A similar situation happened in the 2006 House race in TX-22, after Tom DeLay resigned. The Republican party was unable to replace DeLay on the general election ballot, and Sheila Sekula-Gibbs had to run a candidate as a write-in.

The governor then called a special election to be held at the same time as the general election. As in Mississippi, special elections are run without a party primary and are open to multiple candidates from a party. Sekula-Gibbs and several other Republicans ran in the special election. The Democrat candidate in the general election Nick Lampson did not file for the special election.
So there was no Republican candidate on the general election ballot, and no Democratic candidate on the special election ballot. There was a Libertarian candidate on both.

Sekula-Gibbs won the special election and served the last part of the term (because Congress came back into session after the election, she actually participated in voting etc.).

Lampson won the general election and is currently in office (10 Republicans have filed for the primary to oppose him in November).

Since our special elections are nonpartisan, all candidates run in the same election. Sen. Roger Wicker and the 2 Democrats (Musgrove and Shows) will be together on the Nov. 4 ballot; if no one gets 50%-plus, there will be a runoff a couple of weeks later.

In 1947, the last time Mississippi had a special U. S. Senate election, a majority was not required to win. John Stennis was elected over five opponents with 26.9%. In ’48, the law was changed to require 50%-plus.