Singular Perturbation of the Fourth Order Elliptic Equation When the Limit Equation Is Elliptic-Parabolic

In this paper we cosider the singular perturbation of the fourth order elliptic equation-ε²Δ²u+y^m∂²u/∂y²+∂²u/∂x²+a(x,y)∂u/∂y+b(x,y)∂u/∂x+c(x,y)=0 when the limit equationis elliptic-parabolic, where ε is a positive parameter, Δ is a positive real number, A is Laplacian operator, a,b,c are sufficiently smooth. Under appropriate condition we derive the sufficient condition of solvability and prove the existence of solution and give a uniformly valid asymptotic solution of arbitrary order.