Abstract
Given any closed Riemannian manifold (M, g) we use the Lyapunov–Schmidt finite-dimensional reduction method and the classical Morse and Lusternick–Schnirelmann theories to prove multiplicity results for positive solutions of a subcritical Yamabe type equation on (M, g). If (N, h) is a closed Riemannian manifold of constant positive scalar curvature we obtain multiplicity results for the Yamabe equation on the Riemannian product \((M\times N , g + \varepsilon ^2 h )\), for \(\varepsilon >0\) small. For example, if M is a closed Riemann surface of genus \(\mathbf{g}\) and \((N,h) = (S^2 , g_0)\) is the round 2-sphere, we prove that for \(\varepsilon >0\) small enough and a generic metric g on M, the Yamabe equation on \((M\times S^2 , g + \varepsilon ^2 g_0 )\) has at least \(2 + 2 \mathbf{g}\) solutions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akutagawa, K., Florit, L., Petean, J.: On Yamabe constants of Riemannian products. Commun. Anal. Geom. 15, 947–969 (2007)
Aubin, T.: Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. 55, 269–296 (1976)
Bahri, A., Coron, J.M.: On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect on the topology of the domain. Commun. Pure. Appl. Math. 41, 253–294 (1988)
Bettiol, R., Piccione, P.: Multiplicity of solutions to the Yamabe problem on collapsing Riemannian submersions. Pac. J. Math. 266, 1–21 (2013)
Brendle, S.: Blow-up phenomena for the Yamabe equation. J. Am. Math. Soc. 21, 951–979 (2008)
Brendle, S., Marques, F.C.: Blow-up phenomena for the Yamabe equation II. J. Differ. Geom. 81(2), 225–250 (2009)
De Lima, L.L., Piccione, P., Zedda, M.: A note on the uniqueness of solutions for the Yamabe problem. Proc. Am. Math. Soc. 140, 4351–4357 (2012)
De Lima, L.L., Piccione, P., Zedda, M.: On bifurcation of solutions of the Yamabe problem in product manifolds. Ann. Inst. Henri Poincaré C 29, 261–277 (2012)
Deng, S., Khemiri, Z., Mahmoudi, F.: On spike solutions for a singularly preturbed problem in a compact Riemannian manifold. Commun. Pure Appl. Anal. 17, 2063–2084 (2018)
Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic Schrodinger equation with a bounded potential. J. Funct. Anal. 69, 397–408 (1986)
Gidas, B., Ni, W.-M., Nirenberg, L.: Symmetry of Positive Solutions of Nonlinear Elliptic Equations in \({ R}^n\). Mathematical Analysis and Applications, Advanced Mathematical Supplemented Studies 7A, pp. 369–402. Academic Press, New York (1981)
Henry, G., Petean, J.: Isoparametric hypersurfaces and metrics of constant scalar curvature. Asian J. Math. 18, 53–67 (2014)
Jin, Q., Li, Y.Y., Xu, H.: Symmetry and asymmetry: the method of moving spheres. Adv. Differ. Equ. 13, 601–640 (2008)
Kobayashi, O.: Scalar curvature of a metric with unit volume. Math. Ann. 279, 253–265 (1987)
Kwong, M.K.: Uniqueness of positive solutions of \(\Delta u -u + u^p =0\) in \({ R}^n\). Arch. Rational. Mech. Anal. 105, 243–266 (1989)
Khuri, M.A., Marques, F.C., Schoen, R.M.: A compactness theorem for the Yamabe Problem. J. Differ. Geomet. 81, 143–196 (2009)
Li, Y.Y., Nirenberg, L.: The Dirichlet problem for singularly perturbed elliptic equations. Commun. Pure Appl. Math. 36, 437–477 (1983)
Micheletti, A.M., Pistoia, A.: The role of the scalar curvature in a nonlinear elliptic problem on Riemannian manifolds. Calc. Variations PDE 34, 233–265 (2009)
Micheletti, A.M., Pistoia, A.: Generic properties of critical points of the scalar curvature for a Riemannian manifold. Proc. Am. Math. Soc. 138, 3277–3284 (2010)
Obata, M.: The conjectures on conformal transformations for a conformally invariant scalar equation. J. Diff. Geom. 6, 247–258 (1971)
Petean, J.: Metrics of constant scalar curvature conformal to Riemannian products. Proc. Am. Math. Soc. 138, 2897 (2010)
Petean, J.: Multiplicity results for the Yamabe equation by Lusternik-Schnirelmann Theory. J. Funct. Anal. 276, 1788–1805 (2019)
Pollack, D.: Nonuniqueness and high energy solutions for a conformally invariant scalar equation. Commun. Anal. Geom. 1, 347–414 (1993)
Rey, C., Ruiz, J.M.: Multipeak solutions for the Yamabe equation. J. Geom. Anal. (2019). https://doi.org/10.1007/s12220-019-00258-4
Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20, 479–495 (1984)
Schoen, R.: Variational Theory for the Total Scalar Curvature Functional for Riemannian Metrics and Related Topics in the Calculus of Variations. Lectures Notes in Mathematics, vol. 1365, pp. 120–154. Springer, Berlin (1989)
Trudinger, N.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. 22, 265–274 (1968)
Wei, J., Winter, M.: Mathematics Aspects of Pattern Formations in Biological System. AMS 189. Springer, Berlin (2014)
Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12, 21–37 (1960)
Acknowledgements
Most of the work in this paper was done during the Isidro H. Munive’s visit to the Center for Mathematical Research (CIMAT A.C.), which the author thanks for its hospitality. We also wish to thank Prof. Jimmy Petean for his constant interest and the many helpful conversations on the Yamabe equation.
Funding
Isidro H. Munive was partially supported by Proyecto FORDECYT 265667, CONACYT, México.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Dávila, J., Munive, I.H. Solutions of the Yamabe Equation by Lyapunov–Schmidt Reduction. J Geom Anal 31, 8080–8104 (2021). https://doi.org/10.1007/s12220-020-00570-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-020-00570-4