Normalized ground states for a biharmonic Choquard equation with exponential critical growth

Abstract

In this paper, we consider the normalized ground state solution for the following biharmonic Choquard type problem \begin{align*} \begin{split} \left\{ \begin{array}{ll} \Delta^2u-\beta\Delta u=\lambda u+(I_\mu*F(u))f(u), \quad\mbox{in}\ \ \mathbb{R}^4, \displaystyle\int_{\mathbb{R}^4}|u|^2dx=c^2,\quad u\in H^2(\mathbb{R}^4), \end{array} \right. \end{split} \end{align*} where $\beta\geq0$, $c>0$, $\lambda\in \mathbb{R}$, $I_\mu=\frac{1}{|x|^\mu}$ with $\mu\in (0,4)$, $F(u)$ is the primitive function of $f(u)$, and $f$ is a continuous function with exponential critical growth in the sense of the Adams inequality. By using a minimax principle based on the homotopy stable family, we obtain that the above problem admits at least one ground state normalized solution.