A perturbation method in critical point theory and applications. (English) Zbl 0476.35030

Bahri, Abbas; Berestycki, Henri

Trans. Am. Math. Soc. 267, 1-32 (1981).

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35B20	Perturbations in context of PDEs

Keywords:

nonlinear elliptic equation; critical point theory; multiple solutions; perturbation theorem

References:

[1]	Shmuel Agmon, Lectures on elliptic boundary value problems, Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. · Zbl 0142.37401
[2]	A. Ambrosetti, A perturbation theorem for superlinear boundary value problems, Math. Res. Center, Univ. of Wisconsin-Madison, Tech. Sum. Report # 1446, 1974.
[3]	Antonio Ambrosetti, On the existence of multiple solutions for a class of nonlinear boundary value problems, Rend. Sem. Mat. Univ. Padova 49 (1973), 195 – 204. · Zbl 0273.35037
[4]	Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349 – 381. · Zbl 0273.49063
[5]	Abbas Bahri, Résolution générique d’une équation semi-linéaire, C. R. Acad. Sci. Paris Sér. A-B 291 (1980), no. 4, A251 – A254 (French, with English summary). · Zbl 0461.35037
[6]	-, Topological results on a certain class of functionals and applications, J. Funct. Anal. (in press).
[7]	Abbas Bahri and Henri Berestycki, Points critiques de perturbations de fonctionnelles paires et applications, C. R. Acad. Sci. Paris Sér. A-B 291 (1980), no. 3, A189 – A192 (French, with English summary). · Zbl 0454.35041
[8]	-, Forced vibrations of superquadratic Hamiltonian systems (to appear); See also: Existence d’une infinité de solutions périodiques pour certains systèmes Hamiltoniens en présence d’un terme de contrainte, C. R. Acad. Sci. Paris. Sér. I 292 (1981), 315-318.
[9]	-, Existence of periodic solutions for some second order systems of nonlinear ordinary differential equations (to appear).
[10]	Charles V. Coffman, A minimum-maximum principle for a class of non-linear integral equations, J. Analyse Math. 22 (1969), 391 – 419. · Zbl 0179.15601 · doi:10.1007/BF02786802
[11]	P. E. Conner and E. E. Floyd, Fixed point free involutions and equivariant maps, Bull. Amer. Math. Soc. 66 (1960), 416 – 441. · Zbl 0106.16301
[12]	R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. · Zbl 0051.28802
[13]	J. Dugundji, An extension of Tietze’s theorem, Pacific J. Math. 1 (1951), 353 – 367. · Zbl 0043.38105
[14]	Hans Ehrmann, Über die Existenz der Lösungen von Randwertaufgaben bei gewöhnlichen nichtlinearen Differentialgleichungen zweiter Ordnung, Math. Ann. 134 (1957), 167 – 194 (German). · Zbl 0078.07801 · doi:10.1007/BF01342795
[15]	Svatopluk Fučík and Vladimír Lovicar, Periodic solutions of the equation \?^{\(^{\prime}\)\(^{\prime}\)}(\?)+\?(\?(\?))=\?(\?), Časopis Pěst. Mat. 100 (1975), no. 2, 160 – 175.
[16]	Joachim A. Hempel, Multiple solutions for a class of nonlinear boundary value problems., Indiana Univ. Math. J. 20 (1970/1971), 983 – 996. · Zbl 0225.35045 · doi:10.1512/iumj.1971.20.20094
[17]	M. A. Krasnosel’skii, Topological methods in the theory of nonlinear integral equations, Translated by A. H. Armstrong; translation edited by J. Burlak. A Pergamon Press Book, The Macmillan Co., New York, 1964.
[18]	A. Marino and G. Prodi, Metodi perturbativi nella teoria di Morse, Boll. Un. Mat. Ital. (4) 11 (1975), no. 3, suppl., 1 – 32 (Italian, with English summary). Collection of articles dedicated to Giovanni Sansone on the occasion of his eighty-fifth birthday. · Zbl 0311.58006
[19]	Richard S. Palais, Lusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966), 115 – 132. · Zbl 0143.35203 · doi:10.1016/0040-9383(66)90013-9
[20]	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.
[21]	Paul H. Rabinowitz, Variational methods for nonlinear elliptic eigenvalue problems, Indiana Univ. Math. J. 23 (1973/74), 729 – 754. · Zbl 0278.35040 · doi:10.1512/iumj.1974.23.23061
[22]	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.