Solutions stable outside a compact set for anisotropic quasilinear equations with negative exponent or exponential nonlinearity

We study the equations $Δ_p^H u=u^{-q}$, $u>0$, and $-Δ_p^H u=e^u$ in $\mathbb{R}^N$, where $Δ_p^H u=\operatorname{div}(H(\nabla u)^{p-1}\nabla H(\nabla u))$ is the Finsler $p$-Laplacian associated with a uniformly convex norm $H$, and $p\ge2$. We prove nonexistence of solutions stable outside a com