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Stable solutions to semilinear elliptic equations are smooth up to dimension 9
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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: $\log \left(1 /|x|^{2}\right)$ 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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CABRÉ, Xavier, et al. Stable solutions to semilinear elliptic equations are smooth up to dimension 9. Acta Mathematica. 2020. Vol. 224, num. 2, pags. 187-252. ISSN 0001-5962. [consulted: 17 of August of 2026]. Available at: https://hdl.handle.net/2445/175795