  Recently, the authors of the current paper established in [9] the existence
of a ground-state solution to the following bi-harmonic equation with the
constant potential or Rabinowitz potential: \begin{equation}
(-\Delta)^{2}u+V(x)u=f(u)\ \text{in}\ \mathbb{R}^{4}, \end{equation} when the
nonlinearity has the special form $f(t)=t(\exp(t^2)-1)$ and $V(x)\geq c>0$ is a
constant or the Rabinowitz potential. One of the crucial elements used in [9]
is the Fourier rearrangement argument. However, this argument is not applicable
if $f(t)$ is not an odd function. Thus, it still remains open whether the above
equation with the general critical exponential nonlinearity $f(u)$ admits a
ground-state solution even when $V(x)$ is a positive constant.
  The first purpose of this paper is to develop a Fourier rearrangement-free
approach to solve the above problem. More precisely, we will prove that there
is a threshold $\gamma^{*}$ such that for any $\gamma\in (0,\gamma^*)$, the
above equation with the constant potential $V(x)=\gamma>0$ admits a
ground-state solution, while does not admit any ground-state solution for any
$\gamma\in (\gamma^{*},+\infty)$. The second purpose of this paper is to
establish the existence of a ground-state solution to the above equation with
any degenerate Rabinowitz potential $V$ vanishing on some bounded open set.
Among other techniques, the proof also relies on a critical Adams inequality
involving the degenerate potential which is of its own interest.
