The question whether density profiles in future burning plasmas such as in ITER will be flat or peaked has attracted considerable attention. The reason is that increased density peaking has consequences on the overall characteristics of the plasma. When pedestal density is fixed density peaking increases fusion power, energy confinement and bootstrap current. On the other hand peaking reduces the neoclassical tearing mode beta limit and may also lead to impurity accumulation. In a burning plasma only a turbulence-driven particle pinch can result in peaked density profiles because neo-classical pinch is too weak. The turbulence-driven particle pinch seems to be proven in L-mode plasmas and it is supported by the theory of turbulence equipartition or thermodiffusion. Its existence in ELMy H-modes is still, however, an open question. Also from theory point of view the situation is not clear as the direction of the turbulence-driven particle flux may depend on other parameters such as Te/Ti ratio. This paper reports on a further investigation of the density profiles in stationary H-mode. The numerical studies of the subject and recent L-mode data are presented in the accompanying paper.