  This paper is devoted to several existence results for a generalized version
of the Yamabe problem. First, we prove the remaining global cases for the range
of powers $\gamma\in (0,1)$ for the generalized Yamabe problem introduced by
Gonzalez and Qing. Second, building on a new approach by Case and Chang for
this problem, we prove that this Yamabe problem is solvable in the
Poincar\'{e}-Einstein case for $\gamma\in (1,\min\{2,n/2\})$ provided the
associated fractional GJMS operator satisfies the strong maximum principle.
