= Solution
Use the <fluid displacement> $\boldsymbol\xi$, with velocity perturbation $\delta\mathbf u=\partial_t\boldsymbol\xi$. Denote Eulerian perturbations by $\delta$ and the corresponding <Lagrangian pressure perturbation> by $\Delta_Lp=\delta p+\boldsymbol\xi\cdot\nabla p$. For adiabatic perturbations, the linearized <mass conservation>, momentum, <adiabatic equation of state> and <Poisson equation> are
$$
\boxed{\begin{aligned}
\delta\rho&=-\nabla\cdot(\rho\boldsymbol\xi),\\
\rho\,\partial_t^2\boldsymbol\xi
&=-\nabla\delta p-\delta\rho\,\nabla\Phi-\rho\nabla\delta\Phi,\\
\delta p+\boldsymbol\xi\cdot\nabla p
&=-\gamma p\,\nabla\cdot\boldsymbol\xi,\\
\nabla^2\delta\Phi&=4\pi G\,\delta\rho.
\end{aligned}}
$$
These <self-gravitating adiabatic displacement equations> retain the perturbation of the star's own <Newtonian gravitational potential>. In the uniform-density interior, $\delta\rho=-\rho\nabla\cdot\boldsymbol\xi$, $\nabla\Phi=\omega_d^2\mathbf r$, and $\nabla p=-\rho\omega_d^2\mathbf r$. The <dynamical frequency of a uniform-density star> $\omega_d=(GM/R^3)^{1/2}$ sets the natural timescale. For a <normal mode> with time factor $e^{-i\omega t}$, replace $\partial_t^2$ by $-\omega^2$. The requested interior analysis needs no surface or exterior matching conditions.
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