The scalar curvature equation on $\mathbb R^n$ and $S^n$

Abstract

We study the existence of positive solutions for the equation $\Delta u+ K(x) u^{\frac{n+2}{n-2}}=0$ in $\mathbb{R}^n$ ($n \geq 3$) which decay to $0$ at infinity like $|x|^{2-n}$; $K(x)$ is a function which is bounded from above and below by positive constants, and no symmetry assumption on $K$ is made. We find conditions that guarantee existence for a large class of $K$'s. As a consequence one can explicitly show that the set of coefficients for which a solution exists is dense, in $C^1$ norm, in the space of positive bounded $C^1$ functions. These conditions also allow us to display a radial $K$ such that the previous problem has nonradial solutions but no radial solution. Some new results for the corresponding problem on $\Sn$ are also proved.