Symmetry and monotonicity of positive solutions for a class of general pseudo-relativistic systems

Abstract

<p style='text-indent:20px;'>In this paper, we focus on a class of general pseudo-relativistic systems</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} \begin{aligned} &amp;(-\Delta+m^2)^su(x) = f(u(x), v(x)), \\ &amp;(-\Delta+m^2)^tv(x) = g(u(x), v(x)), \end{aligned} \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ m \in (0, +\infty) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ s, t \in (0,1) $\end{document}</tex-math></inline-formula>. Before giving the main results, we first introduce a decay at infinity and a narrow region principle. Then we implement the direct method of moving planes to show the radial symmetry and monotonicity of positive solutions for the above system in both the unit ball and the whole space.</p>