  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 $\int_M f d\mu_{{\it g}_0} <0$, 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 \[ \max_{x\in M}f_0(x)=0, \quad \int_M
f_0 d\mu_{{\it g}_0} <0 \] 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.

Analysis of PDEs;Differential Geometry