A Singular Quasilinear Anisotropic Elliptic Boundary Value Problem. II

Abstract:

Let $\Omega \subset \mathbf {R}^N$ with $N \geq 2$. We consider the equations \begin{align*} \sum _{i=1}^{N} u^{a_i} \frac {\partial ^2 u}{\partial x_i^2} +p(\mathbf {x})&= 0,\\ u|_{\partial \Omega } & = 0, \end{align*} with $a_1 \geq a_2 \geq \dots \geq a_N \geq 0$ 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.