A variety of engineered systems can encounter critical transitions where the system suddenly shifts from one stable state to another at a threshold. The transition has aroused vital concerns for its potentially disastrous impacts. We validate an often taken-for-granted hypothesis that failure be attributed respective and show how early warning signals are closely associated with transitions. de...

In this paper we study positive solutions to the following nonlocal system of equations: \begin{equation*} \left\{\begin{aligned} &(-\Delta)^s u = \frac{\alpha}{2_s^*}|u|^{\alpha-2}u|v|^{\beta}+f(x)\;\;\text{in}\;\mathbb{R}^{N}, v \frac{\beta}{2_s^*}|v|^{\beta-2}v|u|^{\alpha}+g(x)\;\;\text{in}\;\mathbb{R}^{N}, & \qquad u, \, >0\, \mbox{ in }\,\mathbb{R}^{N}, \end{aligned} \right. \end{equation*...