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

Abstract

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.