We give a structure result for the positive radial solutions of the following equation: $${\Delta }_{p}u+K\left(r\right)u{\left|u\right|}^{q-1}=0$$ with some monotonicity assumptions on the positive function $K\left(r\right)$. Here $r=\left|x\right|$, $x\in {ℝ}^n$; we consider the case when $n>p>1$, and $q>p_{*}=\frac{n(p-1)}{n-p}$. We continue the discussion started by Kawano et al. in [KYY], refining the estimates on the asymptotic behavior of Ground States with slow decay and we state the existence of S.G.S., giving also for them estimates on the asymptotic behavior, both as $r\to 0$ and as $r\to \infty$. We make use of a Emden-Fowler transform...