In this paper we prove the following long-standing conjecture: stable solutions to semi-linear elliptic equations are bounded (and thus smooth) in dimension n ⩽ 9.
This result, that was only known to be true for n ⩽ 4, is optimal: log(1/|x|2) is a W1,2 singular stable solution for n ⩾ 10.
The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension n ⩽ 9, stable solutions are bounded in terms only of their L1 norm, independently of the non-linearity. 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 W1,2 in every dimension and they are smooth in dimension n ⩽ 9. This answers to two famous open problems posed by Brezis and Brezis–Vázquez.