*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.

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