A Singular Quasilinear Anisotropic Elliptic Boundary Value Problem. II

Let $\Omega \subset \mathbf{R}^N$ with $N \geq 2$. We consider the equations

with $a_1 \geq a_2 \geq .... \geq a_N \geq 0$ and $a_1>a_N$. We show that if $\Omega$ is a convex bounded region in $\mathbf{R}^N$, there exists at least one classical solution to this boundary value problem. If the region is not convex, we show the existence of a weak solution. Partial results for the existence of classical solutions for non-convex domains in $\mathbf{R}^2$ are also given.