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| We study the nonlinear fractional equation $(-\Delta)^s u =f(x,u)$, $s\in (0,1)$ in a bounded open set $\Omega$ together with Dirichlet boundary conditions on $R^N \setminus \Omega$. We prove symmetry of nonnegative bounded solutions in the case where the underlying data obeys some symmetry and monotonicity assumptions. This result, which is obtained via a series of new estimates for antisymmetric supersolutions of a corresponding family of linear equations, implies a strong maximum type principle which is not available in the non-fractional case $s=1$. Moreover, we will give possible extensions of this statement for more general nonlocal operators than the fractional Laplacian. |
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