  By using optimal mass transport theory, we provide a direct proof to the
sharp $L^p$-log-Sobolev inequality $(p\geq 1)$ involving a log-concave
homogeneous weight on an open convex cone $E\subseteq \mathbb R^n$. The perk of
this proof is that it allows to characterize the extremal functions realizing
the equality cases in the $L^p$-log-Sobolev inequality. The characterization of
the equality cases is new for $p\geq n$ even in the unweighted setting and
$E=\mathbb R^n$. As an application, we provide a sharp weighted
hypercontractivity estimate for the Hopf-Lax semigroup related to the
Hamilton-Jacobi equation, characterizing also the equality cases.
