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

TL;DR

This paper establishes optimal interior regularity and Liouville properties for stable solutions to semilinear elliptic equations in high dimensions, answering open questions and extending known results.

Contribution

It proves optimal regularity results for stable solutions in dimensions 10 and above, and establishes sharp Liouville theorems under growth conditions, extending prior work by Villegas.

Findings

Interior regularity: BMO for n=10, Morrey space for n≥11.

Liouville property: stable solutions must be constant under growth conditions.

Answers an open question by Cabré, Figalli, Ros-Oton, and Serra.

Abstract

Let $ 0\le f\in C^{0,1}(\mathbb R^n)$. Given a domain $\Omega\subset \mathbb R^n$, we prove that any stable solution to the equation $-\Delta u=f(u)$ in $\Omega$ satisfies a BMO interior regularity when $n=10$, and an Morrey $M^{p_n,4+2/(p_n-2)}$ interior regularity when $n\ge 11$, where $$p_n=\frac{2(n-2\sqrt{n-1}-2)}{n-2\sqrt{n-1}-4}. $$ This result is optimal as hinted by earlier results, and answers an open question raised by Cabr\'e, Figalli, Ros-Oton and Serra. As an application, we show a sharp Liouville property: Any stable solution $u \in C^2(\mathbb R^n)$ to $-\Delta u=f(u)$ in $\mathbb R^n$ satisfying the growth condition, i.e.\ $|u(x)|= o\left( \log|x| \right)$ as $|x|\to+\infty$ when $n=10$; or $|u(x)|= o\left( |x| ^{ -\frac n2+\sqrt{n-1}+2 }\right)$ as $x|\to+\infty$ when $n\ge 11$, must be a constant. This extends the well-known Liouville property for radial stable solutions obtained by Villegas.