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 $(-\mathrm{\Delta }{)}^{1\mathrm{/}2}u=f(u)$ in all the space ${\text{re}}^{2m}$, 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.