标题： 分数Choquard系统的归一化基态在$\mathbb{R}$中 显示英文标题

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

摘要： 在本文中，我们研究以下分数Choquard系统 \begin{align*} \begin{split} \left\{ \begin{array}{ll} (-\Delta)^{1/2}u=\lambda_1 u+(I_\mu*F(u,v))F_u (u,v), \quad\mbox{in}\ \ \mathbb{R}, (-\Delta)^{1/2}v=\lambda_2 v+(I_\mu*F(u,v)) F_v(u,v), \quad\mbox{in}\ \ \mathbb{R}, \displaystyle\int_{\mathbb{R}}|u|^2\mathrm{d}x=a^2,\quad \displaystyle\int_{\mathbb{R}}|v|^2\mathrm{d}x=b^2,\quad u,v\in H^{1/2}(\mathbb{R}), \end{array} \right. \end{split} \end{align*} 其中 $(-\Delta)^{1/2}$表示 $1/2$-拉普拉斯算子， $a,b>0$是给定的， $\lambda_1,\lambda_2\in \mathbb{R}$， $I_\mu(x)=\frac{{1}}{{|x|^\mu}}$与 $\mu\in(0,1)$， $F_u,F_v$是 $F$的偏导数，且 $F_u,F_v$在 $\mathbb{R}$中具有指数临界增长。 通过使用极小极大原理并分析基态能量相对于给定质量的单调性，我们得到了上述系统至少一个归一化基态解。 显示英文摘要

摘要： In this paper, we study the following fractional Choquard system \begin{align*} \begin{split} \left\{ \begin{array}{ll} (-\Delta)^{1/2}u=\lambda_1 u+(I_\mu*F(u,v))F_u (u,v), \quad\mbox{in}\ \ \mathbb{R}, (-\Delta)^{1/2}v=\lambda_2 v+(I_\mu*F(u,v)) F_v(u,v), \quad\mbox{in}\ \ \mathbb{R}, \displaystyle\int_{\mathbb{R}}|u|^2\mathrm{d}x=a^2,\quad \displaystyle\int_{\mathbb{R}}|v|^2\mathrm{d}x=b^2,\quad u,v\in H^{1/2}(\mathbb{R}), \end{array} \right. \end{split} \end{align*} where $(-\Delta)^{1/2}$ denotes the $1/2$-Laplacian operator, $a,b>0$ are prescribed, $\lambda_1,\lambda_2\in \mathbb{R}$, $I_\mu(x)=\frac{{1}}{{|x|^\mu}}$ with $\mu\in(0,1)$, $F_u,F_v$ are partial derivatives of $F$ and $F_u,F_v$ have exponential critical growth in $\mathbb{R}$. By using a minimax principle and analyzing the monotonicity of the ground state energy with respect to the prescribed masses, we obtain at least one normalized ground state solution for the above system.