Assignment to make formula unsatisfiable

Question

Lets imagine we have a satisfiable formula $F(A_0, A_1,...A_k,S_0,...,S_n)$ The problem to solve is "Is there an assignment for variables $(S_0,...,S_n)$ which will make F unsatisfiable?". One way of solving is to find all solutions for F in terms of variables $S_0,...,S_n$ and if the count is < $2^n$, the missing solution will be the answer, but the complexity of this algorithm is huge, if the number of such assignments is small.

My questions are:

• Is there a way to solve the problem with less SAT solver calls?
• Is it a well-known problem in theory (What I should google to read about it)?