Symmetry via antisymmetric maximum principles in nonlocal problems of variable order

We consider the nonlinear problem $$(P)Iu=f(x,u)\text{ in }\mathrm{\Omega }\text{,}u=0\text{ on }{\mathbb{R}}^N\setminus \mathrm{\Omega }$$ in an open bounded set $\mathrm{\Omega }\subset {\mathbb{R}}^N$, where $I$ is a nonlocal operator which may be anisotropic and may have varying order. We assume mild symmetry and monotonicity assumptions on $I$, $\mathrm{\Omega }$ and the nonlinearity $f$ with respect to a fixed direction, say $x_1$, and we show that any nonnegative weak solution $u$ of $(P)$ is symmetric in $x_1$. Moreover, we have the following alternative: Either $u\equiv 0$ in $\mathrm{\Omega }$, or $u$ is strictly decreasing in $\mathrm{\mid }{x}_1\mathrm{\mid }$. The proof relies on new maximum principles for antisymmetric supersolutions of an associated class of linear problems.