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June 2020 Stable solutions to semilinear elliptic equations are smooth up to dimension $9$
Xavier Cabré, Alessio Figalli, Xavier Ros-Oton, Joaquim Serra
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Acta Math. 224(2): 187-252 (June 2020). DOI: 10.4310/ACTA.2020.v224.n2.a1

Abstract

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.

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Xavier Cabré. Alessio Figalli. Xavier Ros-Oton. Joaquim Serra. "Stable solutions to semilinear elliptic equations are smooth up to dimension $9$." Acta Math. 224 (2) 187 - 252, June 2020. https://doi.org/10.4310/ACTA.2020.v224.n2.a1

Information

Received: 22 July 2019; Published: June 2020
First available in Project Euclid: 16 January 2021

Digital Object Identifier: 10.4310/ACTA.2020.v224.n2.a1

Subjects:
Primary: 35B35 , 35B65

Rights: Copyright © 2020 Institut Mittag-Leffler

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Vol.224 • No. 2 • June 2020
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