On a singular quasilinear anisotropic elliptic boundary value problem

Abstract:

We consider the problem \[ {u^a}{u_{xx}} + {u^b}{u_{yy}} + p({\mathbf {x}}) = 0\] with $a \geqslant 0$, $b \geqslant 0$, on a smooth convex bounded region in ${{\mathbf {R}}^2}$ with Dirichlet boundary conditions. We show that if the positive function $p$ is uniformly bounded away from zero, then the problem has a classical solution.