On the Dirichlet Problem for Quasi-linear Elliptic Equation with a Small Parameter

In this paper we deal with the Dirichlet problem for quasilinear ellipticequation with a small parameter at highest derivatives.In case the character-istics of the degenerated equation are curvilinear and the domain,where theproblem is defined,is a bounded convex domain,we offer a method to cons-truct the uniformly valid asymptotic solution of this problem,and prove thatthe solution of this problem really exists,and being uniquely determined asthe small parameter is sufficiently small.