Did Akhenaten live at the Malkata Palace. This palace located on the western bank of the Nile River. It was built late into the reign of Amenhotep III according to Joann Fletcher's book, Chronicle of a Pharaoh, The Intimate Life of Amenhotep III pg 128.
"At the end of his third decade as king, Amenhotep moved his court permanently to Thebes, where he had built "the palace of the dazzling Aten." Later known by its Arabic name, Malkata, "the place where things are picked up" (because it is littered with ancient debris), the palace stood stood on the Theban West Back below the western hills where the sun set each evening, and directly opposite the king's temple of Luxor. Amenhotep's move to Thebes was in direct contrast with the behavior of previous pharaohs, who had based themelves at the traditional capital of Menphis and had only come nouth to Thebes for the annual religious festivals, setting p temporary court in palace building attached to Karnak temple. In constructing an independent palace, not only at the other end of the city from Karnak, but also on the opposite bank of the river, Amenhotep was clearly distancing himself from the Karnak clergy."
Discovered in 1888, the palace is a huge complex sparwling over 30 hectares which was still being added to and emblelished that Amenhotep's death. The king's apartments was located in the southeast section, Queen Tiy's to the south and Sitamun's to the north. There were quarters for the rest of the royal family, numerous royal women, and resiences for hisgh officials and servants.
Now if you don't accept a coregency of any lenght between Amenhotep and his son... there is a possibility that yes, Akhenaten did live for a time at the palace. If he wasn't living in his own estate nearby. However, if you do accept a long period of coregency between father and son. He still may have lived at the palace but the lenght of time shorten because he did move his capital city to Akhetaten, in the 5th year of his reign and settled there about his 9th year.
So where might have Akhenaten grew up? The royal palace at Gurob is a good possibility. It was the palace that Amenhotep had lived at as a young boy. It is located in the southwest of el-Lahun in the Fayuum oasis, on the edge of he desert. It is often referred to as a harem palace. One of Amenhotep's favored guardians came from this area. Then Akhenaten might have grown up in Memphis... i think Gurob is more likely because in Queen Tiy's later years it was one of her favorites.