Acta Mathematica

Volume 224 (2020)

Number 2

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

Pages: 187 – 252



Xavier Cabré (ICREA, Barcelona, Spain; Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain; and BGSMath, Bellaterra, Spain)

Alessio Figalli (Department of Mathematics, ETH Zürich, Switzerland)

Xavier Ros-Oton (Institut für Mathematik, Universität Zürich, Switzerland; ICREA, Barcelona, Spain; and Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Spain)

Joaquim Serra (Department of Mathematics, ETH Zürich, Switzerland)


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.

2010 Mathematics Subject Classification

35B35, 35B65

Received 22 July 2019

Accepted 13 May 2020

Published 23 June 2020