We present results about the growth of GaAs/InAs core-shell nanowires (NWs) using molecular beam epitaxy. The core is grown via the Ga droplet-assisted growth mechanism. For a homogeneous growth of the InAs shell, the As(4) flux and substrate temperature are critical. The shell growth starts with InAs islands along the NW core, which increase in time and merge giving finally a continuous and smooth layer. At the top of the NWs, a small part of the core is free of InAs indicating a crystal phase selective growth. This allows a precise measurement of the shell thickness and the fabrication of InAs nanotubes by selective etching. The strain relaxation in the shell occurs mainly via the formation of misfit dislocations and saturates at ~80%. Additionally, other types of defects are observed, namely stacking faults transferred from the core or formed in the shell, and threading dislocations.