Sliding method for the semi-linear elliptic equations involving the uniformly elliptic nonlocal operators
Sliding method for the semi-linear elliptic equations involving the uniformly elliptic nonlocal operators
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DOI：
10.3934/dcds.2020362
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发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Meng Qu;Jia-Yong Wu;Ting Zhang
中科院分区：
文献类型：
--
作者：
Meng Qu;Jia-Yong Wu;Ting Zhang
本文考虑一致椭圆非局部算子Begin{Document}$A\Alpha}u(X)=C_{n，\α}\Rm{P.V.}\int_{\mathbb{R}^n}\frac{a(x-y)(u(X)-u(Y))}{|x-y|^{n+\α}}dy，$\end{Document}其中Begin{Document}$a(X)$\end{Document}是一致有界的，且满足圆柱条件。我们首先在有界域上建立了窄域原理。然后利用滑动方法，得到了包含有界域和整个空间的半线性方程的解的单调性。此外，我们在无界区域上建立了极大值原理，得到了在上半空间中不存在解的结论。
In this paper, we consider the uniformly elliptic nonlocal operators \begin{document}$ A_{\alpha} u(x) = C_{n,\alpha} \rm{P.V.} \int_{\mathbb{R}^n} \frac{a(x-y)(u(x)-u(y))}{|x-y|^{n+\alpha}} dy, $\end{document} where \begin{document}$ a(x) $\end{document} is positively uniform bounded satisfying a cylindrical condition. We first establish the narrow region principle in the bounded domain. Then using the sliding method, we obtain the monotonicity of solutions for the semi-linear equation involving \begin{document}$ A_{\alpha} $\end{document} in both the bounded domain and the whole space. In addition, we establish the maximum principle in the unbounded domain and get the non-existence of solutions in the upper half space \begin{document}$ \mathbb R^n_+ $\end{document} .