Fuchsian-type singularity for the Finsler $p$-Laplacian with potential

Abstract

Let $\Omega \subseteq {\mathbb{R}}^n$ ($2\le n\in \mathbb{N}$) be a domain and let $\zeta \in \{0,\infty \}$ be an isolated point of the boundary of $\Omega$ in the one-point compactification of ${\mathbb{R}}^n$ with the ideal point $\infty$. Under some further conditions, we study Fuchsian-type singularity at $\zeta$ for the Finsler $p$-Laplace equation with a potential $$-\mathrm{div}\mathcal{A}(x,\nabla u)+V|u|^{p-2}u=0\quad (1<p<\infty)\qquad \mbox{in } Ω,$$ where $\mathcal{A}(x,\xi )\triangleq {\nabla }_{\xi }(H(x,\xi {)}^p/p)$ for almost all $x\in \Omega$ and all $\xi \in {\mathbb{R}}^n$, $H$ is a family of norms on ${\mathbb{R}}^n$ ($n\ge 2$) parameterized by points $x\in \Omega$, and $V$ belongs to a local Morrey space. In particular, we investigate asymptotic behaviors of positive solutions of the equation near $\zeta$ and asymptotic behaviors of their quotients.

30 pages