Consider the following semilinear problem: $$ \begin{cases} - \Delta u + u = u |u|^{p - 2} &\text{in} ~ \mathbb{R}^N; \\ u (x) \to 0 &\text{as} ~ |x| \to \infty, \end{cases} $$ where $N \geq 2$, $2 < p < \infty$ if $N = 2$ and $2 < p < \frac{2 N}{N - 2}$ if $N \geq 3$. It is well-known that there exists $\mu > 0$ such that if $U$ denotes a positive solution to this problem, then $U (x) |x|^{\frac{N - 1}{2}} e^{|x|} \to \mu$ as $|x| \to \infty$ (see Theorem 2 in Gidas, Ni & Nirenberg's 1981 paper as in Zbl 0469.35052 or MR0634248).
Even though this asymptotic behavior is cited in numerous papers and books, I couldn't find the explicit value (or a numerical estimate) of the positive number $\mu$ with the aforementioned property in the literature, so it isn't clear whether this is not already known or if I simply searched for this information at the wrong places.
I appreciate any help.
