On the multiplicity of solutions of a nonlinear elliptic problem on Riemannian manifolds. (English) Zbl 1130.58010

The very interesting paper under review deals with the existence of different solutions to a nonlinear elliptic problem on Riemannian manifolds. Precisely, let \((M,g)\) be a \(C^\infty,\) compact and connected Riemannian manifold without boundary of dimension \(n\geq 3.\) Consider the problem
\[ \begin{cases} -\varepsilon^2\Delta_gu+u-u| u| ^{p-2}=0,\\ 0<u\in H^1_g(M) \end{cases} \tag \(*\) \] for \(p\in(2,2^*)\) with \(2^*\) being the critical exponent for the Sobolev immersion. The authors study the relation between the number of solutions to \((*)\) and the topology of the manifold \(M.\) Precisely, set \(\text{cat}(M)\) for the Ljusternik-Schnirelmann category of \(M\) in itself, and \(P_t(M)\) for its Poincaré polynomial.
The main results of the paper are as follows:
Theorem A. For small enough \(\varepsilon>0\) there exist at least \(\text{cat}(M)+1\) non-constant distinct solutions of the problem \((*)\).
Theorem B. Assume that for small enough \(\varepsilon>0\) all the solutions of \((*)\) are non-degenerate. Then there are at least \(2P_1(M)-1\) solutions.


58J05 Elliptic equations on manifolds, general theory
35J60 Nonlinear elliptic equations
Full Text: DOI


[1] Bahri, A.; Coron, J. M., On a nonlinear elliptic equation involving the Sobolev exponent: The effect of the topology of the domain, Comm. Pure Appl. Math., 41, 253-294 (1988) · Zbl 0649.35033
[2] Bahri, A.; Li, Y. Y., On a min-max procedure for the existence of a positive solution for certain scalar field equations in \(R^N\), Rev. Mat. Iberoamericana, 6, 1-16 (1990) · Zbl 0731.35036
[3] Bahri, A.; Lions, P. L., On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14, 365-413 (1997) · Zbl 0883.35045
[4] 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
[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., Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations, 2, 29-48 (1994) · Zbl 0822.35046
[7] 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. Sup. Pisa, Ed. Norm. Pisa: Publ. Sc. Norm. Sup. Pisa, Ed. Norm. Pisa Pisa), 93-107 · Zbl 0838.35040
[8] Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36, 437-477 (1983) · Zbl 0541.35029
[9] Byeon, J.; Wang, Z. Q., Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 165, 295-316 (2002) · Zbl 1022.35064
[10] Coron, J. M., Topologie et cas limite des injections de Sobolev, C. R. Acad. Sci. Paris Sér. I Math., 299, 209-212 (1984) · Zbl 0569.35032
[11] Dancer, E. N., The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Differential Equations, 74, 120-156 (1988) · Zbl 0662.34025
[12] Ding, W. Y., Positive solutions of \(\Delta u + u^{(n + 2) /(n - 2)} = 0\) on contractible domains, J. Partial Differential Equations, 2, 83-88 (1989) · Zbl 0694.35067
[13] de Figueiredo, D. G., Lectures on the Ekeland Variational Principle with Applications and Detours, Tata Institute of Fundamental Research Lectures on Math. and Phys., vol. 81 (1989), Springer: Springer Berlin · Zbl 0688.49011
[14] Nash, J., The imbedding problem for Riemannian manifolds, Ann. of Math., 63, 20-63 (1956) · Zbl 0070.38603
[15] Passaseo, D., Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains, Manuscripta Math., 65, 147-166 (1989) · Zbl 0701.35068
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.