  In the conformal class of Euclidean space, we give some volume comparison
theorems with help of Q-curvature. Meanwhile, for compact four dimensional
manifolds with non-negative scalar curvature, we give a volume rigidity theorem
with respect to Q-curvature. Finally, we make use of these results to give some
sufficient and necessary conditions for the existence of solutions to some
conformally invariant equations which answers an open problem raised by
Hyder-Martinazzi (2021, JDE).
