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Growth Estimates on Positive Solutions of the Equation

Published online by Cambridge University Press:  20 November 2018

Man Chun Leung
Man Chun Leung*
Affiliation:
Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Republic of Singapore, e-mail: matlmc@math.nus.edu.sg
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Abstract

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We construct unbounded positive ${{C}^{2}}$-solutions of the equation $\Delta u\,+\,K{{u}^{\left( n+2 \right)/\left( n-2 \right)}}\,=\,0$ in ${{\mathbb{R}}^{n}}$ (equipped with Euclidean metric ${{g}_{0}}$) such that $K$ is bounded between two positive numbers in ${{\mathbb{R}}^{n}}$, the conformal metric $g\,=\,{{u}^{4/\left( n-2 \right)}}{{g}_{0}}$ is complete, and the volume growth of $g$ can be arbitrarily fast or reasonably slow according to the constructions. By imposing natural conditions on $u$, we obtain growth estimate on the ${{L}^{2n/\left( n-2 \right)}}$-norm of the solution and show that it has slow decay.

Keywords

35J6058G03positive solutionconformal scalar curvature equationgrowth estimate
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 44 , Issue 2 , 01 June 2001 , pp. 210 - 222
Copyright
Copyright © Canadian Mathematical Society 2001

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