Bulletin (New Series) of the American Mathematical Society

Variational and topological methods in nonlinear problems

L. Nirenberg

Full-text: Open access

Article information

Bull. Amer. Math. Soc. (N.S.) Volume 4, Number 3 (1981), 267-302.

First available in Project Euclid: 4 July 2007

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35-02 35-A15 35A05 35B10: Periodic solutions 4902 4602 58F05 58-G16
Secondary: 35B32: Bifurcation [See also 37Gxx, 37K50] 35J65: Nonlinear boundary value problems for linear elliptic equations 35L20: Initial-boundary value problems for second-order hyperbolic equations 58E15: Application to extremal problems in several variables; Yang-Mills functionals [See also 81T13], etc.


Nirenberg, L. Variational and topological methods in nonlinear problems. Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 3, 267--302. https://projecteuclid.org/euclid.bams/1183548116.

Export citation


  • 1. J. C. Alexander, Bifurcation of zeros of parametrized functions (to appear).
  • 2. J. C. Alexander and J. A. Yorke, Global bifurcation of periodic orbits, Amer. J. Math. 100 (1978), 263-292.
  • 3. J. C. Alexander and J. A. Yorke, Calculating bifurcation invariants as elements in the homotopy of the general linear group, J. Pure Appl. Algebra 13 (1978), 1-8.
  • 4. H. Amann and E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations (to appear).
  • 5. H. Amann and E. Zehnder, Multiple periodic solutions for a class of nonlinear autonomous wave equations (to appear).
  • 6. A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381.
  • 7. A. Ambrosetti and G. Mancini, Sharp nonuniqueness results for some nonlinear problems, Nonlinear Anal. Theory Math. Appl. 3 (1979), 635-645.
  • 8. V. Benci, Some critical point theorems and applications, Comm. Pure Appl. Math. 33 (1980), 147-172.
  • 9. V. Benci, On the critical point theory for indefinite functionals in the presence of symmetries, 1st. Mat. Appl. U. Dini, Univ. di Pisa, March 1980.
  • 10. V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math. 52 (1979), 241-273.
  • 11. M. S. Berger, Nonlinearity and functional analysis, Academic Press, New York, 1978.
  • 12. Yu. G. Borisovich, V. G. Zvyagin and Yu. I. Sapronov, Nonlinear Fredholm maps and the Leray-Schauder theorem, Uspehi Mat. Nauk 32 (1977), 3-54; English transl. in Russian Math. Surveys 32 (1977), 1-54.
  • 13. H. Brézis and R. E. L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977), 601-614.
  • 14. H. Brézis and L. Nirenberg, Forced vibrations for a nonlinear wave equation, Comm. Pure Appl. Math. 31 (1978), 1-30.
  • 15. H. Brézis, J. M. Coron and L. Nirenberg, Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz, Comm. Pure Appl. Math. 33 (1980), 667-684.
  • 16. A. Castro, A two point boundary value problem with jumping nonlinearities, Proc. Amer. Math. Soc. (to appear).
  • 17. A. Castro and A. C. Lazer, Critical point theory and the number of solutions of a nonlinear Dirichlet problem, Ann. Mat. Pura Appl. (IV) 70 (1979), 113-137.
  • 18. K. C. Chang, The obstacle problem and partial differential equations with discontinuous nonlinearities, Comm. Pure Appl. Math. 33 (1980), 117-146.
  • 19. K. C. Chang, Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math. (to appear).
  • 20. F. H. Clarke and I. Ekeland, Hamiltonian trajectories having prescribed minimal period, Comm. Pure Appl. Math. 33 (1980), 103-116.
  • 21. C. C. Conley, Isolated invariant sets and the Morse index, CBMS Regional Conf. Ser. in Math., no. 38, Amer. Math. Soc., Providence, R. I., 1978.
  • 22. D. DeFigueiredo, P. L. Lions and R. D. Nussbaum, Estimations à priori pour les solutions positives de problèmes elliptiques semilinéaires, C. R. Acad. Sci. Paris Ser. A. 290 (1980), 217-220.
  • 23. I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353.
  • 24. I. Ekeland, Periodic solutions of Hamiltonian equations and a theorem of P. Rabinowitz, J. Differential Equations 34 (1979), 523-534.
  • 25. I. Ekeland and R. Témam, Convex analysis and variational problems, Studies in Math. and Appl., Vol. 1, North-Holland, Amsterdam, American Elsevier, New York, 1976.
