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We study the symmetry properties for solutions of elliptic systems of the type (-\Delta)^{s_1} u = F_1(u, v), (-\Delta)^{s_2} v= F_2(u, v), where $F\in C^{1,1}_{loc}(\R^2)$, $s_1,s_2\in (0,1)$ and the operator $(-\Delta)^s$ is the so-called fractional Laplacian. We obtain some Poincar\'e-type formulas for the $\alpha$-harmonic extension in the half-space, that we use to prove a symmetry result both for stable and for monotone solutions.

We consider a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions in a bounded domain Ω ⊂ R n whose boundary has an ( n − 2 ) -dimensional singularity. Assuming 1 p n + 2 n − 2 , we prove that, under suitable geometric conditions on the boundary of the domain, there exist solutions which approach the intersection of the Neumann and the Dirichlet parts as the singular perturbation parameter tends to zero.