Standing waves with prescribed mass for the Schr\"{o}dinger equations with van der Waals type potentials

TL;DR

This paper investigates the existence, multiplicity, and stability of standing wave solutions with prescribed mass for a Schrödinger equation involving nonlocal van der Waals type potentials with different widths, revealing a richer solution structure than the equal-width case.

Contribution

It extends the analysis of Schrödinger equations with nonlocal potentials to the case of different potential widths, providing new existence, multiplicity, and stability results.

Findings

Proved existence of solutions under various conditions on parameters.

Established multiplicity of solutions depending on the prescribed mass.

Analyzed the asymptotic behavior and stability of standing waves.

Abstract

\begin{abstract} In this paper, we focus on the standing waves with prescribed mass for the Schr\"{o}dinger 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} -\Delta u=\lambda u+\mu (|x|^{-\alpha}\ast|u|^{2})u+(|x|^{-\beta}\ast|u|^{2})u,\ \ x\in \R^{N} \end{equation*} under the normalized constraint \[\int_{{\mathbb{R}^N}} {{u}^2}=c>0,\] where $N\geq 3$, $\mu\!>\!0$, $\alpha$, $\beta\in (0,N)$, 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 $\alpha\neq\beta$ 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.