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Growth Estimates on Positive Solutions of the Equation ![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20190730041132095-0974:S0008439500014314:S0008439500014314_inline1.gif?pub-status=live)
Published online by Cambridge University Press: 20 November 2018
Abstract
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.
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- Copyright © Canadian Mathematical Society 2001
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