  In this paper, we first give a necessary and sufficient condition for the
boundedness and the compactness for a class of nonlinear functionals in
$H^{2}(\mathbb{R}^4)$. Using this result and the principle of symmetric
criticality, we can present a relationship between the existence of the
nontrivial solutions to the semilinear bi-harmonic equation of the form \[
(-\Delta)^{2}u+\gamma u=f(u)\ \text{in}\ \mathbb{R}^4 \] and the range of
$\gamma\in \mathbb{R}^{+}$, where $f(s)$ is the general nonlinear term having
the critical exponential growth at infinity.
  Our next goal in this paper is to establish the existence of the ground-state
solutions for the equation \begin{equation}\label{con}
(-\Delta)^{2}u+V(x)u=\lambda s\exp(2|s|^{2}))\ \text{in}\ \mathbb{R}^{4},
\end{equation} when $V(x)$ is a positive constant using the Fourier
rearrangement and the Pohozaev identity. Then we will explore the relationship
between the Nehari manifold and the corresponding limiting Nehari manifold to
derive the existence of the ground state solutions for the above equation when
$V(x)$ is the Rabinowitz type trapping potential, namely it satisfies
$$0<V_{0}=\underset{x\in\mathbb{R}^{4}}{\inf}V(x) <\underset{\ | x\ |
\rightarrow\infty}{\lim}V(x) < +\infty. $$ The same result and proof applies to
the harmonic equation with the critical exponential growth involving the
Rabinowitz type trapping potential in $\mathbb{R}^2$.
