# Stable solutions to semilinear elliptic equations are smooth up to dimension 9

created by figalli on 22 May 2020
modified on 27 May 2020

[BibTeX]

Accepted Paper

Inserted: 22 may 2020
Last Updated: 27 may 2020

Journal: Acta Mathematica
Year: 2020

Abstract:

In this paper we prove the following long-standing conjecture: stable solutions to semilinear elliptic equations are bounded (and thus smooth) in dimension $n \geq 9$. This result, that was only known to be true for $n \leq 4$, is optimal: $\log(1/ x ^2)$ is a $W^{1,2}$ singular stable solution for $n \geq 10$. The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension $n \geq 9$, stable solutions are bounded in terms only of their $L^1$ norm, independently of the nonlinearity. In addition, in every dimension we establish a higher integrability result for the gradient and optimal integrability results for the solution in Morrey spaces. As one can see by a series of classical examples, all our results are sharp. Furthermore, as a corollary we obtain that extremal solutions of Gelfand problems are $W^{1,2}$ in every dimension and they are smooth in dimension n \leq 9. This answers to two famous open problems posed by Brezis and Brezis-Vazquez.