Twenties two memoires, a first one by Morse [63] and a second one by Lusternik and Schnirelman [59], marked the birth of those variational methods known under the name of Calculus of Variation in the Large. These tools are mainly concerned with the existence of critical points, distinct from minima, which give rise to solutions of nonlinear differential equations. The elegance of the abstract tools and the broad range of applications to problems that had been considered of formidable difficulty, such as the existence of closed geodesics on a compact anifold or the problem of minimal surfaces, have rapidly made the Calculus of Variation in the Large a very fruitful field of research.

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References

  1. Adimurthi and Mancini G., Geometry and topology of the boundary in the critical Neumann problem, to appear.

    Google Scholar 

  2. Adimurthi, Pacella, F., and Yadava S.L., Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity, to appear.

    Google Scholar 

  3. Alama, S. and Tarantello, G., On semilinear elliptic equations with indefinite nonlinearities, Calculus of Variat. (to appear)

    Google Scholar 

  4. Ambrosetti, A., A perturbation theorem for superlinear boundary value problems, M.R.C. Tech. Summ. Rep. (1974).

    Google Scholar 

  5. Ambrosetti, A., Elliptic equations with jumping nonlinearities, J. Math. Phys. Sci. 18–1 (1984), 1–12.

    MathSciNet  MATH  Google Scholar 

  6. Ambrosetti, A., Critical points and nonlinear variational problems, Suppl. Bulletin de la Soc. Math, de France, Mem. N.49, Tome 120, 1992.

    Google Scholar 

  7. Ambrosetti, A., Brezis, H., and Cerami, G., Combined effects of concave and convex nonlineatities in some elliptic problems, Jour. Funct. Anal, (to appear)

    Google Scholar 

  8. Ambrosetti, A. and Coti Zelati, V., Critical point with lack of compactness and singular dynamical systems, Ann. Mat. Pura Appl. 149 (1987), 237–259.

    Article  MathSciNet  MATH  Google Scholar 

  9. Ambrosetti A. and Coti Zelati, V., Closed orbits with fixed energy for singular Hamiltonian Systems, Archive Rat. Mech. Analysis, 112 (1990), 339–362.

    Article  MathSciNet  MATH  Google Scholar 

  10. Ambrosetti, A. and Coti Zelati, V., Closed orbits with fixed energy for a class of N-body problems, Ann. Inst. Poincaré Anal. Nonlinéaire. 9 (1992), 187–200, and 337–338.

    MATH  Google Scholar 

  11. Ambrosetti, A. and Coti Zelati, V., Periodic Solutions of Singular Lagrangian Systems, Birkhäuser, 1993.

    Book  MATH  Google Scholar 

  12. Ambrosetti, A. and Prodi, G., A Primer of Nonlinear Analysis, Cambridge Univ. Press, 1993.

    Google Scholar 

  13. Ambrosetti, A. and Rabinowitz, P.H., Dual variational methods in critical points theory and applications, J. Funct. Anal. 14 (1973), 349–381.

    Article  MathSciNet  MATH  Google Scholar 

  14. Ambrosetti, A. and Struwe, M., A note on the problem \( - \Delta u = \lambda u + u\mathop {\left| u \right|}\nolimits^{\mathop 2\nolimits^* - 2} ,u \in \mathop H\nolimits_0^1 ,\lambda > 0,\) Manus. Math. 54 (1986), 373–379.

    Article  MathSciNet  MATH  Google Scholar 

  15. Ambrosetti, A. and Struwe, M., Existence of steady vortex rings in an ideal fluid, Arch. Rat. Mech. Anal. 108, 2 (1989), 97–109.

    Article  MathSciNet  MATH  Google Scholar 

  16. Ambrosetti, A. and Struwe, M., Periodic motions of conservatiove systems with singular potentials, Nonlin. Diff. Equat. Appl. (to appear)

    Google Scholar 

  17. Bahri, A., Topological results on a certain class of functionals and applications, J. Funct. Anal. 41 (1981), 397–427.

    Article  MathSciNet  MATH  Google Scholar 

  18. Bahri, A. and Berestycki, H., A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc. 267 (1981), 1–32.

    Article  MathSciNet  MATH  Google Scholar 

  19. Bahri, A. and Coron, J.M., On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), 253–294.

    Article  MathSciNet  MATH  Google Scholar 

  20. Bahri, A. and Lions, P.L., Morse index of some min-max critical points I. Application to multiplicity results, Comm. Pure Appl. Math. 41 (1988), 1027–1037.

    Article  MathSciNet  MATH  Google Scholar 

  21. Bahri, A. and Rabinowitz, P.H., A minimax method for a class of Hamiltonian systems with singular potentials, J. Funct. Anal. 82 (1989), 412–428.

