In this paper we prove the following long-standing conjecture: stable solutions to semi-linear elliptic equations are bounded (and thus smooth) in dimension $n \leqslant 9$.

This result, that was only known to be true for $n \leqslant 4$, is optimal: $\log (1 / {\lvert x \rvert}^2)$ 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.

In this paper, we show that there is a Cantor set of initial conditions in the planar $4$‑body problem such that all four bodies escape to infinity in a finite time, avoiding collisions. This proves the Painlevé conjecture for the $4$‑body case, and thus settles the last open case of the conjecture.