On solvability of stationary transonic equations in the unbounded domain

Solvability of a boundary value problem in an infinite cylinder is proved for an equation modelling steady-state transonic flows of a chemical mixture:
\begin{gather} u_xu_{xx}-\nabla_yu+\alpha u_x=0, \\ \frac{\partial u}{\partial N}\bigg|_{\partial\Omega\times R^1}=\varphi(x,y),\quad \lim_{|x|\to\infty}u_x=0,\quad \lim_{x\to\infty}|\nabla_yu|=0, \end{gather}
Where $y\in\Omega\subset R^2$, $x\in R^1$, and $\alpha$ is a positive parameter. Conditions on $\varphi (x,y)$ are established under which there exists a classical solution of problem (1), (2) which is unique up to an additive constant.