Sharp Liouville thresholds and endpoint rigidity for finite Morse index solutions of the $p$-Laplace Lane--Emden equation

We study $-{\mathrm{\Delta }}_{p}u=\mathrm{\mid }u{\mathrm{\mid }}^{q-1}u$ in ${\mathbb{R}}^N$ with $N>p\ge 2$, for solutions that are stable outside a compact set, with no assumption of sign, boundedness or symmetry. Damascelli, Farina, Sciunzi and Valdinoci settled the subcritical range for $p>2$, and treated the supercritical range $p^{\ast }-1<q<q_c(N,p)$ only for radial solutions, stating that without radial symmetry they were not able to conclude. The obstruction is that the arguments available for $p=2$ rest on a monotonicity formula with no monotone analogue when $p\mathrm{\ne }2$. We remove the radial hypothesis: every $C^1$ weak solution stable outside a compact set is trivial when $p^{\ast }-1<q<q_c(N,p)$. The whole range $p-1<q<q_c(N,p)$ with $q\mathrm{\ne }{p}^{\ast }-1$ is thereby settled, as it is for $p=2$ by Farina’s theorem, and every finite Morse index solution in it is trivial. In its place we iterate the comparison principle against an explicit two-parameter supersolution, upgrading the decay forced by stability to that of the $p$-harmonic fundamental solution; this step uses no stability, only smallness of the self-similar decay, and is stated on its own. The upper threshold is sharp: for every $q>q_c(N,p)$ we exhibit positive bounded radial solutions stable outside a compact set. At the Sobolev endpoint, every $C^1$ solution stable outside a compact set has finite energy; for $p>2$ this removes the a priori ${\mathcal{D}}^{1,p}$ assumption from the low-Morse-index classification of Farina, Mercuri and Willem and answers their question on the Morse index of the Aubin–Talenti extremals: it is one. At the upper endpoint, $q_c(N,p)$ is the exact threshold for a nonzero homogeneous weak solution on ${\mathbb{R}}^N\setminus \{0\}$ to be stable outside a ball, and at the threshold there are exactly two.