Normalized Ground States for a Biharmonic Choquard System in ℝ⁴

Abstract

In this paper, we study the following biharmonic Choquard system

$\left\{\begin{aligned}\Delta^2 u - \beta \Delta u &= \lambda_1 u + \left(I_\mu * F(u,v)\right) F_u(u,v)\quad \text{in} \mathbb{R}^4, \\[6pt]\Delta^2 v - \beta \Delta v&= \lambda_2 v + \left(I_\mu * F(u,v)\right) F_v(u,v)\quad \text{in} \mathbb{R}^4, \\[6pt]\int_{\mathbb{R}^4} |u|^2 \, dx &= a^2, \quad\int_{\mathbb{R}^4} |v|^2 \, dx = b^2, \quad u,v \in H^2(\mathbb{R}^4).\end{aligned}\right.$

where $\beta \geq 0,\quad a,b>0,\quad \lambda_1,\lambda_2 \in \mathbb{R},\quad I_\mu = \frac{1}{|x|^\mu}$ with μ ∈ (0, 4), F_u, F_v are partial derivatives of F and F_u, F_v have exponential critical growth in the sense of the Adams inequality. By using the minimax principle and analyzing the behavior of the least energy with respect to the masses, we prove the existence of ground state solutions for the above system.

Keywords

Normalized solutions / biharmonic Choquard system / exponential critical growth / 31B30 / 35J35 / 35J61 / 35J91

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