We develop a new, unified approach to the following two classical questions on elliptic PDE:

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the strong maximum principle for equations with non-Lipschitz nonlinearities,

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the at most exponential decay of solutions in the whole space or exterior domains.

Our results apply to divergence and non-divergence operators with locally unbounded lower-order coefficients, in a number of situations where all previous results required bounded ingredients. Our approach, which allows for relatively simple and short proofs, is based on a (weak) Harnack inequality with optimal dependence of the constants in the lower-order terms of the equation and the size of the domain, which we establish.

中文翻译：

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具有无界系数的椭圆偏微分方程的 Vázquez 最大值原理和兰迪斯猜想

我们针对椭圆偏微分方程的以下两个经典问题开发了一种新的、统一的方法：

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非Lipschitz非线性方程的强最大值原理，

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整个空间或外部域中解的至多指数衰减。

我们的结果适用于具有局部无界低阶系数的发散和非发散算子，在许多情况下，所有先前的结果都需要有界成分。我们的方法允许相对简单和简短的证明，它基于（弱）Harnack 不等式，具有方程低阶项中的常数和我们建立的域大小的最佳依赖关系。