Existence results for a degenerated nonlinear elliptic partial differential equation

The aim of this paper is to establish the existence of weak solutions to a steady state two-dimensional irrotational compressible flow around a thin profile. This flow is described by the small disturbance equations. If the speed of sound exceeds the fluid one, the governing equations remain elliptic. But when the fluid speed is beyond the sound one, the flow becomes locally hyperbolic and shock waves arise. For a modified elliptic model, using convexity arguments, we prove the existence of a solution which is the solution to the first model when the flow remains subsonic.