We prove a comparison theorem for super- and sub-solutions with non-vanishing gradients to semilinear PDEs provided a nonlinearity f is L^p function with p > 1. The proof is based on a strong maximum principle for solutions of divergence type elliptic equations with VMO leading coefficients and with lower order coefficients from a Kato class. An application to estimation of periodic water waves profiles is given.

中文翻译：

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非光滑非线性算子的比较定理

我们证明与非零梯度以提供一个非线性偏微分方程半线性超和子解决方案的比较定理˚F被大号^p与函数p > 1.证明是基于用于与发散型椭圆方程的解决方案的强最大原理VMO前导系数和Kato类的低阶系数。给出了估算周期性水浪剖面的应用。