NP Problems with unique solution
Yes, the class is called UP (the U standing for "unambiguous"). David points out in the comments that another answer is US.

UP: If $x \in L$, then there is exactly one "proof" ("witness", "certificate", "accepting path"). If $x \not\in L$, there are exactly zero "proofs".

US: If $x \in L$, then there is exactly one "proof". If $x \not\in L$, there may be zero proofs, or 2+ proofs, as long as there is not exactly one.