Symmetry via antisymmetric maximum principles in nonlocal problems of variable order

We consider the nonlinear problem (P) {Iu = f(x, u) in Omega, u = 0 on R-N backslash Omega in an open bounded set Omega subset of 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 0 in Omega, or u is strictly decreasing in vertical bar x(1)vertical bar . The proof relies on new maximum principles for antisymmetric supersolutions of an associated class of linear problems.