ON DIRICHLET PROBLEMS FOR SECOND ORDER QUASILINEAR DEGENERATE ELLIPTIC EQUATIONS

The purpose of this paper is to study the existence of the classical solutions of someDirichlet problems for quasilinear elliptic equationsa11（x,y,u）（ 2u）/（ x2）+2a12（x,y,u）)（ 2u）/（ x y）+a22（x,y,u）（ 2u）/（ y2）+f（x,y,u,（ u）/（ x）,（ u）/（ y））=0where aij（x,y,u）（i,j=1,2）satisfyλx,y,u丨ξ丨2≤ aij（x,y,u）ξiξj≤Λ（x,y,u）丨ξ丨2for all ξ∈R2 and （x,y,u）∈×[0,+∞),i.e.,λ.（x,y,u）、A（x,y,u）denote the minimumand maximum eigenvalues of the matrix[aij（x,y,u）]respectively,moreoverλ（x,y,0）=0;Λ（x,u,0）=0;Λ（x,y,u）≥λ（x,y,u）>0,（u>0）.Some existence theorems under the “natural conditions” imposed on f（x,y,u,p,q）are obtained.