I am pretty sure the John Ellingsen I found born in Tysvaer is the same person who according to US Social Security Death Index; dies in state of Washington December 1963 under the name of _John Aarvik_ . The same source is givng him the same birthdate 10.Th January 1883 as the John Ellingsen I found born at Aarvik farm in Tysvaer.
In the US census 1920 I think it is this man in Alaska:
John E Arvick age 36
residence: Government Railroad, Fourth, Alaska
In US census 1930:
John E Arvik
Residence: Seattle ; WA.
In US 1940 census:
John E Arvik b 1883 in Norway.
Resicdence : Tract L-5, Seattle, Seattle Election Precinct, King, Washington, United States
There is a John Aarnis (sic) in Fairbanks, Division 2, Alaska in the 1910 census. This is probably a misspelling of Aarvik age born 1883 emigration year 1903.
Aarvik was his farmname or the place he was born . There is yet some work to do connectiong him to your John Ellingsen. However I think this is not quite a shot into the darkness.