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

Abstract

We study the asymptotic behavior of positive groundstate solutions to the quasilinear elliptic equation \begin{equation} -Δ_{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 $\epsilon >0$ is a small parameter. For $\epsilon \to 0$, 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 $p^{\ast }:=\frac{pN}{N-p}$.

38 pages