Geometriae Dedicata

, Volume 155, Issue 1, pp 1–20

Uniform decay estimates for solutions of the Yamabe equation

Original Paper
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Abstract

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 LΓ-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.

Keywords

Nonlinear elliptic partial differential equations Yamabe problem 

Mathematics Subject Classification (2000)

53C21 35J60 

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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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