Published Paper

Inserted: 22 may 2020
Last Updated: 19 aug 2024

Journal: Acta Math.
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 \( n \leq 9 \). This result, which 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 \( n \leq 9 \), 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 from 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 two famous open problems posed by Brezis and Brezis-Vázquez.

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