Mathematische Annalen

, Volume 327, Issue 4, pp 723–744

Blow-up solutions of nonlinear elliptic equations in ℝn with critical exponent Dedicated to Philippe Dufour, whose exquisite horology artwork ‘‘Simplicity’’ inspires so much.

Article

Abstract

For an integer n≥3 and any positive number ɛ, we establish the existence of smooth functions K on ℝn∖{0} with |K−1|≤ɛ, such that the equation \({{ \Delta u + n (n - 2) K u^{{{{n + 2}}}\over{ {{n - 2}}}} = 0 {{{{\rm{ in}}}}} {{{{\mathbb R}}}}^n \setminus \{ 0 \} }}\) has a smooth positive solution which blows up at the origin (i.e., u does not have slow decay near the origin). Furthermore, we show that in some situations K can be extended as a Lipschitz function on ℝn. These provide counter-examples to a conjecture of C.-S. Lin when n>4, and a question of Taliaferro.

Keywords

nonlinear differential equation scalar curvature blow-up solutions decay estimates 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Department of MathematicsNational University of SingaporeSingaporeRepublic of Singapore

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