  The Yamabe problem in compact closed Riemannian manifolds is concerned with
finding a metric with constant scalar curvature in the conformal class of a
given metric. This problem was solved by the combined work of Yamabe,
Trudinger, Aubin, and Schoen. In particular, Aubin solved the case when the
Riemannian manifold is compact, is nonlocally conformally flat and has a
dimension equal to or greater than $6$. In $2015$, Case considered a
Yamabe-type problem in the setting of smooth measure space in manifolds and for
a parameter $m$, which generalizes the original Yamabe problem when $m=0$.
Additionally, Case solved this problem when the parameter $m$ is a natural
number. In the context of the Yamabe-type problem, we generalize Aubin's result
for nonlocally conformally flat manifolds, with dimension equal and greater
than 6 and parameter $m$ close to nonnegative integers.
