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 $$-\text{div}(|\nabla u{|}_A^{p-2}A\nabla u)+V|u{|}^{p-2}u=0\text{ in }\Omega ,$$ where $\Omega$ is a domain in ${\mathbb{R}}^d$, $d\ge 2$, $1<p<d$, and $A=(a_{ij})\in L_{\mathrm{loc}}^{\infty }(\Omega ;{\mathbb{R}}^{d\times d})$ 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 $\zeta \in \partial \Omega \cup \{\infty \}$.

40 pages