Optimal regularity & Liouville property for stable solutions to semilinear elliptic equations in ${\mathbb{R}}^n$ with $n\ge 10$

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 Cabré, 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 $|u(x)|=o\left(|x{|}^{-\frac{n}{2}+\sqrt{n-1}+2}\right)$ 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.

21 pages