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.

We study the existence of positive solutions for the nonlinear Schrödinger equation with the fractional LaplacianFurthermore, we analyse the regularity, decay and symmetry properties of these solutions.

Abstract. We establish existence and non-existence results to the BrezisNirenberg type problem involving the square root of the Laplacian in a bounded domain with zero Dirichlet boundary condition.

Abstract. We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is different from the one studied, among others, by Caffarelli and Silvestre.