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Proceedings of the American Mathematical Society

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Arbitrarily large solutions of the conformal scalar curvature problem at an isolated singularity


Authors: Steven D. Taliaferro and Lei Zhang
Journal: Proc. Amer. Math. Soc. 131 (2003), 2895-2902
MSC (2000): Primary 35J60, 53C21
Published electronically: January 28, 2003
MathSciNet review: 1974347
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Abstract: We study the conformal scalar curvature problem

\begin{displaymath}k(x) u^{\frac{n+2}{n-2}} \le -\Delta u \le u^{\frac{n+2}{n-2}}\qquad \hbox {in} \qquad \mathbf{R}^{n}, n\ge 3,\end{displaymath}

where $k : \mathbf{R}^{n} \to (0,1]$ is a continuous function. We show that a necessary and sufficient condition on $k$ for this problem to have $C^{2}$positive solutions which are arbitrarily large at $\infty $ is that $k$ be less than 1 on a sequence of points in $\mathbf{R}^{n}$ which tends to $\infty $.


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Additional Information

Steven D. Taliaferro
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: stalia@math.tamu.edu

Lei Zhang
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: lzhang@math.tamu.edu

DOI: https://doi.org/10.1090/S0002-9939-03-06932-6
Received by editor(s): March 1, 2002
Received by editor(s) in revised form: April 11, 2002
Published electronically: January 28, 2003
Communicated by: Bennett Chow
Article copyright: © Copyright 2003 American Mathematical Society