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

Abstract

Let $\Omega\subseteq\mathbb{R}^{n}$ ($2\leq n\in\mathbb{N}$) 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 $$-\mathrm{div}\mathcal{A}(x,\nabla u)+V|u|^{p-2}u=0\quad (1<p<\infty)\qquad \mbox{in } \Omega,$$ 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 $\mathbb{R}^{n}$ ($n\geq 2$) 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.