    Article  MathSciNet  MATH  Google Scholar 

  22. Bahri, A. and Rabinowitz, P.H., Solutions of the three-body problem via critical points of infinity, Ann. Inst. Poincaré Anal. Nonlinéaire 8 (1991), 561–649.

    MathSciNet  MATH  Google Scholar 

  23. 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.

    Article  MathSciNet  MATH  Google Scholar 

  24. Benci, V., Closed geodesics fior the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems, Ann. Inst. Poincaré Anal. Nonlinéaire 1 (1984), 401–412.

    MathSciNet  MATH  Google Scholar 

  25. Benci, V. and Cerami, G., Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rat. Mech. Anal. 99 (1987), 283–300.

    Article  MathSciNet  MATH  Google Scholar 

  26. Benci, V. and Cerami, G., The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rat. Mech. Anal. 114 (1991), 79–93.

    Article  MathSciNet  MATH  Google Scholar 

  27. Benci, V. and Fortunato, D., Subharmonic solutions of prescribed minimal period for non-autonomous differential equations, in: Recent Advances in Hamiltonian systems, World Scientific, 1987.

    Google Scholar 

  28. Benci, V. and Giannoni, F., A new proof of the existence of a brake orbit, in: Advanced topics in the theory of dynamical systems, Academic Press, 1990.

    Google Scholar 

  29. Benci, V. and Rabinowitz, P.H., Critical point theorems for indefinite junctionals, Invent. Math. 52 (1979), 241–273.

    Article  MathSciNet  MATH  Google Scholar 

  30. Berestycki, H. and Lions, P.L., Nonlinear scalar field equations, Arch. Rat. Mech. Anal. 82 (1983), 313–376.

    MathSciNet  MATH  Google Scholar 

  31. Berestycki, H. and Lions, P.L., Existence of stationary states in nonlinear scalar field equations, in: Bifurcation Phenomena in Mathematical Physics and related topics, Ridel Publ. Co., 1980.

    Google Scholar 

  32. Boccardo, L., Escobedo, M., and Peral, I., A Dirichlet problem involving critical exponent, to appear.

    Google Scholar 

  33. Brezis, H. and Coron, J.M., Multiple solutions of H-systems and the Rellich conjecture, Comm. Pure Appl. Math 37 (1984), 149–187.

    Article  MathSciNet  MATH  Google Scholar 

  34. Brezis, H. and Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477.

    Article  MathSciNet  MATH  Google Scholar 

  35. Browder, F., Infinite dimensional manifolds and nonlinear eigenvalue problems, Ann. of Math. 82 (1965), 459–477.

    Article  MathSciNet  MATH  Google Scholar 

  36. Capozzi, A., Fortunato, D., and Palmieri, G., An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. Poincaré Anal. Nonlinéaire 2 (1985), 463–470.

    MathSciNet  MATH  Google Scholar 

  37. Capozzi, A., Greco, C., and Salvatore, A., Lagrangian systems in presence of singularities, Proc. Amer. Math. Soc. 102 (1988), 125–130.

    Article  MathSciNet  MATH  Google Scholar 

  38. Cerami, G., Solimini, S., and Struwe, M., Some existence results for superlinear elliptic boundary value problems involving critical Sobolev exponent, J. Funct. Anal. 69 (1986), 209–212.

    Article  MathSciNet  Google Scholar 

  39. Coron, J.M., Topologie et cas limite des injections de Sobolev, C. R. Acad. Sci. Paris 299 (1984), 209–212.

    MathSciNet  MATH  Google Scholar 

  40. Coti Zelati, V., Periodic solutions for N-body type problems, Ann. Inst. Poincaré Anal. Nonlinéaire 7 (1990), 477–492.

    MathSciNet  MATH  Google Scholar 

  41. Coti Zelati, V. and Serra, E., Collision and noncollision solutions for a class of Keplerian-like dynamical systems, to appear.

    Google Scholar 

  42. Dancer, E.N., The effect of the domain shape on the number of positive solutions of certain nonlinear equations, J. Diff. Equat. 74 (1988), 120–156.

    Article  MathSciNet  MATH  Google Scholar 

  43. Ding, W.Y., Positive solutions of -Δu + u (N+2)/(N-2) = 0 on contractible domains, to appear.

    Google Scholar 

  44. Ekeland, I., Convexity Methods in Hamiltonian Mechanics, Springer, 1990.

    MATH  Google Scholar 

  45. Ekeland, I., On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353.

    Article  MathSciNet  MATH  Google Scholar 

  46. Fadell, E. and Husseini, S., A note on the category of free loop space, Proc. Amer. Math. Soc. 107 (1989), 527–536.

    Article  MathSciNet  MATH  Google Scholar 

  47. Fraenkel, L.E. and Berger, M.S., A global theory of steady vortex rings in an ideal fluid, Acta Math. 132 (1974), 13–51.

