The 37-year-old woman and her son, who were both Thai nationals, were found severely injured in a Kirkenes residence shortly before 5am on Monday. Both died from their wounds shortly thereafter.
The woman's 59-year-old husband was also severely injured in the incident. He was brought to hospital in Tromsø by air ambulance and is now facing murder charges.
The University Hospital of North Norway told NTB that the man's condition is stable and not life threatening.
According to broadcaster NRK, an impending breakup of the couple's marriage is likely behind the double murder. The 37-year-old woman reportedly wanted out of the marriage.
One of the woman's close friends told VG that he brought her to a shelter last week before reporting her husband to the police for psychological abuse.
Norasenteret in Kirkenes, a centre for abuse victims, confirmed to NRK that the deceased woman was in contact with the centre just days before she was murdered. According to the woman's friend, she was scheduled to move into the centre on the same day she was killed.
The friend said that the woman spoke briefly with police, but he claims that they did not take any action. He is now accusing officers of not doing their job.
But Finnmark Police Commissioner Ellen Katrine Hætta told NRK that the woman denied having been abused by her husband and was primarily concerned with her son's legal status in Norway.
“She was in contact with the police last Friday. At that point, she was mostly worried about the status of her son's residence permit if she were to leave her husband,” Hætta said.
According to VG, the couple was married in 2013. The woman had a Norwegian residence permit but it was not immediately known if her son also had a valid residence permit.
Police are working on the theory that the 59-year-old man shot his wife and stepson and then turned the gun on himself.
“We believe it is likely that it is the accused man who did it, based on the findings we have so far,” chief investigator Torstein Pettersen said.
Pettersen said questioning the man would be the top priority once the man's physical condition allows it.
The National Criminal Investigation Service (Kripos) is assisting Finnmark Police with the investigation.