In this paper we prove the following long-standing conjecture: stable solutions to semilinear elliptic equations are bounded (and thus smooth) in dimension n \leq 9.
This result, that was only known to be true for n\leq4, is optimal: \log(1/|x|^2) is a W^{1,2} singular stable solution for n\geq10.
The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension n \leq 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-Vázquez.
