Normalized ground state solutions for a biharmonic equation involving critical exponential growth in $\mathbb{R}^{4}$

In this paper, we study the existence of normalized ground state solutions to the following biharmonic equation: $$ \left \{ \textstyle\begin{array}{l} \Delta ^{2} u = \lambda u+ \mu |u|^{q-2}u +f(u) \quad \text{in } \mathbb{R}^{4}, \\ \displaystyle \int _{\mathbb{R}^{4}} |u|^{2} \: dx = a^{2}, \end{array}\displaystyle \right . $$ where $a,\mu > 0 $ , $q>4$ , $\lambda \in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, and f is a nonlinear function that possesses critical exponential growth motivated by the Adams inequality. To prove the existence of solutions, we construct an augmented functional that possesses a mountain-pass-type geometry.