Interior Regularity for Stable Solutions of Semilinear Elliptic Equations with Sign-Changing Nonlinearities
Abstract
We investigate interior regularity for stable solutions of the semilinear elliptic equation −Δu=f(u) in an open set Ω⊆ ℝn, where f is locally Lipschitz and may change sign. We establish sharp interior Hölder estimates for stable classical solutions in every dimension n≤ 9, thereby fully resolving a question raised by Cabré concerning interior regularity without the nonnegativity assumption on f. For n≤ 5, we also give a short alternative proof of the corresponding Hölder regularity result due to Peng, Zhang, and Zhou. We further establish higher-integrability estimates for the gradient and L1 bounds for the Hessian in all dimensions. For n≥ 10, we obtain BMO and Morrey estimates, extending related results previously established for nonnegative nonlinearities. When f is convex, we extend the interior regularity estimates to stable weak solutions through an approximation theorem. As applications, we prove Liouville-type theorems for entire stable solutions of equations with sign-changing nonlinearities across all dimensions.
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BibTeX
@misc{huang2026interior,
title = {{Interior Regularity for Stable Solutions of Semilinear Elliptic Equations with Sign-Changing Nonlinearities}},
author = {Long-Han Huang and Yuxuan Zhou and Wenming Zou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21641},
eprint = {2610.21641},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21641},
}
RIS
TY - UNPB
TI - Interior Regularity for Stable Solutions of Semilinear Elliptic Equations with Sign-Changing Nonlinearities
AU - Long-Han Huang
AU - Yuxuan Zhou
AU - Wenming Zou
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21641
UR - https://arxiv.org/abs/2610.21641
AB - We investigate interior regularity for stable solutions of the semilinear elliptic equation −Δu=f(u) in an open set Ω⊆ ℝ^n, where f is locally Lipschitz and may change sign. We establish sharp interior Hölder estimates for stable classical solutions in every dimension n≤ 9, thereby fully resolving a question raised by Cabré concerning interior regularity without the nonnegativity assumption on f. For n≤ 5, we also give a short alternative proof of the corresponding Hölder regularity result due to Peng, Zhang, and Zhou. We further establish higher-integrability estimates for the gradient and L^1 bounds for the Hessian in all dimensions. For n≥ 10, we obtain BMO and Morrey estimates, extending related results previously established for nonnegative nonlinearities. When f is convex, we extend the interior regularity estimates to stable weak solutions through an approximation theorem. As applications, we prove Liouville-type theorems for entire stable solutions of equations with sign-changing nonlinearities across all dimensions.
ER -