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

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 variational methods under certain conditions.

Contribution

It introduces a novel approach to find normalized solutions for a biharmonic Choquard equation with exponential critical growth, expanding the understanding of such nonlinear PDEs.

Findings

Existence of ground state normalized solutions proven.

Solutions exist under specific conditions on the nonlinearity.

The approach handles 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} \gamma\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>0,\quad u\in H^2(\mathbb{R}^4), \end{array} \right. \end{split} \end{align*} where $\gamma>0$, $\beta\geq0$, $\lambda\in \mathbb{R}$, $I_\mu=\frac{1}{|x|^\mu}$ 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.