Decision variation of max sat
Given a $\overline{SAT}$ instance $\psi$ with $m$ clauses, let $\phi$ be the formula formed by adding to each clause a new variable $x$ (the same for all clauses), and a new clause $\bar{x}$. If $\psi$ is satisfiable then $\phi$ is satisfiable, whereas if $\psi$ is unsatisfiable then the maximum number of clauses that can be satisfied is $m$.