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 $n\leq4$, is optimal: $\log(1/|x|^2)$ is a $W^{1,2}$ singular stable solution for $n\geq10$. 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.

中文翻译：

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半线性椭圆方程的稳定解在维度上是平滑的 $9$

在本文中，我们证明了以下长期存在的猜想：半线性椭圆方程的稳定解在维度 $n \leq 9$ 上有界（因此是平滑的）。这个结果，只知道对 $n\leq4$ 是正确的，是最优的： $\log(1/|x|^2)$ 是 $n 的 $W^{1,2}$ 奇异稳定解\geq10$。这个猜想的证明是一个新的普遍估计的结果：我们证明，在维度 $n \leq 9$ 中，稳定的解决方案仅在其 $L^1$ 范数方面有界，与非线性无关。此外，在每个维度上，我们都建立了更高的梯度可积性结果和莫雷空间中解的最优可积性结果。从一系列经典例子可以看出，我们所有的结果都很清晰。此外，作为推论，我们得到 Gelfand 问题的极值解是 $W^{1，2}$ 在每个维度上，它们在维度 $n \leq 9$ 上都是平滑的。这是对 Brezis 和 Brezis-Vazquez 提出的两个著名的公开问题的回答。