From the introduction: 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 \[ S(\Omega_\lambda) = S(\mathbb{R}^n_+) - \lambda^{-1} \gamma \max_{x\in \partial \Omega}H(x) + o(\lambda^{-1}) \] as \(\lambda \to \infty\), where \(\gamma\) is a positive constant depending on \(p,q\) and \(N\). The behavior of associated extremals \(u_\lambda\), which satisfy the Euler-Lagrange equation \[ \Delta_p u_\lambda-|u_\lambda|^{p-2} u_\lambda+|u_\lambda|^{q-2} u_\lambda=0 \quad\text{in }\Omega_\lambda, \]
\[ u>0 \quad\text{in }\Omega_\lambda, \qquad \frac{\partial u}{\partial\eta}=0\quad\text{on }\partial\Omega_\lambda, \] where \(\Delta_p\) stands for the \(p\)-Laplacian operator, \(\Delta_p u= \text{div}(|\nabla u|^{p-2} \nabla u)\), is also analyzed.