PreprintCC BY 4.0arXiv
Universal Hölder estimates for stable solutions to semilinear equations of fractional order
Abstract
We establish universal interior Hölder and fractional energy estimates for stable solutions of (−Δ)su=f(u), 0<s<1, with f∈ C1loc(ℝ) nonnegative and convex. The bounds depend only on a weighted L1 norm of the solution and hold under optimal conditions on the dimension and the fractional order. In particular, the half-Laplacian estimate holds through dimension eight. The estimates extend to stable finite-energy Dirichlet L1-weak solutions without an a priori local boundedness assumption, and an explicit logarithmic solution proves optimality in this class. Applications include global bounds in convex domains, bounded extremal solutions, and a Liouville theorem.
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BibTeX
@misc{xiao2026universal,
title = {{Universal Hölder estimates for stable solutions to semilinear equations of fractional order}},
author = {Yiyang Xiao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24727},
eprint = {2610.24727},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24727},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Universal Hölder estimates for stable solutions to semilinear equations of fractional order
AU - Yiyang Xiao
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24727
UR - https://arxiv.org/abs/2610.24727
AB - We establish universal interior Hölder and fractional energy estimates for stable solutions of (−Δ)^su=f(u), 0<s<1, with f∈ C^1_loc(ℝ) nonnegative and convex. The bounds depend only on a weighted L^1 norm of the solution and hold under optimal conditions on the dimension and the fractional order. In particular, the half-Laplacian estimate holds through dimension eight. The estimates extend to stable finite-energy Dirichlet L^1-weak solutions without an a priori local boundedness assumption, and an explicit logarithmic solution proves optimality in this class. Applications include global bounds in convex domains, bounded extremal solutions, and a Liouville theorem.
ER -