We consider the problem
$ \cases{ - \Delta u = N(N-2) u^p +\varepsilon u & \mbox{in} $\varOmega$,\cr u >0 & \mbox{in} $\varOmega$,\cr u =0 & \mbox{on} $\curpartial\varOmega$,\cr} $
where
$\varOmega$
is a bounded smooth domain of
$ \mathbf{R}^N \ (N \geq 5)$
which is symmetric with respect to the coordinate hyperplanes
$\{x_i = 0\}$
,
$i = 1, \ldots, N ,$
and it is convex in the
$x_i$
‐directions for
$i = 1, \ldots, N$
; here
$0 < \varepsilon < \lambda_1$
(
$\lambda_1 $
being the first eigenvalue of the Laplace operator in
$ H_0^1 (\varOmega)$
) and
$p = ({N+2})/({N-2})$
. For
$ \varepsilon $
small, we prove uniqueness and nondegeneracy of the solution
$ u_\varepsilon $
with the property that
$ \lim_{\varepsilon \rightarrow0} \frac{\int_\varOmega|\nabla u_\varepsilon|^2 \,\mathrm{d} x }{ (\int_\varOmega|u_\varepsilon|^{p+1}\,\mathrm{d} x)^{2/p+1} } = S_N, $
where
$S_N$
is the best Sobolev constant in
$ \mathbf{R}^N$
.
