Geometriae Dedicata

, Volume 155, Issue 1, pp 1–20 | Cite as

Uniform decay estimates for solutions of the Yamabe equation

Original Paper


We study positive solutions u of the Yamabe equation \({c_{m} \Delta u-s\left( x\right) u+k\left( x\right) u^{\frac{m+2}{m-2}}=0}\), when k(x) > 0, on manifolds supporting a Sobolev inequality. In particular we get uniform decay estimates at infinity for u which depend on the behaviour at infinity of k, s and the L Γ-norm of u, for some \({\Gamma\geq\tfrac{2m}{m-2}}\). The required integral control, in turn, is implied by further geometric conditions. Finally we give an application to conformal immersions into the sphere.


Nonlinear elliptic partial differential equations Yamabe problem 

Mathematics Subject Classification (2000)

53C21 35J60 


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  1. 1.
    Hoffman D., Spruck J.: Sobolev and isoperimetric inequalities for Riemannian submanifolds. Comm. Pure Appl. Math. 27, 715–727 (1974)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Kim S.T.: The Yamabe problem and applications on noncompact complete Riemannian manifolds. Geom. Dedicata 64(3), 373–381 (1997)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Leung M.C.: Asymptotic behavior of positive solutions of the equation Δg u + ku p = 0 in a complete Riemannian manifold and positive scalar curvature. Commun. Partial Diff. Equ. 24, 425–462 (1999)CrossRefMATHGoogle Scholar
  4. 4.
    Pigola, S., Veronelli, G.: Uniform decay estimates for finite-energy solutions of semi-linear elliptic inequalities and geometric applications. Differ. Geom. Appl. (to appear)Google Scholar
  5. 5.
    Schoen, R., Yau, S.T.: Lectures on differential geometry. In: Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge (1994)Google Scholar
  6. 6.
    Shen Y.B., Zhu X.H.: On complete hypersurfaces with constant mean curvature and finite L p-norm curvature in \({\mathbb{R} ^{m+1}}\). Acta Math. Sin. 21, 631–642 (2005)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Zhang Q.S.: Finite energy solutions to the Yamabe equation. Geom. Dedicata 101, 153–165 (2003)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di MilanoMilanoItaly

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