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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Variational and topological methods in nonlinear problems


Author: L. Nirenberg
Journal: Bull. Amer. Math. Soc. 4 (1981), 267-302
MSC (1980): Primary 35-02, 35-A15, 35A05, 35B10, 4902, 4602, 58F05, 58-G16; Secondary 35B32, 35J65, 35L20, 58E15
DOI: https://doi.org/10.1090/S0273-0979-1981-14888-6
MathSciNet review: 609039
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  • 1. J. C. Alexander, Bifurcation of zeroes of parametrized functions, J. Funct. Anal. 29 (1978), no. 1, 37–53. MR 499933, https://doi.org/10.1016/0022-1236(78)90045-9
  • 2. J. C. Alexander and J. A. Yorke, Global bifurcation of periodic orbits, Amer. J. Math. 100 (1978), 263-292. MR 474406
  • 3. J. C. Alexander and James A. Yorke, Calculating bifurcation invariants as elements in the homotopy of the general linear group, J. Pure Appl. Algebra 13 (1978), no. 1, 1–8. MR 508723, https://doi.org/10.1016/0022-4049(78)90035-X
  • 4. H. Amann 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), no. 4, 539–603. MR 600524
  • 5. H. Amann and E. Zehnder, Multiple periodic solutions for a class of nonlinear autonomous wave equations, Houston J. Math. 7 (1981), no. 2, 147–174. MR 638944
  • 6. A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. MR 370183
  • 7. Antonio Ambrosetti and Giovanni Mancini, Sharp nonuniqueness results for some nonlinear problems, Nonlinear Anal. 3 (1979), no. 5, 635–645. MR 541874, https://doi.org/10.1016/0362-546X(79)90092-0
  • 8. Vieri Benci, Some critical point theorems and applications, Comm. Pure Appl. Math. 33 (1980), no. 2, 147–172. MR 562548, https://doi.org/10.1002/cpa.3160330204
  • 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. Vieri Benci and Paul H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math. 52 (1979), no. 3, 241–273. MR 537061, https://doi.org/10.1007/BF01389883
  • 11. Carl L. DeVito, Functional analysis, Pure and Applied Mathematics, vol. 81, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506238
  • 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. MR 464282
  • 13. H. Brézis and R. E. L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977), 601-614. MR 509489
  • 14. H. Brézis and L. Nirenberg, Forced vibrations for a nonlinear wave equation, Comm. Pure Appl. Math. 31 (1978), no. 1, 1–30. MR 470377, https://doi.org/10.1002/cpa.3160310102
  • 15. Haïm Brézis, Jean-Michel Coron, and Louis Nirenberg, Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz, Comm. Pure Appl. Math. 33 (1980), no. 5, 667–684. MR 586417, https://doi.org/10.1002/cpa.3160330507
  • 16. Alfonso Castro B., A two-point boundary value problem with jumping nonlinearities, Proc. Amer. Math. Soc. 79 (1980), no. 2, 207–211. MR 565340, https://doi.org/10.1090/S0002-9939-1980-0565340-1
  • 17. Alfonso Castro and A. C. Lazer, Critical point theory and the number of solutions of a nonlinear Dirichlet problem, Ann. Mat. Pura Appl. (4) 120 (1979), 113–137. MR 551063, https://doi.org/10.1007/BF02411940
  • 18. K. C. Chang, The obstacle problem and partial differential equations with discontinuous nonlinearities, Comm. Pure Appl. Math. 33 (1980), no. 2, 117–146. MR 562547, https://doi.org/10.1002/cpa.3160330203
  • 19. Kung Ching Chang, Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math. 34 (1981), no. 5, 693–712. MR 622618, https://doi.org/10.1002/cpa.3160340503
  • 20. Frank H. Clarke and Ivar Ekeland, Hamiltonian trajectories having prescribed minimal period, Comm. Pure Appl. Math. 33 (1980), no. 2, 103–116. MR 562546, https://doi.org/10.1002/cpa.3160330202
  • 21. Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133
  • 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. MR 346619
  • 24. Ivar Ekeland, Periodic solutions of Hamiltonian equations and a theorem of P. Rabinowitz, J. Differential Equations 34 (1979), no. 3, 523–534. MR 555325, https://doi.org/10.1016/0022-0396(79)90034-2
  • 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. MR 463994
