In this paper, we propose mathematical optimization models to solve the satellite constellation design problem for discontinuous coverage. In such a design problem, the aim is to determine the minimal number of satellites (and, incidentally, their 3D placements) in order to observe a fixed Earth region within a given revisiting time. Two Mixed Integer Nonlinear formulations are introduced. The first one is a feasibility problem based on the direct mathematical definition of pixel observability. The second one consists in introducing a set of indicator variables which specify if a satellite observes a pixel at a given time-stamp. In order to obtain a linear problem, the possible positions of the satellites are discretized. Finally, computational results show the potential and limitations of the proposed approaches.