Monotonicity results for quasilinear fractional systems in epigraphs

We prove the monotonicity of positive solutions to the quasilinear fractional system \begin{cases} (-\Delta)^s_p u = f(x,u,v) &\text{in } \Omega,\\ (-\Delta)^t_q v = g(x,u,v) &\text{in } \Omega,\\ u=v=0 &\text{in } \mathbb{R}^n\setminus \Omega, \end{cases} where \Omega is a coercive or non-coercive epigraph, such as the halfspace \mathbb{R}^n_+ , 0<s,t<1 and p,q\ge2 . By combining a new decay at infinity principle with the direct method of moving planes, we improve recent works in the literature even in the case of a single equation.