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تسجيل مستخدم جديدStable 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 semilinear elliptic equations are bounded (and thus smooth) in dimension $n leq 9$. This result, that was only known to be true for $nleq4$, is optimal: $log(1/|x|^2)$ is a $W^{1,2}$ singular stable solution for $ngeq10$. 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 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 $n leq 9$. This answers to two famous open problems posed by Brezis and Brezis-Vazquez.
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