标题： 带有双调和 Choquard 系统的归一化基态解$\mathbb{R}^4$ 显示英文标题

标题： Normalized ground states for a biharmonic Choquard system in $\mathbb{R}^4$

摘要： 本文研究如下双调和Choquard系统的归一化基态解的存在性问题\begin{align*} \begin{split} \left\{ \begin{array}{ll} \Delta^2u=\lambda_1 u+(I_\mu*F(u,v))F_u (u,v), \quad\mbox{in}\ \ \mathbb{R}^4, \Delta^2v=\lambda_2 v+(I_\mu*F(u,v)) F_v(u,v), \quad\mbox{in}\ \ \mathbb{R}^4, \displaystyle\int_{\mathbb{R}^4}|u|^2dx=a^2,\quad \displaystyle\int_{\mathbb{R}^4}|v|^2dx=b^2,\quad u,v\in H^2(\mathbb{R}^4), \end{array} \right. \end{split} \end{align*}，其中$a,b>0$为已知，$\lambda_1,\lambda_2\in \mathbb{R}$，$I_\mu=\frac{1}{|x|^\mu}$满足$\mu\in (0,4)$，$F_u,F_v$是$F$的偏导数，且$F_u,F_v$在Adams不等式的意义下具有指数量子临界或临界增长。 利用极小极大原理并分析基态能量关于规定质量的行为，我们得到了上述问题基态解的存在性。 显示英文摘要

摘要： In this paper, we study the existence of normalized ground state solutions for the following biharmonic Choquard system \begin{align*} \begin{split} \left\{ \begin{array}{ll} \Delta^2u=\lambda_1 u+(I_\mu*F(u,v))F_u (u,v), \quad\mbox{in}\ \ \mathbb{R}^4, \Delta^2v=\lambda_2 v+(I_\mu*F(u,v)) F_v(u,v), \quad\mbox{in}\ \ \mathbb{R}^4, \displaystyle\int_{\mathbb{R}^4}|u|^2dx=a^2,\quad \displaystyle\int_{\mathbb{R}^4}|v|^2dx=b^2,\quad u,v\in H^2(\mathbb{R}^4), \end{array} \right. \end{split} \end{align*} where $a,b>0$ are prescribed, $\lambda_1,\lambda_2\in \mathbb{R}$, $I_\mu=\frac{1}{|x|^\mu}$ with $\mu\in (0,4)$, $F_u,F_v$ are partial derivatives of $F$ and $F_u,F_v$ have exponential subcritical or critical growth in the sense of the Adams inequality. By using a minimax principle and analyzing the behavior of the ground state energy with respect to the prescribed mass, we obtain the existence of ground state solutions for the above problem.