Singular perturbation of the fourth order elliptic equation when the limit equation is elliptic-parabolic (Q1192838)

(From author's summary.) In this paper the author considers the singular perturbation of the fourth order elliptic equation \[ -\varepsilon^ 2 \Delta^ 2 u+y^ m{\partial^ 2u \over\partial y^ 2}+{\partial^ 2 u\over\partial x^ 2}+a(x,y){\partial u\over\partial y}+b(x,y){\partial u\over\partial x}+c(x,y)=0, \] when the limit equation is elliptic- parabolic, where \(\varepsilon\) is a positive parameter, \(m\) is a positive number, \(\Delta\) is the Laplace operator, \(a,b,c\) are sufficiently smooth. Under an appropriate condition applying a boundary layer function, he derives at the sufficient condition of solvability and proves the existence of a solution and gives a uniformly valid asymptotic solution of arbitrary order.