论文标题
涉及$ p $ -laplacian的非线性方程的稳定解决方案的最佳规律性
Optimal regularity of stable solutions to nonlinear equations involving the $p$-Laplacian
论文作者
论文摘要
我们考虑在$ \ mathbb {r}^n $的平滑有限域中的方程$-Δ_Pu = f（u）$，其中$Δ_p$是$ p $ -laplace运算符。如果$ n \ geq p+4p/（p-1）$，知道无限稳定能量解决方案的明确示例。取而代之的是，当$ n <p+4p/（p-1）$时，事实证明，稳定的解决方案仅在径向案例中或在$ f $上的强烈假设下受到限制。 在本文中，我们解决了一个长期的开放问题：我们证明了一个内部$ c^α$用于稳定解决方案，每当$ p \ geq2 $和最佳条件$ n <p+4p/（p-1）$时，它适用于c^1 $中的每一个非负$ f \。当$ p \ in（1,2）$中时，我们根据非sharp假设$ n <5p $获得相同的结果。这些内部估计导致稳定和极端解决方案严格凸出时，稳定和极端解决方案对相关的Dirichlet问题的界限。 我们的工作扩展到了$ p $ -laplacian，最新的figalli，Ros-oton，Serra和古典Laplacian的第一作者的一些结果，当$ n <10 $ $ n <10 $中时，$ p = 2 $。
We consider the equation $-Δ_p u=f(u)$ in a smooth bounded domain of $\mathbb{R}^n $, where $Δ_p$ is the $p$-Laplace operator. Explicit examples of unbounded stable energy solutions are known if $n\geq p+4p/(p-1)$. Instead, when $n<p+4p/(p-1)$, stable solutions have been proved to be bounded only in the radial case or under strong assumptions on $f$. In this article we solve a long-standing open problem: we prove an interior $C^α$ bound for stable solutions which holds for every nonnegative $f\in C^1$ whenever $p\geq2$ and the optimal condition $n<p+4p/(p-1)$ holds. When $p\in(1,2)$, we obtain the same result under the non-sharp assumption $n<5p$. These interior estimates lead to the boundedness of stable and extremal solutions to the associated Dirichlet problem when the domain is strictly convex. Our work extends to the $p$-Laplacian some of the recent results of Figalli, Ros-Oton, Serra, and the first author for the classical Laplacian, which have established the regularity of stable solutions when $p=2$ in the optimal range $n<10$.
