Optimal regularity and the Liouville property for stable solutions to semilinear elliptic equations in ℝn with n ≥ 10

Peng, Fa; Zhang, Yi Ru-Ya; Zhou, Yuan

doi:10.2140/apde.2024.17.3335

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1948-206X (online)

ISSN 2157-5045 (print)

Author index

To appear

Other MSP journals

Optimal regularity and the Liouville property for stable solutions to semilinear elliptic equations in $\mathbb{R}^n$ with $n\ge10$

Fa Peng, Yi Ru-Ya Zhang and Yuan Zhou

Vol. 17 (2024), No. 9, 3335–3353

DOI: 10.2140/apde.2024.17.3335

Abstract

Let $0 \leq f \in C^{0, 1} (ℝ)$ . Given a domain $Ω \subset ℝ^{n}$ , we prove that any stable solution to the equation $- Δ u = f (u)$ in $Ω$ satisfies

a BMO interior regularity, when $n = 10$ ,
a Morrey $M^{p_{n}, 4 + 2 ∕ (p_{n} - 2)}$ interior regularity, when $n \geq 11$ , where
$p_{n} = \frac{2 (n - 2 \sqrt{n - 1} - 2)}{n - 2 \sqrt{n - 1} - 4} .$

This result is optimal as hinted by, e.g., Brezis and Vázquez (1997), Cabré and Capella (2006), and Dupaigne (2011), and answers an open question raised by Cabré, Figalli, Ros-Oton and Serra (2020). As an application, we show a sharp Liouville property: any stable solution $u \in C^{2} (ℝ^{n})$ to $- Δ u = f (u)$ in $ℝ^{n}$ satisfying the growth condition

| u (x) | = {\begin{matrix} o (\log | x |) & as | x | \to + \infty, when n = 10, \\ o (| x |^{- n ∕ 2 + \sqrt{n - 1} + 2}) & as | x | \to + \infty, when n \geq 11, \end{matrix}

must be a constant. This extends the well-known Liouville property for radial stable solutions obtained by Villegas (2007).

Keywords

elliptic PDE, semilinear elliptic equation, stable solution, BMO regularity, Morry regularity

Mathematical Subject Classification

Primary: 35J61

Milestones

Received: 3 July 2022

Accepted: 13 June 2023

Published: 1 November 2024

Authors

Fa Peng
School of Mathematical Science Beihang University Beijing China	Academy of Mathematics and Systems Science The Chinese Academy of Sciences Beijing China

Yi Ru-Ya Zhang
Academy of Mathematics and Systems Science The Chinese Academy of Sciences Beijing China

Yuan Zhou
School of Mathematical Science Beijing Normal University Beijing China

Open Access made possible by participating institutions via Subscribe to Open.

Vol. 17, No. 9, 2024