In all the examples above the decision problem is in P and the search problem is not known to be in P but not known to be NP-hard either. I want to point out that it is possible to have an NP-hard search problem whose decision version is easy.
Consider the generalized satisfiability problem for given relations over Boolean domain . An instance is an expression of the form
's are either variables or constants in
are the arities of
(this is the same framework as in Schaeffer's dichotomy theorem with constants, in case you know what it is). The search problem is: given such an expression, find a lexicographically minimal solution, if there is one.
It was shown by Reith and Vollmer here that there exists a choice of relations that make this problem NP-hard (actually OptP-complete) but keep the satisfiability problem easy (quite trivial actually). An example given in the paper is (here ). Once the satisfiability problem is solvable in polynomial-time, the question whether there exists a lexicographically minimal satisfying assignment is trivial.
See Corollary 13 and the example following it in the paper above (at least in this on-line version).