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

TL;DR

This paper establishes the existence of normalized ground state solutions for a biharmonic Choquard equation with exponential critical growth in four-dimensional space, using a minimax approach and Adams inequality.

Contribution

It introduces a novel variational method to find normalized ground states for a biharmonic Choquard equation with exponential critical growth.

Findings

Existence of at least one normalized ground state solution.

Application of minimax principle based on homotopy stable family.

Extension of results to equations 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.