Normalized solutions for Schrödinger system with subcritical Sobolev exponent and combined nonlinearities

Abstract

In this paper, we look for solutions to the following coupled Schrödinger system \begin{equation*} \begin{cases} -Δu+λ_{1}u=α_{1}|u|^{p-2}u+μ_{1}u^{3}+ρv^{2}u & \text{in} \ \ \mathbb{R}^{N}, -Δv+λ_{2}v=α_{2}|v|^{p-2}v+μ_{2}v^{3}+ρu^{2}v& \text{in} \ \ \mathbb{R}^{N}, \end{cases} \end{equation*} with the additional conditions ${\int }_{{\mathbb{R}}^N}{u}^{2}dx=b_1^2$ and ${\int }_{{\mathbb{R}}^N}{v}^{2}dx=b_2^2.$ Here $b_1,b_2>0$ are prescribed, $N\le 3$, ${\mu }_1,{\mu }_2,{\alpha }_1,{\alpha }_2,\rho >0$, $p\in (2,4)$ 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 ${\alpha }_1,{\alpha }_2\to 0$. 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 $N=4$.

substantial text overlap with our old paper arXiv:2108.09461