OPTIMAL REGULARITY AND THE LIOUVILLE PROPERTY FOR STABLE SOLUTIONS TO SEMILINEAR ELLIPTIC EQUATIONS IN Rⁿ WITH n ≥ 10

Let 0 ≤ f ∈ C^0,1(R). Given a domain Ω ⊂ Rⁿ, we prove that any stable solution to the equation −Δu = f (u) in Ω satisfies • a BMO interior regularity, when n = 10, • a Morrey M^{pn,4+2/(pn−2)} interior regularity, when n ≥ 11, where (Formula presented.) 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 ∈ C²(Rⁿ) to −Δu = f (u) in Rⁿ satisfying the growth condition (Formula presented.) must be a constant. This extends the well-known Liouville property for radial stable solutions obtained by Villegas (2007).