Rigidity of minimizers in nonlocal phase transitions II

In this paper we extend the results of \cite{S3} to the borderline case $s = \frac 12$. We obtain the classification of global bounded solutions with asymptotically flat level sets for semilinear nonlocal equations of the type $${\mathrm{\Delta }}^{\frac{1}{2}}u=W^{\mathrm{\prime }}(u)\text{in}{R}^n,$$ where $W$ is a double well potential.