Cabré, XavierFigalli, AlessioRos, XavierSerra Montolí, Joaquim2021-03-262021-03-262020-09-010001-5962https://hdl.handle.net/2445/175795In this paper we prove the following long-standing conjecture: stable solutions to semi-linear elliptic equations are bounded (and thus smooth) in dimension $n \leqslant 9$. This result, that was only known to be true for $n \leqslant 4,$ is optimal: $\log \left(1 /|x|^{2}\right)$ is a $W^{1,2}$ singular stable solution for $n \geqslant 10$. The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension $n \leqslant 9,$ stable solutions are bounded in terms only of their $L^{1}$ 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 $W^{1,2}$ in every dimension and they are smooth in dimension $n \leqslant 9$. This answers to two famous open problems posed by Brezis and Brezis-Vázquez.66 p.application/pdfeng(c) Institut Mittag-Leffler , 2020Equacions en derivades parcialsEquacions diferencials el·líptiquesPartial differential equationsElliptic differential equationsinfo:eu-repo/semantics/article7085502021-03-26info:eu-repo/semantics/openAccess