On Dirichlet problems for second order quasilinear degenerate elliptic equations (Q802080)

This paper studies the quasi-linear degenerate elliptic equation \[ a_{11}(x,y,u)\partial^ 2u/\partial x^ 2+2a_{12}(x,y,u)\partial^ 2u/\partial x\partial y+ \] \[ a_{22}(x,y,u)\partial^ 2u/\partial y^ 2+f(x,y,u,\partial u/\partial x,\quad \partial u/\partial y)=0, \] together with the boundary condition \(u(x,y)|_{\partial \Omega}=\phi (x,y)\), where \(a_{ij}(x,y,u)\) \((i,j=1,2)\) satisfy \[ \lambda (x,y,u) | \xi |^ 2\leq \sum^{2}_{i,j=1}a_{ij}(x,y,u)\xi_ i\xi_ j\leq \Lambda (x,y,u) | \xi |^ 2 \] for all \(\xi \in {\mathbb{R}}^ 2\) and (x,y,u)\(\in {\bar \Omega}\times [0,+\infty)\), \(\lambda\) (x,y,u), \(\Lambda\) (x,y,u) are the minimum and maximum eigenvalues of the matrix \([a_{ij}(x,y,u)]\) with \(\lambda (x,y,0)=0\), \(\Lambda (x,y,0)>0\), \(\Lambda (x,y,u)\geq \lambda (x,y,u)>0\) \((u>0)\), the non-negative function \(\phi\) (x,y) is vanishing at a part of \(\partial \Omega\). The equation is elliptic when \(u>0\), but is degenerate when \(u=0\). Some existence theorems under the natural conditions imposed on f(x,y,u,p,q) are obtained.