Symmetry via antisymmetric maximum principles in nonlocal problems of variable order

We consider the nonlinear problem $$(P) \;\; I u=f(x,u) \text{ in $\Omega$,} \;\; u=0 \text{ on $\mathbb{R}^{N}\setminus\Omega$ }$$ in an open bounded set $\Omega\subset\mathbb{R}^{N}$, where $I$ is a nonlocal operator which may be anisotropic and may have varying order. We assume mild symmetry and monotonicity assumptions on $I$, $\Omega$ 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 $\Omega$, or $u$ is strictly decreasing in $|x_1|$. The proof relies on new maximum principles for antisymmetric supersolutions of an associated class of linear problems.