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

Abstract

In this paper, we study the following biharmonic Choquard type equation \begin{align*} \begin{split} \left\{ \begin{array}{ll} γΔ^2u-βΔu=λu+(I_μ*F(u))f(u), \quad\mbox{in}\ \ \mathbb{R}^4, \displaystyle\int_{\mathbb{R}^4}|u|^2dx=c^2>0,\quad u\in H^2(\mathbb{R}^4), \end{array} \right. \end{split} \end{align*} where $\gamma >0$, $\beta \ge 0$, $\lambda \in \mathbb{R}$, $I_{\mu }=\frac{1}{|x{|}^{\mu }}$ with $\mu \in (0,4)$, $F(u)$ is the primitive function of $f(u)$, and $f$ is a continuous function with exponential critical growth. We can prove the existence of ground state normalized solutions for the above problem when the nonlinearity $f$ satisfies some conditions.