For reference, this discussion concerns the paper "Displacement and stress produced by a pressurized, spherical magma chamber, surrounded by a viscoelastic shell" by Michele Dragoni and Cecilia Magnanensi (Physics of the Earth and Planetary Interiors, 56 (1989) 316—328).
Very important in the whole discussion is that the materials are assumed to be compressible, so the $1/r^2$ behaviour has nothing to do with incompressibility (as is often the case in these problems). If we read Landau Lifshitz Vol7: Theory of Elasticity, §7 The equations of equilibrium for isotropic bodies and there look at equation (7.7), we see that the displacement function $\mathbf{u}$ needs to be a biharmonic function, i.e.
$$\Delta \Delta \mathbf{u}=0$$
According to A. Oberbeck: Ueber stationäre Flüssigkeitsbewegungen mit
Berücksichtigung der inneren Reibung (Journal für die reine und angewandte Mathematik (Crelle's Journal), Band 1876, Heft 81, Seiten 62–80)$^{1}$, the most general solution to this is:
$$u=\sum_{n=0}^{\infty}\frac{A_n+B_nr^2}{r^{2n+1}}r^n P_n (\vartheta,\varphi) $$
It is however clear that a subset of the solution needs to be the solutions of
$$\Delta \mathbf{u}=0$$
If we either believe Oberbeck or use only the solutions to this Laplace equation and consider the fact that due to the symmetry of the problem, only solutions without dependence on the angles are acceptable solutions, the most general solution for the radial component is:
$$ u = A r + B r^{-2} $$
For $r > R_2$, the coefficient $A$ needs to equal zero so that the displacement goes to zero as $r \to \infty$. With this, we can now I think understand why this form of the displacement field was assumed as most general form possible.
Physically, Dragoni and Magnanensi write in their paper that your $a \propto \mu_1 - \mu_2$, the difference between the elastic constants of both layers, due to the boundary conditions between the layers. So the $a$ term is important if they have different elastic constants. And this immediately makes sense if we think about the fact that the stress at the surface between the layers needs to be continuous - therefore, in case of a difference in elasticity, the deformation needs to vary throughout the first layer not only by radially decreasing in the expected $r^{-2}$ form but also in another form to guarantee the continuity of the stress across the interlayer boundary.
$^1$: nope, couldn't find a less obscure source
