Sharp Liouville thresholds and endpoint rigidity for finite Morse index solutions of the p-Laplace Lane--Emden equation
Abstract
We study −Δp u = |u|q−1u in ℝN with N>p≥ 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*−1<q<qc(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≠2. We remove the radial hypothesis: every C1 weak solution stable outside a compact set is trivial when p*−1<q<qc(N,p). The whole range p−1<q<qc(N,p) with q≠ p*−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>qc(N,p) we exhibit positive bounded radial solutions stable outside a compact set. At the Sobolev endpoint, every C1 solution stable outside a compact set has finite energy; for p>2 this removes the a priori 𝒟1,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, qc(N,p) is the exact threshold for a nonzero homogeneous weak solution on ℝN∖{0} to be stable outside a ball, and at the threshold there are exactly two.
Related
- A Liouville type rigidity result for entire solutions of the k-Hessian equation
- Nonexistence of Stable Solutions to the Liouville Equation on Graphs with Polynomial Volume Growth
- A Parametrization of Solutions to the Liouville Equation in a Punctured Disc
- Small-energy regularity for stationary weak solutions of the Liouville equation
- Liouville Rigidity and Universal Spacelikeness Estimates for a Lorentzian Prescribed Mean Curvature Equation
BibTeX
@misc{le2026sharp,
title = {{Sharp Liouville thresholds and endpoint rigidity for finite Morse index solutions of the $p$-Laplace Lane--Emden equation}},
author = {Phuong Le},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02751},
eprint = {2610.02751},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02751},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp Liouville thresholds and endpoint rigidity for finite Morse index solutions of the p-Laplace Lane--Emden equation
AU - Phuong Le
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02751
UR - https://arxiv.org/abs/2610.02751
AB - We study −Δ_p u = |u|^q−1u in ℝ^N with N>p≥ 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^*−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≠2. We remove the radial hypothesis: every C^1 weak solution stable outside a compact set is trivial when p^*−1<q<q_c(N,p). The whole range p−1<q<q_c(N,p) with q≠ p^*−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 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 ℝ^N∖{0} to be stable outside a ball, and at the threshold there are exactly two.
ER -