We report on the growth of GaAs-AlGaAs core-multishell nanowire quantum heterostructures by metalorganic vapor phase epitaxy, and their photoluminescence (PL) properties. Dense arrays of vertically-aligned GaAs nanowires were fabricated onto (111)B-GaAs wafers by Au-catalyzed self-assembly, and radially overgrown by two AlGaAs shells between which a few-nm thin GaAs shell was introduced to form a quantum well tube (QWT). Besides the GaAs nanowire core emission band peaked at around 1.503 eV, 7K PL spectra showed an additional broad peak in the 1.556- 1.583 eV energy interval, ascribed to the transition between electron and hole confined states within the QWT. The emission blue-shifts with the shrinkage of as-grown GaAs well tubes, as the nanowire local (on the substrate) density and height change.