Corner boundary layer in nonlinear singularly perturbed elliptic problems

The Dirichlet problem in a rectangle is considered for the elliptic equation $\varepsilon^2\Delta u=F(u,x,y,\varepsilon)$, where $F(u,x,y,\varepsilon)$ is a nonlinear function of $u$. The method of corner boundary functions is applied to the problem. Assuming that the leading term of the corner part of the asymptotics exists, an asymptotic expansion of the solution is constructed and the remainder is estimated.