Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Multiple critical points of perturbed symmetric functionals

Author: Paul H. Rabinowitz
Journal: Trans. Amer. Math. Soc. 272 (1982), 753-769
MSC: Primary 35J60; Secondary 35B20, 49B50, 58E05
MathSciNet review: 662065
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Variational problems which are invariant under a group of symmetries often possess multiple solutions. This paper studies the effect of perturbations which are not small and which destroy the symmetry for two classes of such problems and shows how multiple solutions persist despite the perturbation.

References [Enhancements On Off] (What's this?)

  • [1] C. V. Coffman, A minimum-maximum principle for a class of nonlinear integral equations, J. Analyse Math. 22 (1969), 391-419. MR 0249983 (40:3224)
  • [2] J. Hemple, Superlinear variational boundary value problems and nonuniqueness, Thesis, Univ. of New England, Australia, 1970.
  • [3] A. Ambrosetti, On the existence of multiple solutions for a class of nonlinear boundary value problems, Rend. Sem. Mat. Univ. Padova 49 (1973), 195-204. MR 0336068 (49:844)
  • [4] P. H. Rabinowitz, Variational methods for nonlinear elliptic eigenvalue problems, Indiana Univ. Math. J. 23 (1974), 729-754. MR 0333442 (48:11767)
  • [5] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. MR 0370183 (51:6412)
  • [6] V. Benci, Some critical point theorems and applications, Comm. Pure Appl. Math. 33 (1980), 147-172. MR 562548 (81f:58015)
  • [7] P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), 157-184. MR 0467823 (57:7674)
  • [8] -, On large norm periodic solutions of some differential equations, Ergodic Theory and Dynamical Systems, Proc. Sympos. Yi (Maryland) (A. Katok, ed.), pp. 79-80 (to appear).
  • [9] A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc. 267 (1981), 1-32. MR 621969 (82j:35059)
  • [10] M. Struwe, Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems, Manuscripta Math. 32 (1980), 335-364. MR 595426 (82e:58030)
  • [11] G.-C. Dong and S. Li, On the existence of infinitely many solutions of the Dirichlet problem for some nonlinear elliptic equations, Univ. of Wisconsin Math. Research Center Tech. Rep. N02161, Dec., 1980.
  • [12] A. Bahri, Topological results on a certain class of functionals and applications (preprint). MR 619960 (84c:58017)
  • [13] A. Bahri and H. Berestycki, Existence d'une infinité de solutions periodiques pour certains systèmes hamiltoniens en présence d'un terme de contrainte, C. R. Acad. Sci. Sér. I 292, 315-318. MR 608843 (82b:58023)
  • [14] P. H. Rabinowitz, Variational methods for nonlinear eigenvalue problems, Proc. Sympos. on Eigenvalues of Nonlinear Problems, Edizioni Cremonese, Rome, 1974, pp. 141-195. MR 0464299 (57:4232)
  • [15] E. 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), 139-174. MR 0478189 (57:17677)
  • [16] J. T. Schwartz, Nonlinear functional analysis, Lecture Notes, Courant Inst. of Math. Sci., New York Univ., 1965. MR 0433481 (55:6457)
  • [17] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 1-48. MR 0109940 (22:823)
  • [18] R. Courant and D. Hilbert, Methods of mathematical physics, Vol. I, Interscience, New York, 1953. MR 0065391 (16:426a)
  • [19] D. C. Clark, A variant of the Ljusternick-Schnirelman theory, Indiana Univ. Math. J. 22 (1972), 65-74. MR 0296777 (45:5836)
  • [20] S. I. Pohozaev, Eigenfunctions of the equation $ - \Delta u + \lambda f(u) = 0$, Soviet Math. Dokl. 5 (1965), 1408-1411. MR 0192184 (33:411)
  • [21] E. R. Fadell, S. Husseini and P. H. Rabinowitz, On $ S^1$ versions of the Borsuk-Ulam Theorem, Univ. of Wisconsin Math. Res. Center Tech. Rep., 1981.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35J60, 35B20, 49B50, 58E05

Retrieve articles in all journals with MSC: 35J60, 35B20, 49B50, 58E05

Additional Information

Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society