Growth estimates on positive solutions of the equation $\Delta u+K{u}^{\frac{n+2}{n-2}}=0$ in ${\mathbb{R}}^n$

Abstract

We construct unbounded positive $C^2$-solutions of the equation $\Delta u+K{u}^{(n+2)/(n-2)}=0$ in ${\mathbb{R}}^n$ (equipped with Euclidean metric $g_o$) such that $K$ is bounded between two positive numbers in ${\mathbb{R}}^n$, 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.

15 pages