  The vortex-wave system describes the motion of a two-dimensional ideal fluid
in which the vorticity includes continuously distributed vorticity, which is
called the background vorticity, and a finite number of concentrated vortices.
In this paper we restrict ourselves to the case of a single point vortex in
bounded domains. We prove the existence of steady vortex patch solutions to
this system with prescribed distribution for the background vorticity.
Moreover, we show that the supports of these solutions "shrink" to a minimum
point of the Kirchhoff-Routh function as the strength parameter of the
background vorticity goes to infinity.
