Lower Hybrid Current Drive (LHCD) experiments performed at density close to that required for the steady state scenario are reported. On C-Mod, FTU and Tore Supra, a strong decay of the brehmsstrahlung emission is observed when the density is increased, much faster that the prediction of LHCD modelling. On JET, LH power deposition is also found to be sensitive to the plasma density: LH power modulation indicates that the power deposition moves to the very edge of the plasma (r/a~0.9) when the density approaches the requirement of the JET SS scenario. From this experiment but also from the reconstruction of the electron cyclotron emission spectrum, the decrease of the LHCD efficiency with density is also found. From LHCD modelling of different JET pulses performed at different densities and wave parallel refraction indexes, it is concluded that the wave accessibility condition is not the key parameter for explaining the decrease of the efficiency. C-Mod, FTU and Tore Supra experiments indicate that the plasma edge parameters, namely density and temperature but also fluctuations, are affecting the efficiency via loss mechanisms which are likely to be collisional damping (C-Mod), parametric decay instabilities or wave scattering (FTU/Tore Supra).