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

We study the asymptotic behaviour of positive groundstate solutions to the quasilinear elliptic equation Open image in new window where \(1 0 \) is a small parameter. For \({\varepsilon }\rightarrow 0\), we give a characterization of asymptotic regimes as a function of the parameters q, l and N. In particular, we show that the behaviour of the groundstates is sensitive to whether q is less than, equal to, or greater than the critical Sobolev exponent \(p^{*} :=\frac{pN}{N-p}\).