We study the sharp constant for the embedding of
W
0
1
,
p
(
Ω
)
W^{1,p}_0(\Omega )
into
L
q
(
Ω
)
L^q(\Omega )
, in the case
2
>
p
>
q
2>p>q
. We prove that for smooth connected sets, when
q
>
p
q>p
and
q
q
is sufficiently close to
p
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
−
p-
Laplace–type equations by L. Damascelli and B. Sciunzi.