The Serrin exponent for Lane-Emden inequalities driven by sums of fractional p-Laplacians
Abstract
We study positive weak super-solutions of the Lane-Emden inequality ℒu := ∑i=1n(−Δpi)siu ≥ uq in ℝN, where each (−Δpi)si is a fractional pi-Laplacian of order si∈(0,1] and exponent pi∈(1,∞), and maxi sipi<N. Let κi := (N−sipi)/(pi−1) be the decay exponent of the fundamental solution of the i-th operator, and let κ̅ := maxiκi. We show that the Serrin exponent of ℒ is qℒ = N/κ̅. More precisely, for 0<q<qℒ the only non-negative continuous weak super-solution is u≡ 0, while for every q>qℒ we construct explicit positive weak super-solutions of the form A(λ+|x|)−κ. In other words, the Serrin exponent of the sum is the smallest of the Serrin exponents N/κi of the single operators, that is, the one of the operator whose fundamental solution decays fastest. No structural assumption relating the different operators is needed. The endpoint q=qℒ is left open. The local order si=1 is treated together with the nonlocal orders, and all the arguments are carried out in the weak setting.
Related
- Optimal Hardy Inequalities for Fractional p-Laplacians on the Integers
- Strict rearrangement inequalities for logarithmic p-Laplacians
- Heintze--Karcher-Type Inequalities for the p-Laplacian and Serrin-Type Rigidity
- Fractional gradients and nonlocal PDEs in variable exponent spaces
- Revisiting Positive Exponent power weighted Stein-Weiss inequalities in L1
BibTeX
@misc{pezzo2026serrin,
title = {{The Serrin exponent for Lane-Emden inequalities driven by sums of fractional p-Laplacians}},
author = {Leandro M. Del Pezzo and Bryan Pichucho and Alexander Quaas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30415},
eprint = {2610.30415},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30415},
}
RIS
TY - UNPB
TI - The Serrin exponent for Lane-Emden inequalities driven by sums of fractional p-Laplacians
AU - Leandro M. Del Pezzo
AU - Bryan Pichucho
AU - Alexander Quaas
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.30415
UR - https://arxiv.org/abs/2610.30415
AB - We study positive weak super-solutions of the Lane-Emden inequality ℒu := ∑_i=1^n(−Δ_p_i)^s_iu ≥ u^q in ℝ^N, where each (−Δ_p_i)^s_i is a fractional p_i-Laplacian of order s_i∈(0,1] and exponent p_i∈(1,∞), and max_i s_ip_i<N. Let κ_i := (N−s_ip_i)/(p_i−1) be the decay exponent of the fundamental solution of the i-th operator, and let κ̅ := max_iκ_i. We show that the Serrin exponent of ℒ is q_ℒ = N/κ̅. More precisely, for 0<q<q_ℒ the only non-negative continuous weak super-solution is u≡ 0, while for every q>q_ℒ we construct explicit positive weak super-solutions of the form A(λ+|x|)^−κ. In other words, the Serrin exponent of the sum is the smallest of the Serrin exponents N/κ_i of the single operators, that is, the one of the operator whose fundamental solution decays fastest. No structural assumption relating the different operators is needed. The endpoint q=q_ℒ is left open. The local order s_i=1 is treated together with the nonlocal orders, and all the arguments are carried out in the weak setting.
ER -