Uniqueness and instability of subsonic-sonic potential flow in a convergent approximate nozzle

We proved uniqueness and instability of the symmetric subsonic- sonic flow solution of the compressible potential flow equation in a surface with convergent areas of cross-sections. Such a surface may be regarded as an approximation of a two-dimensional convergent nozzle in aerodynamics. Mathematically these are uniqueness and nonexistence results of a nonlinear degenerate elliptic equation with Bernoulli type boundary conditions. The proof depends on maximum principles and a generalized Hopf boundary point lemma which was proved in the paper.