  We study the sharp constant for the embedding of $W^{1,p}_0(\Omega)$ into
$L^q(\Omega)$, in the case $2<p<q$. We prove that for smooth connected sets,
when $q>p$ and $q$ is sufficiently close to $p$, extremal functions attaining
the sharp constant are unique, up to a multiplicative constant. This in turn
gives the uniqueness of solutions with minimal energy to the Lane-Emden
equation, with super-homogeneous right-hand side.
  The result is achieved by suitably adapting a linearization argument due to
C.-S. Lin. We rely on some fine estimates for solutions of $p-$Laplace--type
equations by L. Damascelli and B. Sciunzi.
