Orbital stability of standing waves for fractional Hartree equation with unbounded potentials
Orbital stability of standing waves for fractional Hartree equation with unbounded potentials
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DOI：
10.1090/conm/725/14561
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发表时间：
2019-08
期刊：
影响因子：
--
通讯作者：
Jian Zhang;Shijun Zheng;Shihui Zhu
中科院分区：
文献类型：
--
作者：
Jian Zhang;Shijun Zheng;Shihui Zhu
我们证明了在适当的能量空间$\Sigma^s=\{u：\int_{\mathbb{R}^N} \bar{u}（-\Delta+m^2）^s u+V中基态集合的存在性|u| ^2<\infty\}$，$s\in（0，\frac {N}{2}）$的无界位势质量亚临界非线性分数阶Hartree方程.作为结果，我们得到，作为先验结果，轨道稳定性的一组驻波。主要的成分是观察到$\Sigma^s$是紧密嵌入在$L^2$中的。这使我们能够应用Cazenave-Lions和Zhang的著作中的集中紧性论证，即能量空间中任何极小化序列的相对紧性。
We prove the existence of the set of ground states in a suitable energy space $\Sigma^s=\{u: \int_{\mathbb{R}^N} \bar{u}(-\Delta+m^2)^s u+V |u|^2<\infty\}$, $s\in (0,\frac{N}{2})$ for the mass-subcritical nonlinear fractional Hartree equation with unbounded potentials. As a consequence we obtain, as a priori result, the orbital stability of the set of standing waves. The main ingredient is the observation that $\Sigma^s$ is compactly embedded in $L^2$. This enables us to apply the concentration compactness argument in the works of Cazenave-Lions and Zhang, namely, relative compactness for any minimizing sequence in the energy space.