Asymptotics of Sobolev embeddings and singular perturbations for the $p$-Laplacian

We consider the best constant $S(\Omega_\lambda)$for the embedding of $W^{1,p} (\Omega_\lambda)$ into $L^q(\Omega_\lambda)$ where $1<p<2$, $p<q< {Np\over N-p}$. Here $\Omega_\lambda = \lambda \Omega$ with $\Omega$a smooth, bounded domain in $\mathbb{R} ^n$ and $\lambda$ a large positive number. It is proven by the validity of the expansion

as $\lambda \to \infty$, where $\gamma$ is a positive constant depending on $p,q$ and $N$. The behavior of associated extremals, which satisfy an equation involving the $p$-Laplacian operator, is also analyzed.