Nonexistence Results for a Singular Quasilinear Anisotropic Equation

Abstract

We establish the nonexistence of positive stable solutions $u\in W_{\mathrm{loc}}^{1,p}({\mathbb{R}}^N)$ to the singular quasilinear anisotropic equation ${\mathrm{\Delta }}_B^{H}u=u^{-q}$ in ${\mathbb{R}}^N$ , where $q>0$ , in low dimensions. Under similar assumptions, we further prove that, for the pure Finsler $p$ -Laplacian, there exist no solutions that are stable outside a compact set. The proofs rely on several integral estimates and asymptotic analysis of the solutions.