    Article  MathSciNet  MATH  Google Scholar 

  48. Garcia, J. and Peral, I., Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc. 323 (1991), 877–895.

    Article  MathSciNet  MATH  Google Scholar 

  49. Gordon, W.B., Conservative dynamical systems involving strong forces, Trans. Amer. Math. Soc. 204 (1975), 113–135.

    Article  MathSciNet  MATH  Google Scholar 

  50. Ghoussoub, N. and Preiss, D., A general mountain pass principle for locating and classifying critcal points, Ann. Inst. Poincaré Anal. Nonlinéaire, 6 (1989), 321–330.

    MathSciNet  MATH  Google Scholar 

  51. Greco, C., Periodic solutions of a class of singular Hamiltonian systems, J. Nonlinear Analysis T.M.A. 12 (1988), 259–270.

    Article  MathSciNet  MATH  Google Scholar 

  52. Hayashi, K., Periodic solutions of classical Hamiltonian systems, Tokio J. Math. 6 (1983), 473–486.

    Article  MATH  Google Scholar 

  53. Hofer, H., A note on the topological degree at a critical point of mountain-pass type, Proc. Amer. Math. Soc. 90 (1984), 309–315.

    Article  MathSciNet  MATH  Google Scholar 

  54. Hofer, H. and Zehnder, E., Periodic solutions on hypersurfaces and a result by C. Viterbo, Inv. Math. 90 (1987), 1–9.

    Article  MathSciNet  MATH  Google Scholar 

  55. Krasnoselskii, M.A., Topological Methods in the theory of nonlinear integral equations, Pergamon, Oxford, 1965.

    Google Scholar 

  56. Lazer, A.C. and Solimini, S., Nontrivial solutions of operator equations and Morse indices of critical points of min-max type, J. Nonlin. Anal. T.M.A. 12 (1988), 761–775.

    Article  MathSciNet  MATH  Google Scholar 

  57. Lions, P.L., The concentration-compactness principle in the Calculus of Variaitons. The locally compact case (part 1 and 2), Ann. Inst. Poincaré Anal. Nonlinéaire (1984), 109–145 and 223–283.

    Google Scholar 

  58. Lions, P.L., The concentration-compactness principle in the Calculus of Variaitons. The limit case (part 1 and 2), Riv. Iberoamer. 1 (1985), 45–121 and 145–201.

    MATH  Google Scholar 

  59. Lusternik, L. and Schnirelman, L., Méthode topologique dans les problémes varationelles, Hermann, Paris, 1934.

    Google Scholar 

  60. Majer, P., Ljusternik-Schnirelman theory without Palais-Smale condition and singular dynamical systems, Ann. Inst. Poincaré Anal. Nonlinéaire 8 (1991), 459–476.

    MathSciNet  MATH  Google Scholar 

  61. Majer, P. and Terracini, S., Periodic solutions to some N-body type problems, to appear.

    Google Scholar 

  62. Mawhin, J. and Willem, M., Critical Point Theory and Hamiltonian Systems, Springer, 1989.

    MATH  Google Scholar 

  63. Morse, M., Relations between the critical points of a real function of n independent variables, Trans. Amer. Math. Soc. 27 (1925), 345–396.

    MathSciNet  MATH  Google Scholar 

  64. Nirenberg, L., Variational and topological methods in nonlinear problems, Bull. Amer. Math. Soc. 4-3 (1981), 267–302.

    Article  MathSciNet  MATH  Google Scholar 

  65. Palais, R., Lusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966), 115–132.

    Article  MathSciNet  MATH  Google Scholar 

  66. Palais, R. and Smale, S., A generalized Morse theory, Bull. Amer. Math. Soc. 70 (1964), 165–171.

    Article  MathSciNet  MATH  Google Scholar 

  67. Pohozaev, S.I., Eigenfunctions of the equations Δu + λf(u) = 0, Soviet Math. 5 (1965), 1408–1411.

    Google Scholar 

  68. Passaseo, D., Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains, Manus. Math. 65 (1989), 147–166.

    Article  MathSciNet  MATH  Google Scholar 

  69. Pucci, P. and Serrin, J., Extension of the mountain pass theorem, J. Funct. Anal. 59 (1984), 185–210.

    Article  MathSciNet  MATH  Google Scholar 

  70. Pucci, P. and Serrin, J., A mountain pass theorem, J. Diff. Equat. 60 (1985), 142–149.

    Article  MathSciNet  MATH  Google Scholar 

  71. Rabinowitz, P.H., Some minmax theorems and applications to nonlinear partial differential equations, in: Nonlinear Analysis, Academic Press, 1978.

