  The aim of this paper is to develop the general $L_p$ theory for the
barotropic compressible Navier-Stokes equations with the free boundary
condition in the exterior domain in $\mathbb{R}^N$ ($N\geq 3$). By the spectral
analysis, we obtain the classical $L_p$-$L_q$ decay estimate for the linearized
model problem (with variable coefficients) in view of the partial Lagrangian
transformation.
