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 equation

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

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 $\zeta$.