Normalized solutions for a biharmonic Choquard equation with exponential critical growth in \(\mathbb {R}^4\)

In this paper, we study the following biharmonic Choquard-type problem \(\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} \Delta ^2u-\beta \Delta u=\lambda u+(I_\mu *F(u))f(u), \quad \text{ in }\ \ \mathbb {R}^4,\\ \displaystyle \int \limits _{\mathbb {R}^4}|u|^2\textrm{d}x=c^2>0,\quad u\in H^2(\mathbb {R}^4), \end{array} \right. \end{aligned} \end{aligned}\) $$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{\mathrm{\Delta }}^{2}u-\beta \mathrm{\Delta }u=\lambda u+(I_{\mu }\ast F(u))f(u),\text{in}{\mathbb{R}}^4,\hfill \\ {\displaystyle \underset{{\mathbb{R}}^4}{\int }{|u|}^2\text{d}x=c^2>0,u\in H^2({\mathbb{R}}^4),}\hfill \end{array}\right)\end{array}\end{array}$$ where \(\beta \ge 0\) $\beta \ge 0$ , \(\lambda \in \mathbb {R}\) $\lambda \in \mathbb{R}$ , \(I_\mu =\frac{1}{|x|^\mu }\) $I_{\mu }=\frac{1}{{|x|}^{\mu }}$ with \(\mu \in (0,4)\) $\mu \in (0,4)$ , F(u) is the primitive function of f(u), and f is a continuous function with exponential critical growth. By using the mountain-pass argument, we prove the existence of radial ground-state solutions for the above problem.