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1. CMB 2001 (vol 44 pp. 210)
Growth Estimates on Positive Solutions of the Equation $\Delta u+K u^{\frac{n+2}{n-2}}=0$ in $\R^n$ We construct unbounded positive $C^2$-solutions of the equation
$\Delta u + K u^{(n + 2)/(n - 2)} = 0$ in $\R^n$ (equipped
with Euclidean metric $g_o$) such that $K$ is bounded between two
positive numbers in $\R^n$, the conformal metric $g=u^{4/(n-2)}g_o$
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 $L^{2n/(n-2)}$-norm
of the solution and show that it has slow decay.
Keywords:positive solution, conformal scalar curvature equation, growth estimate Categories:35J60, 58G03 |