On the asymptotics of a solution to an equation with a small parameter at some of the highest derivatives

We study the asymptotic behavior of a solution of the first boundary value problem for a second-order elliptic equation in a nonconvex domain with smooth boundary in the case where the small parameter is a factor at only some of the highest derivatives and the limit equation is an ordinary differential equation. Although the limit equation has the same order as the initial equation, the problem under consideration is singulary perturbed. The asymptotic behavior of a solution of this problem is studied by the method of matched asymptotic expansions.