  • 26. I. Ekeland and J. M. Lasry, On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, MRC Technical Report 2050, Math. Research Center, Univ. of Wisconsin, 1980.
  • 27. K. D. Elworthy and A. J. Tromba, Differential structures and Fredholm maps, Proc. Sympos. Pure Math. (Berkeley, California, 1968), vol. 15, Amer. Math. Soc., Providence, R. I., 1970, pp. 45-94.
  • 28. K. D. Elworthy and A. J. Tromba, Degree theory on Banach manifolds, Proc Sympos. Pure Math., vol. 18, part 1, Nonlinear functional analysis, Amer. Math. Soc., Providence, R. I., 1970, pp. 86-94.
  • 29. F. R. Fadell and P. H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math. 45 (1978), 134-174.
  • 30. F. B. Fuller, An index of fixed point type for periodic orbits, Amer. J. Math. 89 (1967), 133-148.
  • 31. K. Geba and A. Granas, Infinite dimensional cohomology theories, J. Math. Pure Appl. 52 (1973), 145-270.
  • 32. B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations (submitted).
  • 33. M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory, Comm. Pure Appl. Math. 32 (1979), 21-98.
  • 34. L. Hörmander, The boundary value problems of physical geodesy, Arch. Rational Mech. Anal. 62 (1976), 1-52.
  • 35. J. Ize, Bifurcation theory for Fredholm operators, Mem. Amer. Math. Soc No. 174 (1976).
  • 36. S. Klainerman, Global existence for nonlinear wave equations, Comm. Pure Appl. Math. 33 (1980), 43-101.
  • 37. M. A. Krasnoselskii, Topological methods in the theory of nonlinear integral equations, Macmillan, New York, 1964.
  • 38. J. Leray and J. Schauder, Topologie et équations fonctionelles, Ann. Sci. École Norm. Sup. 51 (1934), 45-78.
  • 39. N. G. Lloyd, Degree theory, Cambridge Tracts in Math., No. 73, Cambridge Univ. Press, London, 1978.
  • 40. J. Mallet-Paret and J. A. Yorke, Snakes: Oriented families of periodic orbits, their sources, sinks and continuation (to appear).
  • 41. J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Appl. Math. Sciences, Vol. 19, Springer-Verlag, New York, 1976.
  • 42. J. Moser, A new technique for the construction of solutions of nonlinear differential equations, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 1824-1831.
  • 43. J. Moser, A rapidly convergent iteration method and nonlinear partial differential equations. I, II, Ann. Scuola Norm. Sup. Pisa 20 (1966), 265-315; 499-535.
  • 44. J. Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. (2) 63 (1956), 20-63.
  • 45. W. M. Ni, Some minimax principles and their applications in nonlinear elliptic equations, J. d'Analyse Math. 37 (1980), 248-275.
  • 46. L. Nirenberg, An application of generalized degree to a class of nonlinear problems, 3rd Colloq. Anal. Fonct. Liège Centre Belge de Rech. Math., 1971, pp. 57-73.
  • 47. L. Nirenberg, Topics in nonlinear functional analysis, Lecture Notes, Courant Inst., 1974.
  • 48. L. Nirenberg, Remarks on nonlinear problems, The Chern Symposium, 1979 (W. Y. Hsiang et al., eds.) Springer-Verlag, Berlin and New York, 1980, pp. 189-197.
  • 49. R. S. Palais, Critical point theory and the minimax principle, Proc. Sympos. Pure Math., vol. 15, Amer. Math. Soc., Providence, R. I., 1970, pp. 185-212.
  • 50. P. H. Rabinowitz, Variational methods for nonlinear eigenvalue problems (CIME, Verona, 1974), Ediz. Cremonese Rome, 1974, pp. 141-195.
  • 51. P. H. Rabinowitz, Théorie du degré topologique et applications à des problèmes aux limites nonlinéaires, Lecture Notes, Analyse Numerique Fonctionelle, Univ. Paris VI, 1975.
  • 52. P. H. Rabinowitz, Free vibrations for a semilinear wave equation, Comm. Pure Appl. Math. 31 (1978), 31-68.
  • 53. P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), 157-184.
  • 54. P. H. Rabinowitz, A variational method for finding periodic solutions of differential equations. Nonlinear Evolution Equations (M. G. Crandall, ed.), Academic Press, New York, 1978, pp. 225-251.
  • 55. D. H. Sattinger, Topics in stability and bifurcation theory, Lecture Notes in Math. no. 309, Springer-Verlag, Berlin and New York, 1973.
  • 56. M. Schechter, Principles of functional analysis, Academic Press, New York, 1971.
  • 57. J. Schwartz, Nonlinear functional analysis, Gordon and Breach, New York, 1969.
  • 58. J. Sylvester, Ph.D. Thesis, Courant Inst. Math. Sci., New York Univ., 1980.