Comparison principles for 3-D steady potential flow in spherical coordinates

In this paper, we consider the 3-D steady potential flow for a compressible gas with pressure satisfying $p'(\rho)=\rho^{\gamma-1}$, where $\rho$ is the density and $\gamma\geq-1$ is a constant. In spherical coordinates, the potential equation is of mixed type in the unit sphere. We establish a strong comparison principle for elliptic solutions of the equation. The main difference from the classical case is that the coefficients of this equation depend fully on the potential function itself. We overcome this difficulty by the sufficient analysis on the structure of the equation itself, and finally derive the result. The result obtained here can be applied to the problem of supersonic flow over a delta wing and other problems related to gas dynamics.