Positive Solutions of Quasilinear Elliptic Equations with Fuchsian Potentials in Wolff Class

Abstract

Using Harnack’s inequality and a scaling argument we study Liouville-type theorems and the asymptotic behaviour of positive solutions near an isolated singular point $\zeta \in \partial \Omega \cup \{\infty \}$ for the quasilinear elliptic equation

$$\begin{aligned} -\text {div}(|\nabla u|_A^{p-2}A\nabla u)+V|u|^{p-2}u =0\quad \text { in } \Omega , \end{aligned}$$

where $\Omega$ is a domain in $\mathbb {R}^d$, $d\ge 2$, $1<p<d$, and $A=(a_{ij})\in L_{\textrm{loc}}^{\infty }(\Omega ; \mathbb {R}^{d\times d})$ is a symmetric and locally uniformly positive definite matrix. It is assumed that the potential V belongs to a certain Wolff class and has a generalized Fuchsian-type singularity at an isolated point $\zeta \in \partial \Omega \cup \{\infty \}$.