  • 26. Ivar Ekeland and Jean-Michel Lasry, On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, Ann. of Math. (2) 112 (1980), no. 2, 283–319. MR 592293, https://doi.org/10.2307/1971148
  • 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. MR 264708
  • 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. MR 277009
  • 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. MR 478189
  • 30. F. B. Fuller, An index of fixed point type for periodic orbits, Amer. J. Math. 89 (1967), 133-148. MR 209600
  • 31. K. Geba and A. Granas, Infinite dimensional cohomology theories, J. Math. Pure Appl. 52 (1973), 145-270. MR 380865
  • 32. B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), no. 8, 883–901. MR 619749, https://doi.org/10.1080/03605308108820196
  • 33. M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory, Comm. Pure Appl. Math. 32 (1979), no. 1, 21–98. MR 508917, https://doi.org/10.1002/cpa.3160320103
  • 34. L. Hörmander, The boundary value problems of physical geodesy, Arch. Rational Mech. Anal. 62 (1976), 1-52. MR 602181
  • 35. J. Ize, Bifurcation theory for Fredholm operators, Mem. Amer. Math. Soc No. 174 (1976). MR 425696
  • 36. Sergiu Klainerman, Global existence for nonlinear wave equations, Comm. Pure Appl. Math. 33 (1980), no. 1, 43–101. MR 544044, https://doi.org/10.1002/cpa.3160330104
  • 37. M. A. Krasnoselskii, Topological methods in the theory of nonlinear integral equations, Macmillan, New York, 1964. MR 159197
  • 38. Jean Leray and Jules Schauder, Topologie et équations fonctionnelles, Ann. Sci. École Norm. Sup. (3) 51 (1934), 45–78 (French). MR 1509338
  • 39. N. G. Lloyd, Degree theory, Cambridge Tracts in Math., No. 73, Cambridge Univ. Press, London, 1978. MR 493564
  • 40. John Mallet-Paret and James A. Yorke, Snakes: oriented families of periodic orbits, their sources, sinks, and continuation, J. Differential Equations 43 (1982), no. 3, 419–450. MR 649847, https://doi.org/10.1016/0022-0396(82)90085-7
  • 41. J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Appl. Math. Sciences, Vol. 19, Springer-Verlag, New York, 1976. MR 494309
  • 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. MR 132859
  • 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. MR 75639
  • 45. Wei Ming Ni, Some minimax principles and their applications in nonlinear elliptic equations, J. Analyse Math. 37 (1980), 248–275. MR 583639, https://doi.org/10.1007/BF02797687
  • 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. MR 413207
  • 47. L. Nirenberg, Topics in nonlinear functional analysis, Lecture Notes, Courant Inst., 1974. MR 488102
  • 48. Louis Nirenberg, Remarks on nonlinear problems, The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), Springer, New York-Berlin, 1980, pp. 189–197. MR 609561
  • 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. MR 264712
  • 50. P. H. Rabinowitz, Variational methods for nonlinear eigenvalue problems (CIME, Verona, 1974), Ediz. Cremonese Rome, 1974, pp. 141-195. MR 464299
  • 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. MR 470773
  • 52. Paul H. Rabinowitz, Free vibrations for a semilinear wave equation, Comm. Pure Appl. Math. 31 (1978), no. 1, 31–68. MR 470378, https://doi.org/10.1002/cpa.3160310103
  • 53. P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), 157-184. MR 467823
  • 54. Paul H. Rabinowitz, A variational method for finding periodic solutions of differential equations, Nonlinear evolution equations (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1977) Publ. Math. Res. Center Univ. Wisconsin, vol. 40, Academic Press, New York-London, 1978, pp. 225–251. MR 513821
  • 55. D. H. Sattinger, Topics in stability and bifurcation theory, Lecture Notes in Math. no. 309, Springer-Verlag, Berlin and New York, 1973. MR 463624
  • 56. M. Schechter, Principles of functional analysis, Academic Press, New York, 1971. MR 445263
  • 57. J. Schwartz, Nonlinear functional analysis, Gordon and Breach, New York, 1969. MR 433481
  • 58. J. Sylvester, Ph.D. Thesis, Courant Inst. Math. Sci., New York Univ., 1980.

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DOI: https://doi.org/10.1090/S0273-0979-1981-14888-6

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