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

Abstract: We study the equations $Δ_p^H u=u<sup>{-q}$, $u&gt;0$, and $-Δ_p^H u=e<sup>u$ in $\mathbb{R}<sup>N$, 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 compact set for the negative-power equation when $p\le N&lt;\frac{p(p+3)}{p-1}$ and $q$ lies above the sharp critical threshold, and for the exponential equation when $p&lt;N&lt;\frac{p(p+3)}{p-1}$. Consequently, neither equation admits solutions of finite Morse index in these ranges. In the borderline case $N=p$, we prove that the solutions of $-Δ_N^H u=e^u$ that are stable outside a compact set are precisely the finite-mass solutions classified by Ciraolo and Li. A main ingredient is independent of stability: if $N&gt;qτ$, where $τ=\frac{p}{q+p-1}$, a positive exterior solution of $Δ_p^H u=u<sup>{-q}$ cannot remain above $θ$ times the singular solution near infinity for any $θ&gt;1$. An analogous obstruction holds for the exponential equation. Both obstructions are proved by comparison with explicit radial barriers having frozen right-hand sides. We also prove sharpness when $N&gt;p$, and for the negative-power equation when $N=p&gt;2$. Finally, for operators $\operatorname{div}(B&#39;(H(\nabla u))\nabla H(\nabla u))$ with $p$-growth, we establish the integral estimates needed for the stability arguments with fully justified truncation and absorption procedures. These arguments also repair gaps in earlier proofs for the corresponding isotropic and anisotropic problems.