Multiplicity of solutions of a zero mass nonlinear equation on a Riemannian manifold. (English) Zbl 1152.58018

The paper deals with relation between the number of solutions of a nonlinear equation on a Riemannian manifold and the topology of the manifold. The machinery used is based on the Ljusternik-Schnirelmann category and Morse theory.


58J05 Elliptic equations on manifolds, general theory
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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[1] Bahri, A.; Coron, J.-M., On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain, Comm. Pure Appl. Math., 41, 3, 253-294 (1988) · Zbl 0649.35033
[2] Benci, V., Introduction to Morse theory: A new approach, (Topological Nonlinear Analysis. Topological Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl., vol. 15 (1995), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 37-177 · Zbl 0823.58008
[3] Benci, V.; Bonanno, C.; Micheletti, A. M., On the multiplicity of solutions of a nonlinear elliptic problem on Riemannian manifolds, J. Funct. Anal., 252, 2, 464-489 (2007) · Zbl 1130.58010
[4] Benci, V.; Cerami, G., Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations, 2, 1, 29-48 (1994) · Zbl 0822.35046
[5] Benci, V.; Cerami, G., The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Ration. Mech. Anal., 114, 79-93 (1991) · Zbl 0727.35055
[6] Benci, V.; Cerami, G.; Passaseo, D., On the number of the positive solutions of some nonlinear elliptic problems, (Ambrosetti, A.; Marino, A., Nonlinear Analysis. A Tribute in Honour of Giovanni Prodi (1991), Publ. Sc. Norm. Super. Pisa), 93-107 · Zbl 0838.35040
[7] Benci, V.; Micheletti, A. M., Solutions in exterior domains of null mass nonlinear field equations, Adv. Nonlinear Stud., 6, 2, 171-198 (2006) · Zbl 1114.35143
[8] Berestycki, H.; Lions, P.-L., Existence d’états multiples dans des équations de champs scalaires non linéaires dans le cas de masse nulle, C. R. Acad. Sci. Paris Sér. I Math., 297, 4, 267-270 (1983) · Zbl 0542.35072
[9] Berestycki, H.; Lions, P.-L., Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal., 82, 4, 313-345 (1983) · Zbl 0533.35029
[10] Dancer, E. N., A note on an equation with critical exponent, Bull. London Math. Soc., 20, 6, 600-602 (1988) · Zbl 0646.35027
[11] de Figueiredo, D. G., Lectures on the Ekeland Variational Principle with Applications and Detours, Tata Inst. Fund. Res. Lect. Math. Phys., vol. 81 (1989), Springer-Verlag: Springer-Verlag Berlin · Zbl 0688.49011
[12] Hebey, E., Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lect. Notes Math., vol. 5 (1999), Courant Institute of Mathematical Sciences, New York University
[13] Ljusternik, L.; Schnirelmann, L., Méthodes topologiques dans les problèmes variationelles, Actualites Sci. Industr., vol. 188 (1934)
[14] Milnor, J., Morse Theory, Ann. of Math. Stud., vol. 51 (1963), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ, Based on lecture notes by M. Spivak and R. Wells · Zbl 0108.10401
[15] Strauss, W. A., Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55, 2, 149-162 (1977) · Zbl 0356.35028
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