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In this paper, we first give a necessary and sufficient condition for the boundedness and the compactness for a class of nonlinear functionals in $H^{2}\ ( \mathbb{R}^{4}\right)$. 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 $f\ ( s\right)$ 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 $$ 0<V_{0}=\underset{x\in\mathbb{R}^{4}}{\inf}V\ ( x\right) <\underset{\ | x\ | \rightarrow\infty}{\lim}V\ ( x\right) <+\infty. $$ The same result and proof applies to the harmonic equation with the critical exponential growth involving the Rabinowitz type trapping potential in $\mathbb{R}^2$.

Ground states of bi-harmonic equations with critical exponential growth involving constant and trapping potentials