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

## TL;DR

This paper proves that stable solutions to semilinear elliptic equations are bounded and smooth in dimensions up to 9, resolving a long-standing conjecture and establishing sharp universal estimates.

## Contribution

It establishes the optimal dimension bound for regularity of stable solutions and introduces a universal estimate linking boundedness to the L^1 norm, independent of the nonlinearity.

## Key findings

- Stable solutions are bounded in dimensions n ≤ 9.
- All stable solutions are smooth in dimensions n ≤ 9.
- Extremal solutions of Gelfand problems are W^{1,2} in all dimensions.

## Abstract

In this paper we prove the following long-standing conjecture: stable solutions to semilinear elliptic equations are bounded (and thus smooth) in dimension $n \leq 9$. 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-V\'azquez.