Variational and topological methods in nonlinear problems

Nirenberg, L.

doi:10.1090/S0273-0979-1981-14888-6

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

References

J. C. Alexander, Bifurcation of zeroes of parametrized functions, J. Functional Analysis 29 (1978), no. 1, 37–53. MR 499933, DOI 10.1016/0022-1236(78)90045-9
J. C. Alexander and James A. Yorke, Global bifurcations of periodic orbits, Amer. J. Math. 100 (1978), no. 2, 263–292. MR 474406, DOI 10.2307/2373851
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, DOI 10.1016/0022-4049(78)90035-X
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
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
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
Antonio Ambrosetti and Giovanni Mancini, Sharp nonuniqueness results for some nonlinear problems, Nonlinear Anal. 3 (1979), no. 5, 635–645. MR 541874, DOI 10.1016/0362-546X(79)90092-0
Vieri Benci, Some critical point theorems and applications, Comm. Pure Appl. Math. 33 (1980), no. 2, 147–172. MR 562548, DOI 10.1002/cpa.3160330204
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
Carl L. DeVito, Functional analysis, Pure and Applied Mathematics, vol. 81, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506238
Ju. G. Borisovič, V. G. Zvjagin, and Ju. I. Sapronov, Nonlinear Fredholm mappings, and Leray-Schauder theory, Uspehi Mat. Nauk 32 (1977), no. 4(196), 3–54, 287 (Russian). MR 0464282
H. Brézis and R. E. L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977), no. 6, 601–614. MR 509489, DOI 10.1080/03605307708820041
H. Brézis and L. Nirenberg, Forced vibrations for a nonlinear wave equation, Comm. Pure Appl. Math. 31 (1978), no. 1, 1–30. MR 470377, DOI 10.1002/cpa.3160310102
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
Alfonso Castro B., A two-point boundary value problem with jumping nonlinearities, Proc. Amer. Math. Soc. 79 (1980), no. 2, 207–211. MR 565340, DOI 10.1090/S0002-9939-1980-0565340-1
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, DOI 10.1007/BF02411940
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, DOI 10.1002/cpa.3160330203
Kung Ching Chang, Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math. 34 (1981), no. 5, 693–712. MR 622618, DOI 10.1002/cpa.3160340503
Frank H. Clarke and Ivar Ekeland, Hamiltonian trajectories having prescribed minimal period, Comm. Pure Appl. Math. 33 (1980), no. 2, 103–116. MR 562546, DOI 10.1002/cpa.3160330202
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, DOI 10.1090/cbms/038
I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353. MR 346619, DOI 10.1016/0022-247X(74)90025-0
Ivar Ekeland, Periodic solutions of Hamiltonian equations and a theorem of P. Rabinowitz, J. Differential Equations 34 (1979), no. 3, 523–534. MR 555325, DOI 10.1016/0022-0396(79)90034-2
Ivar Ekeland and Roger Temam, Analyse convexe et problèmes variationnels, Collection Études Mathématiques, Dunod, Paris; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974 (French). MR 0463993
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
K. D. Elworthy and A. J. Tromba, Differential structures and Fredholm maps on Banach manifolds, Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 45–94. MR 0264708
K. D. Elworthy and A. J. Tromba, Degree theory on Banach manifolds, Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 1, Chicago, Ill., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 86–94. MR 0277009
Edward R. Fadell and Paul H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math. 45 (1978), no. 2, 139–174. MR 478189, DOI 10.1007/BF01390270
F. Brock Fuller, An index of fixed point type for periodic orbits, Amer. J. Math. 89 (1967), 133–148. MR 209600, DOI 10.2307/2373103
Kazimierz Gęba and Andrzej Granas, Infinite dimensional cohomology theories, J. Math. Pures Appl. (9) 52 (1973), 145–270. MR 380865
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, DOI 10.1080/03605308108820196
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, DOI 10.1002/cpa.3160320103
Lars Hörmander, The boundary problems of physical geodesy, Arch. Rational Mech. Anal. 62 (1976), no. 1, 1–52. MR 602181, DOI 10.1007/BF00251855
Jorge Ize, Bifurcation theory for Fredholm operators, Mem. Amer. Math. Soc. 7 (1976), no. 174, viii+128. MR 425696, DOI 10.1090/memo/0174
Sergiu Klainerman, Global existence for nonlinear wave equations, Comm. Pure Appl. Math. 33 (1980), no. 1, 43–101. MR 544044, DOI 10.1002/cpa.3160330104
M. A. Krasnosel’skii, Topological methods in the theory of nonlinear integral equations, A Pergamon Press Book, The Macmillan Company, New York, 1964. Translated by A. H. Armstrong; translation edited by J. Burlak. MR 0159197
Jean Leray and Jules Schauder, Topologie et équations fonctionnelles, Ann. Sci. École Norm. Sup. (3) 51 (1934), 45–78 (French). MR 1509338, DOI 10.24033/asens.836
N. G. Lloyd, Degree theory, Cambridge Tracts in Mathematics, No. 73, Cambridge University Press, Cambridge-New York-Melbourne, 1978. MR 0493564
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, DOI 10.1016/0022-0396(82)90085-7
J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Applied Mathematical Sciences, Vol. 19, Springer-Verlag, New York, 1976. With contributions by P. Chernoff, G. Childs, S. Chow, J. R. Dorroh, J. Guckenheimer, L. Howard, N. Kopell, O. Lanford, J. Mallet-Paret, G. Oster, O. Ruiz, S. Schecter, D. Schmidt and S. Smale. MR 0494309, DOI 10.1007/978-1-4612-6374-6
Jürgen 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, DOI 10.1073/pnas.47.11.1824
John Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. (2) 63 (1956), 20–63. MR 75639, DOI 10.2307/1969989
Wei Ming Ni, Some minimax principles and their applications in nonlinear elliptic equations, J. Analyse Math. 37 (1980), 248–275. MR 583639, DOI 10.1007/BF02797687
L. Nirenberg, An application of generalized degree to a class of nonlinear problems, Troisième Colloque sur l’Analyse Fonctionnelle (Liège, 1970) Vander, Louvain, 1971, pp. 57–74. MR 0413207
L. Nirenberg, Topics in nonlinear functional analysis, Courant Institute of Mathematical Sciences, New York University, New York, 1974. With a chapter by E. Zehnder; Notes by R. A. Artino; Lecture Notes, 1973–1974. MR 0488102
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
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
P. H. Rabinowitz, Variational methods for nonlinear eigenvalue problems, Eigenvalues of non-linear problems (Centro Internaz. Mat. Estivo (C.I.M.E.), III Ciclo, Varenna, 1974) Edizioni Cremonese, Rome, 1974, pp. 139–195. MR 0464299
Paul H. Rabinowitz, A note on topological degree for potential operators, J. Math. Anal. Appl. 51 (1975), no. 2, 483–492. MR 470773, DOI 10.1016/0022-247X(75)90134-1
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
Paul H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), no. 2, 157–184. MR 467823, DOI 10.1002/cpa.3160310203
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
David H. Sattinger, Topics in stability and bifurcation theory, Lecture Notes in Mathematics, Vol. 309, Springer-Verlag, Berlin-New York, 1973. MR 0463624, DOI 10.1007/BFb0060079
Martin Schechter, Principles of functional analysis, Academic Press, New York-London, 1971. MR 0445263
J. T. Schwartz, Nonlinear functional analysis, Notes on Mathematics and its Applications, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Notes by H. Fattorini, R. Nirenberg and H. Porta, with an additional chapter by Hermann Karcher. MR 0433481