Groundstate asymptotics for a class of singularly perturbed $p$-Laplacian problems in $\mathbb {R}^N$

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 $p^{*} :=\frac{pN}{N-p}$.