Standing waves with prescribed mass for the Schrödinger equations with van der Waals type potentials

Abstract

\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 $N\ge 3$, $\mu >0$, $\alpha$, $\beta \in (0,N)$, and the frequency $\lambda \in \mathbb{R}$ is unknown and appears as Lagrange multiplier. Compared with the well studied case $\alpha =\beta$, the solution set of the above problem with different width of two body potentials $\alpha \ne \beta$ is much richer. Under different assumptions on $c$, $\alpha$ and $\beta$, 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.

Part 7 in the present paper need to be modified

References in corpus (1)

• Lifshitz theory of van der Waals pressure in dissipative media