On strongly indefinite functionals with applications

Hofer, Helmut

doi:10.1090/S0002-9947-1983-0678344-2

On strongly indefinite functionals with applications
HTML articles powered by AMS MathViewer

by Helmut Hofer PDF

Trans. Amer. Math. Soc. 275 (1983), 185-214 Request permission

Abstract:

Recently, in their remarkable paper Critical point theory for indefinite functionals, V. Benci and P. Rabinowitz gave a direct approach—avoiding finite-dimensional approximations—to the existence theory for critical points of indefinite functionals. In this paper we develop under weaker hypotheses a simpler but more general theory for such problems. In the second part of the paper the abstract results are applied to a class of resonance problems of the Landesman and Lazer type, and moreover they are illustrated by an application to a wave equation problem.

References

Paul H. Rabinowitz, Some minimax theorems and applications to nonlinear partial differential equations, Nonlinear analysis (collection of papers in honor of Erich H. Rothe), Academic Press, New York, 1978, pp. 161–177. MR 0501092
Paul H. Rabinowitz, Free vibrations for a semilinear wave equation, Comm. Pure Appl. Math. 31 (1978), no. 1, 31–68. MR 470378, DOI 10.1002/cpa.3160310103
Herbert Amann, Saddle points and multiple solutions of differential equations, Math. Z. 169 (1979), no. 2, 127–166. MR 550724, DOI 10.1007/BF01215273
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, DOI 10.1002/cpa.3160330507
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, DOI 10.2307/1971148
Vieri Benci and Paul H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math. 52 (1979), no. 3, 241–273. MR 537061, DOI 10.1007/BF01389883
Richard S. Palais, Critical point theory and the minimax principle, Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 185–212. MR 0264712
Peter Hess, On a theorem by Landesman and Lazer, Indiana Univ. Math. J. 23 (1973/74), 827–829. MR 352687, DOI 10.1512/iumj.1974.23.23068
H. Brézis and L. Nirenberg, Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 2, 225–326. MR 513090
M. M. Vainberg, Variational methods for the study of nonlinear operators, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964. With a chapter on Newton’s method by L. V. Kantorovich and G. P. Akilov. Translated and supplemented by Amiel Feinstein. MR 0176364
Helmut Hofer, On the range of a wave operator with nonmonotone nonlinearity, Math. Nachr. 106 (1982), 327–340. MR 675766, DOI 10.1002/mana.19821060128
Helmut Hofer, A multiplicity result for a class of nonlinear problems with application to a nonlinear wave equation, Nonlinear Anal. 5 (1981), no. 1, 1–11. MR 597275, DOI 10.1016/0362-546X(81)90064-X
Peter Hess, Nonlinear perturbations of linear elliptic and parabolic problems at resonance: existence of multiple solutions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 3, 527–537. MR 506999
Svatopluk Fučík and Peter Hess, Nonlinear perturbations of linear operators having nullspace with strong unique continuation property, Nonlinear Anal. 3 (1979), no. 2, 271–277. MR 525977, DOI 10.1016/0362-546X(79)90082-8
Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. MR 0370183, DOI 10.1016/0022-1236(73)90051-7