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

Xavier Cabré ICREA Barcelona, Spain and Universitat Polit`ecnica 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

Analysis of PDEs mathscidoc:2203.03003

Acta Mathematica, 224, (2), 187-252, 2020.6
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⩽9. This result, that was only known to be true for n⩽4, is optimal: log(1/|x|^2) is a W^{1,2} singular stable solution for n⩾10. The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension n⩽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⩽9. This answers to two famous open problems posed by Brezis and Brezis–Vázquez.
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@inproceedings{xavier2020stable,
  title={Stable solutions to semilinear elliptic equations are smooth up to dimension 9},
  author={Xavier Cabré, Alessio Figalli, Xavier Ros-Oton, and Joaquim Serra},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220310102128396778912},
  booktitle={Acta Mathematica},
  volume={224},
  number={2},
  pages={187-252},
  year={2020},
}
Xavier Cabré, Alessio Figalli, Xavier Ros-Oton, and Joaquim Serra. Stable solutions to semilinear elliptic equations are smooth up to dimension 9. 2020. Vol. 224. In Acta Mathematica. pp.187-252. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220310102128396778912.
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