Saddle-shaped solutions of bistable elliptic equations involving the half-Laplacian

We establish existence and qualitative properties of saddle-shaped solutions of the elliptic fractional equation (−Δ)1/2u=f(u) in all the space R2m, where f is of bistable type. These solutions are odd with respect to the Simons cone and even with respect to each coordinate. More precisely, we prove the existence of a saddle-shaped solution in every even dimension 2m, as well as its monotonicity properties, asymptotic behaviour, and instability in dimensions 2m=4 and 2m=6. These results are relevant in connection with the analog for fractional equations of a conjecture of De Giorgi on the 1-D symmetry of certain solutions. Saddle-shaped solutions are the simplest candidates, besides 1-D solutions, to be global minimizers in high dimensions, a property not yet established.