Multiplicity of ground states for the scalar curvature equation

We study existence and multiplicity of radial ground states for the scalar curvature equation $$ Delta u+ K(|x|), u^{rac{n+2}{n-2}}=0, quad xinRR^n,, quad n>2, $$ when the function $K:RR^+ o RR^+$ is bounded above and below by two positive constants, i.e. $0<\underline{K} leq K(r) leq overline{K}$ for every $r > 0$, it is decreasing in $(0,1)$ and increasing in $(1,+infty)$. Chen and Lin in cite{CL} had shown the existence of a large number of bubble tower solutions if $K$ is a sufficiently small perturbation of a positive constant. Our main purpose is to improve such a result by considering a non-perturbative situation: we are able to prove multiplicity assuming that the ratio $overline{K}/\underline{K}$ is smaller than some computable values.