The authors investigate the steady compressible Navier-Stokes equations \[ -\mu_1\Delta v-(\mu_1+\mu_2)\nabla\text{div} v +\nabla \rho+\rho(v\cdot\nabla)v=\rho(\nabla U+f), \] \(\text{div} (\rho v)=0\) for \(x\in\Omega\), \(v=0\) for \(x\in\partial\Omega\). Here \(\Omega\) is a bounded domain of \(\mathbb{R}^n\) \((n=2,3)\), \(\mu_1, \mu_2\) are given constants, \(v\) and \(\rho\) are unknown velocity and density. The potential \(U\) can be arbitrarily “large”. The rest state \((0,\rho_0)\) is the solution corresponding to the potential force \(\nabla U\). Here \(\rho_0(x)=\rho_M\exp U(x)\), where \(\rho_M\) is a certain positive constant.
The solution of the problem is found near the rest state \((0,\rho_0)\). It is proved that if \(f\) is sufficiently small, then there exists a unique solution in a neighborhood of \((0,\rho_0)\). The solution is obtained by the method of decomposition: the velocity \(v\) is split into a non-homogeneous incompressible part \(u (\text{div} (\rho_0 u)=0)\) and a compressible part \(\nabla \psi\).