Multiple solutions of differential equations without the Palais-Smale condition. (English) Zbl 0506.35034


35J60 Nonlinear elliptic equations
35A15 Variational methods applied to PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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[1] Bahri, A.: Topological results on a certain class of functionals and applications (preprint) · Zbl 0499.35050
[2] Bahri, A., Berestycki, H.: A perturbation method in critical point theory and applications. Trans. AMS (to appear) · Zbl 0476.35030
[3] Berkowitz, L.D.: Lower semicontinuity of integral functionals. Trans. AMS192, 51-57 (1974) · Zbl 0294.49001
[4] Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. Arch. Rat. Mech. Anal. (to appear) · Zbl 0707.35143
[5] Berestycki, H., Lions, P.-L.: Existence of infinitely many solutions in the ?zero mass? case for nonlinear field equations (in preparation)
[6] Berger, M.S.: On the existence and structure of stationary states for a nonlinear Klein-Gordon equation. J. Functional Analysis9, 249-261 (1872) · Zbl 0224.35061
[7] Bongers, A., Heinz, H.-P., K?ppers, T.: Existence and bifurcation theorems for nonlinear elliptic eigenvalue problems on unbounded domains (preprint)
[8] Br?zis, H., Kato, T.: Remarks on the Schr?dinger operators with singular complex potentials. J. Math. Pures Appl.59, 137-151 (1979) · Zbl 0408.35025
[9] Browder, F.E.: Existence theorems for nonlinear partial differential equations. Proc. Symp. Pure App. Math.16, 1-60 (1968) · Zbl 0165.49803
[10] Chang, K.C.: Variational methods for nondifferentiable functionals and its applications to partial differential equations (to appear)
[11] Coffman, C.V.: A minimum-maximum principle for a class of nonlinear integral equations. J. Analyse Math.22, 391-419 (1960) · Zbl 0179.15601
[12] Coleman, S., Glaser, V., Martin, A.: Action minima among solutions to a class of Euclidian scalar field equations. Comm. Math. Phys.58, 211-221 (1978)
[13] Eisen, G.: A selection lemma for sequences of measurable sets and lower semicontinuity of multiple integrals. Manuscripta Math.27, 73-79 (1979) · Zbl 0404.28004
[14] Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems. SFB 72 Vorlesungsreihe 6, University of Bonn (1981) · Zbl 0498.49001
[15] Hempel, J.A.: Superlinear boundary value problems and non-uniqueness. Thesis, Univ. of New England, Australia (1970)
[16] Krasnoselskii, M.A.: Topological methods in the theory of nonlinear integral equations. New York: Macmillan 1964
[17] Lusternik, L.A., Schnirelman, L.: Methodes topologiques dans les probl?mes variationnels. Actualit?s Sci. Indust.188 (1934)
[18] Morrey, C.B.: Multiple integrals in the calculus of variations. Berlin, Heidelberg, New York: Springer 1966 · Zbl 0142.38701
[19] Nehari, Z.: Characteristic values associated with a class of nonlinear second order differential equations. Acta Math.105, 141-175 (1961) · Zbl 0099.29104
[20] Palais, R.S.: Morse theory on Hilbert manifolds. Topology2, 299-340 (1963) · Zbl 0122.10702
[21] Poho?aev, S.I.: Eigenfunctions of the equation ?u+?f(u)=0. Doklady165.1, 1408-1412 (1965)
[22] Rabinowitz, P.H.: Variational methods for nonlinear elliptic eigenvalue problems. Ind. Univ. Math. J.23, 729-754 (1974) · Zbl 0278.35040
[23] Smale, S.: Morse theory and a nonlinear generalization of the Dirichlet problem. Ann. Math.80, 382-396 (1964) · Zbl 0131.32305
[24] Strauss, W.A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55, 149-162 (1977) · Zbl 0356.35028
[25] Struwe, M.: Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems. Manuscripta Math.32, 335-364 (1980) · Zbl 0456.35031
[26] Struwe, M.: Quasilinear eigenvalue problems (in preparation) · Zbl 0531.35035
[27] Struwe, M.: On a weakened Palais-Smale condition (in preparation) · Zbl 0604.58017
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