  We consider nonlinear elliptic problems involving a nonlocal operator: the
square root of the Laplacian in a bounded domain with zero Dirichlet boundary
conditions. For positive solutions to problems with power nonlinearities, we
establish existence and regularity results, as well as a priori estimates of
Gidas-Spruck type. In addition, among other results, we prove a symmetry
theorem of Gidas-Ni-Nirenberg type.
