  In this paper we are concerned with a class of double phase energy
functionals arising in the theory of transonic flows. Their main feature is
that the associated Euler equation is driven by the Baouendi-Grushin operator
with variable coefficient. This partial differential equation is of mixed type
and possesses both elliptic and hyperbolic regions. After establishing a
weighted inequality for the Baouendi-Grushin operator and a related compactness
property, we establish the existence of stationary waves under arbitrary
perturbations of the reaction.
