Normalized solutions for Schr\"{o}dinger system with subcritical Sobolev exponent and combined nonlinearities

TL;DR

This paper investigates the existence and behavior of normalized solutions to a coupled Schrödinger system with subcritical nonlinearities across different spatial dimensions, providing new results on ground states, excited states, and limit behaviors.

Contribution

It extends the analysis of normalized solutions for Schrödinger systems to dimensions N ≤ 3, including existence, multiplicity, and asymptotic properties under various conditions.

Findings

Existence of normalized ground states for N=1 and N=2.

Existence of at least two solutions for N=3, including ground and excited states.

Analysis of the limit behavior of solutions as nonlinear parameters tend to zero.

Abstract

In this paper, we look for solutions to the following coupled Schr\"{o}dinger system \begin{equation*} \begin{cases} -\Delta u+\lambda_{1}u=\alpha_{1}|u|^{p-2}u+\mu_{1}u^{3}+\rho v^{2}u & \text{in} \ \ \mathbb{R}^{N}, -\Delta v+\lambda_{2}v=\alpha_{2}|v|^{p-2}v+\mu_{2}v^{3}+\rho u^{2}v& \text{in} \ \ \mathbb{R}^{N}, \end{cases} \end{equation*} with the additional conditions $\int_{\mathbb{R}^{N}}u^{2}dx=b^{2}_{1}$ and $\int_{\mathbb{R}^{N}}v^{2}dx=b^{2}_{2}.$ Here $b_1, b_2>0$ are prescribed, $N\leq3$, $\mu_{1}, \mu_{2}, \alpha_{1},\alpha_{2},\rho>0$, $p\in (2,4)$ and the frequencies $\lambda_{1},\lambda_{2}$ are unknown and will appear as Lagrange multipliers. In the one dimension case, the energy functional is bounded from below on the product of $L^2$-spheres, normalized ground states exist and are obtained as global minimizers. When $N=2$, the energy functional is not always bounded on the product of $L^2$-spheres, we prove the existence of normalized ground states under suitable conditions on $b_1$ and $b_2$, which are obtained as global minimizers. When $N=3$, we show that under suitable conditions on $b_1$ and $b_2$, at least two normalized solutions exist, one is a ground state and the other is an excited state. We also shows the limit behavior of the normalized solutions as $\alpha_{1},\alpha_{2}\rightarrow 0$. The first solution will disappear and the second solution will converge to the normalized solution of system (1.1) with $\alpha_{1}=\alpha_{2}=0$, which has been studied by T. Bartsch, L. Jeanjean and N. Soave (J. Math. Pures Appl. 2016). Furthermore, by refining the upper bound of the ground state energy, we provide a precise mass collapse behavior of the ground states. The results in this paper complement the main results established by X. Luo, X. Yang and W. Zou (arXiv:2107.08708), where the authors considered the case $N=4$.