Arbitrarily large solutions of the conformal scalar curvature problem at an isolated singularity

Abstract:

We study the conformal scalar curvature problem \begin{equation*}k(x) u^{\frac {n+2}{n-2}} \le -\Delta u \le u^{\frac {n+2}{n-2}}\qquad \mathrm {in} \qquad \mathbf {R}^{n},\ n\ge 3,\end{equation*} where $k : \mathbf {R}^{n} \to (0,1]$ is a continuous function. We show that a necessary and sufficient condition on $k$ for this problem to have $C^{2}$ positive solutions which are arbitrarily large at $\infty$ is that $k$ be less than 1 on a sequence of points in $\mathbf {R}^{n}$ which tends to $\infty$.