Positive Liouville theorem and asymptotic behaviour for $(p,A)$-Laplacian type elliptic equations with Fuchsian potentials in Morrey space

Abstract

We study Liouville-type theorems and the asymptotic behaviour of positive solutions near an isolated singular point $\zeta \in \partial \Omega \cup \{\infty \}$ of the quasilinear elliptic equations $$-\text{div}(|\nabla u{|}_A^{p-2}A\nabla u)+V|u{|}^{p-2}u=0\text{in }\Omega \setminus \{\zeta \},$$ where $\Omega$ is a domain in ${\mathbb{R}}^d$ ($d\ge 2$), and $A=(a_{ij})\in L_{\mathrm{loc}}^{\infty }(\Omega ;{\mathbb{R}}^{d\times d})$ is a symmetric and locally uniformly positive definite matrix. The potential $V$ lies in a certain local Morrey space (depending on $p$) and has a Fuchsian-type isolated singularity at $\zeta$.

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