= Solution
For repulsive interactions $g>0$, the <Thomas–Fermi approximation for a condensate> neglects the density-gradient <kinetic energy> away from boundaries and vortex cores. The stationary <Gross–Pitaevskii equation> then gives $\mu=V_{\rm ext}(r)+gn(r)$ wherever the condensate is present. At the axis $\mu=gn_0$, hence
$$
\boxed{n(r)=n_0\left(1-\frac{r^2}{R_r^2}\right)_+,\qquad
R_r=\sqrt{\frac{2gn_0}{m\omega_r^2}}.}
$$
Here $(q)_+=\max(q,0)$ and the support is also restricted to the physical bucket $r\leq R_0$. The next parts' explicit outer cutoff $R_r$ assumes $R_r\leq R_0$; if $R_0<R_r$ the wall truncates the parabola and the corresponding <integrals> must use $R_0$, not the natural trap radius. A hard-wall boundary layer or the smooth trap-edge healing region is beyond the <Thomas–Fermi approximation for a condensate>. For an untruncated radial profile, the total particle number is $N=\pi H_0n_0R_r^2/2$.
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