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

Abstract:

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 \begin{equation} S( \Omega _\lambda ) = S(\mathbb {R}^n_+) - \lambda ^{-1} \gamma \max _{x\in \partial \Omega } H(x) + o ( \lambda ^{-1} ), \nonumber \end{equation} 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.