论文标题
带有规定质量的施罗丁方程的常规波浪具有范德华型电势
Standing waves with prescribed mass for the Schrödinger equations with van der Waals type potentials
论文作者
论文摘要
\ begin {Abstract}在本文中，我们专注于具有范德华类型电位的Schrödinger方程的常规质量，即具有不同宽度的两体势。这导致研究以下非局部椭圆方程\ begin {qore*} \ label {1}-ΔU=λu=λu+μ（| x | x |^{ - α} \ ast | u |^{2} {2} {2} {2}） \ r^{n} \ end {equation*}在归一化的约束\ [\ int _ {{\ Mathbb {\ Mathbb {r}^n}}} {{u}^2} = c> 0，\]中$λ\ in \ mathbb {r} $是未知的，并且以Lagrange乘法器为单位。与所研究的情况$α=β$相比，上述问题的解决方案集具有不同的两个体电位$α\neqβ$的宽度的溶液集更丰富。在$ C $，$α$和$β$的不同假设下，我们证明解决方案解决上述问题的几种存在，多重性和渐近行为。另外，讨论了相关时间依赖性问题的相应驻波的稳定性。
\begin{abstract} In this paper, we focus on the standing waves with prescribed mass for the Schrödinger equations with van der Waals type potentials, that is, two-body potentials with different width. This leads to the study of the following nonlocal elliptic equation \begin{equation*}\label{1} -Δu=λu+μ(|x|^{-α}\ast|u|^{2})u+(|x|^{-β}\ast|u|^{2})u,\ \ x\in \R^{N} \end{equation*} under the normalized constraint \[\int_{{\mathbb{R}^N}} {{u}^2}=c>0,\] where $N\geq 3$, $μ\!>\!0$, $α$, $β\in (0,N)$, and the frequency $λ\in \mathbb{R}$ is unknown and appears as Lagrange multiplier. Compared with the well studied case $α=β$, the solution set of the above problem with different width of two body potentials $α\neqβ$ is much richer. Under different assumptions on $c$, $α$ and $β$, we prove several existence, multiplicity and asymptotic behavior of solutions to the above problem. In addition, the stability of the corresponding standing waves for the related time-dependent problem is discussed.
