Corner Boundary Layer in Nonlinear Elliptic Problems Containing Derivatives of First Order

In a rectangular domain the first boundary value problem is considered for a singularly perturbed elliptic equation
$$ \varepsilon^2\Delta u-\varepsilon^\alpha A(x, y)\frac{\partial u}{\partial y}= F(u,x,y,\varepsilon) $$
with a nonlinear on $u$ function $F$. The complete asymptotic solution expansion uniform in a closed rectangle is constructed for $\alpha> 1$. If $0<\alpha< 1$, the uniform asymptotic approximation is constructed in zero and first approximations. The features of the asymptotic behavior are noted in the case $\alpha=1$.