Bubbling of the prescribed $Q$-curvature equation on $4$-manifolds in the null case

Analog to the classical result of Kazdan-Warner for the existence of solutions to the prescribed Gaussian curvature equation on compact 2-manifolds without boundary, it is widely known that if $(M,g_0)$ is a closed 4-manifold with zero $Q$-curvature and if $f$ is any non-constant, smooth, sign-changing function with ${\int }_{M}fd{\mu }_{g_0}<0$, then there exists at least one solution $u$ to the prescribed $Q$-curvature equation $${\mathbf{P}}_{g_0}u=f{e}^{4u},$$ where ${\mathbf{P}}_{g_0}$ is the Paneitz operator which is positive with kernel consisting of constant functions. In this paper, we fix a non-constant smooth function $f_0$ with $$\underset{x\in M}{\max }{f}_0(x)=0,{\int }_M{f}_{0}d{\mu }_{g_0}<0$$ and consider a family of prescribed $Q$-curvature equations $${\mathbf{P}}_{g_0}u=(f_0+\lambda )e^{4u},$$ where $\lambda >0$ is a suitably small constant. A solution to the equation above can be obtained from a minimizer $u_{\lambda }$ of certain energy functional associated to the equation. Firstly, we prove that the minimizer $u_{\lambda }$ exhibits bubbling phenomenon in a certain limit regime as $\lambda \searrow 0$. Then, we show that the analogous phenomenon occurs in the context of $Q$-curvature flow.