When was Mathilda born? It is hard to help without dates.
Emihamn was made from registrations with Swedish port police before leaving Sweden. (Many Swedes, however, left through ports in Norway, Denmark, and Germany.)
Here are two possibilities for you.
First name: MATHILDA Last name: SOLBERG Age: 29Gender: K Parish: HÖGSBYCounty: H (Kalmar) Port: GÖTEBORG Date: 1872 05 31 Destination: NEW YORK Fellows: NEJ Source: 6:358:7303
First name: MATHILDA Head: SOLBERG T Age: 55Gender: K Parish: BETTNACounty: D (Södermanland) Title/Note: H (wife) Port: GÖTEBORG Date: 1883 05 11 (May 11) Destination: WINNIPEG Fellows: JA Source: 22:585:4864
She traveled with:
First name: T Last name: SOLBERG Age: 45Gender: M Parish: BETTNACounty: D Port: GÖTEBORG Date: 1883 05 11 Destination: WINNIPEG Fellows: JA Source: 22:585:4864
First name: JOSEFINA Head: SOLBERG T Age: 18Gender: K Parish: BETTNACounty: D Port: GÖTEBORG Date: 1883 05 11 Destination: WINNIPEG Fellows: JA Source: 22:585:4864
First name: CARL Head: SOLBERG T Age: 11Gender: M Parish: BETTNACounty: D Port: GÖTEBORG Date: 1883 05 11 Destination: WINNIPEG Fellows: JA Source: 22:585:4864
(Of course, most of our ancestors had patronymic last names (ending with son or dotter) and only later adopted a surname such as Sohlberg.)
There is another emigration database called the Emibas CD. Emibas was made from registrations in the Swedish parish to go to a foreign country. It is about 75% complete.
I don't see the two people listed above but there are a couple other possibilities.
This one moved to Great Britain (Storbritannien) but could have later moved to another country.
Solberg, Alma Matilda Unmarried woman
b. 10/28/1883 in Forshälla, Göteborgs och Bohus län (Bohuslän)
Emigrated 9/8/1899 from Solberg, Giljan, Forshälla, Göteborgs och Bohus län (Bohuslän) to Storbritannien
Emigrated 8/2/1868 from Gruvan 220, Höganäs, Malmöhus län (Skåne) to Amerika
Source: Household Examination Roll, p. 210
Emibas migration file ID: Höganäs M 1868 017
Solberg (and Sohlberg, with the decorative h) are NOT just Norwegian names. I see 215 hits for those names on the Emibas CD. There are 263 hits on the Emihamn database. There are 346 hits for those names on the 1880 Swedish Census CD.(Wives usually kept their maiden names and children were usually listed without a last name.) There are 367 hits on the 1890 Swedish Census CD. There are 406 hits on the 1900 Swedish Census CD.
This modern database for names shows 697 people with the Sohlberg spelling and 688 people with the Solberg spelling, as of 31 December 2010.