  We study asymptotic behaviour of positive ground state solutions of the
nonlinear Schr\"odinger equation $$ -\Delta u+ u=u^{2^*-1}+\lambda u^{q-1}
\quad {\rm in} \ \ \mathbb{R}^N, $$ where $N\ge 3$ is an integer,
$2^*=\frac{2N}{N-2}$ is the Sobolev critical exponent, $2<q<2^*$ and
$\lambda>0$ is a parameter. It is known that as $\lambda\to 0$, after a
rescaling the ground state solutions of the equation converge to a particular
solution of the critical Emden-Fowler equation $-\Delta u=u^{2^*-1}$. We
establish a sharp asymptotic characterisation of such a rescaling, which
depends in a non-trivial way on the space dimension $N=3$, $N=4$ or $N\ge 5$.
