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

Let 0 ≤ f ∈ C(R). Given a domain Ω ⊂ R, we prove that any stable solution to the equation −∆u = f(u) in Ω satisfies • a BMO interior regularity when n = 10, • an Morrey Mnn interior regularity when n ≥ 11, where pn = 2(n− 2 √ n− 1− 2) n− 2 √ n− 1− 4 . This result is optimal as hinted by e.g. [3, 7, 13], and answers an open question raised by Cabré, Figalli, Ros-Oton and Serra [8]. As an application, we show a sharp Liouville property: Any stable solution u ∈ C(R) to −∆u = f(u) in R satisfying the growth condition |u(x)| = { o (log |x|) as |x| → +∞, when n = 10; o ( |x|−n2 + √ n−1+2 ) as |x| → +∞, when n ≥ 11 must be a constant. This extends the well-known Liouville property for radial stable solutions obtained by Villegas [24].