    Google Scholar 

  72. Rabinowitz, P.H., Multiple critical points for perturbed symmetric functional, Trans. Amer. Math. Soc. 272 (1982), 753–770.

    Article  MathSciNet  MATH  Google Scholar 

  73. Rabinowitz, P.H., A note on periodic solutions of prescribed energy for singular Hamiltonian systems, to appear.

    Google Scholar 

  74. Rabinowitz, P.H. and Tanaka, K., Some results on connecting orbits for a class of Hamiltonian systems, Math. Zeit 206 (1991), 473–499.

    Article  MathSciNet  MATH  Google Scholar 

  75. Salvatore, A., Multiple periodic solutions for Hamiltonian systems with singular potentials, Rend. Mat. Acc. Lincei, Ser. IX, 3 (1992), 111–120.

    MathSciNet  MATH  Google Scholar 

  76. Schwartz, J.T., Generalizing the Lusternik-Schnirelman theory of critical points, Comm. Pure Appl. Math. 17 (1964), 307–315.

    Article  MathSciNet  MATH  Google Scholar 

  77. Serra, E., Dynamical systems with singular potentials: existence and qualitative properties of periodic motions, Ph. D. Thesis, SISSA, Trieste, 1991.

    Google Scholar 

  78. Serra, E. and Terracini, S., Collisionless periodic solutions to some three-body like problems, Arch. Rat. Mech Anal, (to appear).

    Google Scholar 

  79. Srikanth, P.N., Uniqueness of solutions of nonlinear Dirichlet problems, Diff. Int. Equations, 6 (1993), 663–670.

    MathSciNet  MATH  Google Scholar 

  80. Struwe, M., Infinitely many critical points for functional which are not even and applications to superlinear boundary value problems, Manus. Math. 32 (1980), 335–364.

    Article  MathSciNet  MATH  Google Scholar 

  81. Struwe, M., Large H-surfaces via the mountain-pass- lemma, Math. Annalen 270 (1985), 441–459.

    Article  MathSciNet  MATH  Google Scholar 

  82. Struwe, M., Variational Methods, Springer, 1990.

    MATH  Google Scholar 

  83. Szulkin, A., Ljusternik-Schnirelmann theory on C 1 manifolds, Ann. Inst. Poincaré Anal. Nonlinéaire 5 (1988), 119–139.

    MathSciNet  MATH  Google Scholar 

  84. Tanaka, K., Prescribed energy problem for a singular Hamiltonian system with a weak force, to appear.

    Google Scholar 

  85. Tanaka, K., Non-collision solutions for a second order singular Hamiltonian system with weak forces, J. Funct. Anal, (to appear).

    Google Scholar 

  86. Terracini, S., Periodic solutions for dynamical systems with Keplerian type potentials, Ph. D. Thesis, SISSA, Trieste, 1990.

    Google Scholar 

  87. Vainberg, M.M., Variational Methods for the Study of Nonlinear Operators, Hoden-Day, San Francisco, 1964.

    MATH  Google Scholar 

  88. Viterbo, C., Indice de Morse des points critiques obtenus par minmax, Ann. Inst. Poincaré Anal. Nonlinéaire 5 (1988), 221–225.

    MathSciNet  MATH  Google Scholar 

  89. Wang, Z.Q., On a superlinear elliptic equation, Ann. Inst. Poincaré Anal. Nonlinéaire 8 (1991), 43–58.

    MATH  Google Scholar 

  90. Wang, Z.Q., Uniqueness of positive solutions of Δu + u + u p = 0 in a ball, Comm. P.D.E. 17 (1992), 1141–1164.

    Article  Google Scholar 

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  1. Scuola Normale Superiore, 56100, Pisa, Italy

    Antonio Ambrosetti

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  1. Department of Mathematics, University of Rome, Tor Vergata, 00133, Rome, Italy

    Michele Matzeu  & Alfonso Vignoli  & 

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Ambrosetti, A. (1995). Variational Methods and Nonlinear Problems: Classical Results and Recent Advances. In: Matzeu, M., Vignoli, A. (eds) Topological Nonlinear Analysis. Progress in Nonlinear Differential Equations and Their Applications, vol 15. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2570-6_1

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  • DOI: https://doi.org/10.1007/978-1-4612-2570-6_1

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