  In this paper, we first consider global well-posedness and long time behavior
of compressible Navier-Stokes equations with Yukawa-type potential in
$L^p$-framework under the stability condition $P'(\bar\rho)+\gamma\bar\rho>0$.
Here $\bar\rho>0$ is the background density, P is the pressure and
$\gamma\in\mathbb{R}$ is Yukawa coefficient. This is a continuity work of
Chikami \cite{chikami1} concerning on local existence and blow-up criterion. On
the other hand, we study the instability of the linear and nonlinear problem of
the system when $P'(\bar\rho)+\gamma\bar\rho<0$ in the Hadamard sense.
