  We study the asymptotic behavior of positive groundstate solutions to the
quasilinear elliptic equation \begin{equation} -\Delta_{p} u + \varepsilon
u^{p-1} - u^{q-1} +u^{\mathit{l}-1} = 0 \qquad \text{in} \quad \mathbb{R}^{N},
\end{equation} where $1<p<N $, $p<q<l<+\infty$ and $\varepsilon> 0 $ is a small
parameter. For $\varepsilon\rightarrow 0$, we give a characterisation of
asymptotic regimes as a function of the parameters $q$, $l$ and $N$. In
particular, we show that the behavior of the groundstates is sensitive to
whether $q$ is less than, equal to, or greater than the critical Sobolev
exponent $p^{*} :=\frac{pN}{N-p}$.
