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

Abstract

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.