= Solution
The spatial factor $r^lY_l^m$ is a <regular solid harmonic>. Besides the <Laplace equation> $\nabla^2F=0$, homogeneity gives $\mathbf r\cdot\nabla F=lF$. The divergence of the <fluid displacement> is
$$
\begin{aligned}
\nabla\cdot(UF\mathbf r)
&=[rU'+(l+3)U]F,\\
\nabla\cdot(V\nabla F)
&=\frac{l}{r}V'F.
\end{aligned}
$$
Define the scalar dilation amplitude $D_r=rU'+(l+3)U+(l/r)V'$; this is the quantity denoted $\Delta$ in the question, not the notation $\Delta_L$ for a <Lagrangian pressure perturbation>. Then <mass conservation> gives $\widehat\rho=-\rho D_r$.
The equilibrium pressure gradient and homogeneity identity give
$$
\boldsymbol\xi\cdot\nabla p
=-\rho\omega_d^2(Ur^2+lV)F.
$$
Consequently the <adiabatic equation of state> yields
$$
\boxed{\widehat\rho=-\rho D_r,\qquad
\widehat p=\rho\omega_d^2(Ur^2+lV)-\gamma pD_r.}
$$
For the force equations, the product rule gives
$$
\nabla(\widehat pF)=\widehat p'F\mathbf e_r+\widehat p\nabla F,\qquad
\nabla(\widehat\Phi F)=\widehat\Phi'F\mathbf e_r+\widehat\Phi\nabla F.
$$
Equating the coefficients of $F\mathbf e_r$ and $\nabla F$ in the <self-gravitating adiabatic displacement equations> gives
$$
\boxed{\rho\omega^2Ur
=\widehat\rho\,\omega_d^2r+\rho\widehat\Phi'+\widehat p',
\qquad
\rho\omega^2V=\rho\widehat\Phi+\widehat p.}
$$
Finally, applying the <Laplacian> to the gravitational perturbation gives
$$
\nabla^2(\widehat\Phi F)
=\left(\widehat\Phi''+\frac{2(l+1)}r\widehat\Phi'\right)F,
$$
so the <Poisson equation> becomes
$$
\boxed{\widehat\Phi''+\frac{2(l+1)}r\widehat\Phi'
=4\pi G\,\widehat\rho,\qquad
D_r=rU'+(l+3)U+\frac lrV'.}
$$
These are the required interior equations for <uniform-density stellar oscillation>.
There is a radial degeneracy at $l=0$: $F$ is spatially constant and $\nabla F=0$, so the displacement is independent of $V$. The second force equation then cannot be inferred by equating independent vectors. For nonzero $\omega$ it may be imposed as an auxiliary definition of $V$, but it is not an additional physical radial equation. At zero frequency the radial equations should be used directly. This distinction matters for interpreting the zero factor in (c).
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