Subsonic solutions to compressible transonic potential problems for isothermal self-similar flows and steady flows

We establish boundary and interior gradient estimates, and show that no supersonic bubble appears inside of a subsonic region for transonic potential flows for both self-similar isothermal and steady problems. We establish an existence result for the self-similar isothermal problem, and improve the Hopf maximum principle to show that the flow is strictly elliptic inside of the subsonic region for the steady problem.