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

TL;DR

This paper constructs positive solutions to a nonlinear PDE in Euclidean space with bounded coefficient and explores their growth behavior, providing estimates on their norms and decay properties.

Contribution

It introduces new unbounded positive solutions with controlled volume growth and establishes growth estimates under natural conditions.

Findings

Solutions can have arbitrarily fast or slow volume growth.

The $L^{2n/(n-2)}$-norm of solutions exhibits slow decay.

Constructed solutions are complete and bounded between positive constants.

Abstract

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.