= Solution
Stability requires $\lambda>0$. The classical potential $r\rho+\lambda\rho^2$ has a symmetry-breaking minimum when $r<0$, at
$$
\bar\rho=-\frac r{2\lambda}>0.
$$
Writing $\rho=\bar\rho+\delta\rho$ and discarding constants and <total derivatives> gives the <quadratic Lagrangian>
$$
\mathcal L_2=\delta\rho\,\dot\theta-
\frac{(\nabla\delta\rho)^2}{8m\bar\rho}-
\lambda(\delta\rho)^2-
\frac{\bar\rho}{2m}(\nabla\theta)^2.
$$
The density fluctuation is a gapped <amplitude mode>, while the phase is the prospective <Goldstone boson>.
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