Sliding method for the semi-linear elliptic equations involving the uniformly elliptic nonlocal operators

Abstract

<p style='text-indent:20px;'>In this paper, we consider the uniformly elliptic nonlocal operators <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \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, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ a(x) $\end{document}</tex-math></inline-formula> is positively uniform bounded satisfying a cylindrical condition. We first establish the narrow region principle in the bounded domain. Then using the sliding method, we obtain the monotonicity of solutions for the semi-linear equation involving <inline-formula><tex-math id="M2">\begin{document}$ A_{\alpha} $\end{document}</tex-math></inline-formula> in both the bounded domain and the whole space. In addition, we establish the maximum principle in the unbounded domain and get the non-existence of solutions in the upper half space <inline-formula><tex-math id="M3">\begin{document}$ \mathbb R^n_+ $\end{document}</tex-math></inline-formula>.