Geodesic graphs with constant mean curvature in spheres

We study the existence and unicity of graphs with constant mean curvature in the Euclidean sphere Sn+1(a) of radius a. Given a compact domain Omega, with some conditions, contained in a totally geodesic sphere S of Sn+1(a) and a real differentiable function chi on Omega, we define the graph of chi in Sn+1(a) considering the 'height' chi(x) on the minimizing geodesic joining the point x of Omega to a fixed pole of Sn+1(a). For a real number H such that \H\ is bounded for a constant depending on the mean curvature of the boundary Gamma of Omega and on a fixed number delta in (0,1), we prove that there exists a unique graph with constant mean curvature H and with boundary Gamma, whenever the diameter of Omega is smaller than a constant depending on delta. If we have conditions on Gamma, that is, let Gamma' be a graph over Gamma of a function, if \H\ is bounded for a constant depending only on the mean curvature of Gamma and if the diameter of Omega is smaller than a constant depending on H and Gamma, then we prove that there exists a unique graphs with mean curvature H and boundary Gamma'. The existence of such a graphs is equivalent to assure the existence of the solution of a Dirichlet problem envolving a nonlinear elliptic operator.