The QQ-curvature on a 4-dimensional Riemannian manifold (M,g)(M,g) with ∫MQdVg=8π2∫MQdVg=8π2

We deal with the QQ-curvature problem on a 4-dimensional compact Riemannian manifold (M,g)(M,g) with ∫MQgdVg=8π2∫MQgdVg=8π2 and positive Paneitz operator PgPg. Let Q̃ be a positive smooth function. The question we consider is, when can we find a metric g̃ which is conformal to gg, such that Q̃ is just the QQ-curvature of g̃. A sufficient condition to this question is given in this paper.