Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case

In this paper, we study the following coupled nonlocal system

$ \begin{equation*} \begin{cases} (-\Delta)^{s}u-\lambda_{1}u = \mu_{1}|u|^{\alpha}u+\beta|u|^{\frac{\alpha-2}{2}}u|v|^{\frac{\alpha+2}{2}} & \text{in} \ \ \mathbb{R}^{N},\\ (-\Delta)^{s}v-\lambda_{2}v = \mu_{2}|v|^{\alpha}v+\beta|u|^{\frac{\alpha+2}{2}}|v|^{\frac{\alpha-2}{2}}v& \text{in} \ \ \mathbb{R}^{N}, \end{cases} \end{equation*} $

satisfying the additional conditions

$ \int_{\mathbb{R}^{N}}u^{2}dx = b^{2}_{1}\ \text{and} \ \int_{\mathbb{R}^{N}}v^{2}dx = b^{2}_{2}, $

where $ (-\Delta)^{s} $ is the fractional Laplacian, $ 0<s<1 $, $ \mu_{1},\, \mu_{2}>0 $, $ N>2s $, and $ \frac{4s}{N}<\alpha\leq \frac{2s}{N-2s} $. We are concerned with the attractive case, which corresponds to $ \beta>0 $. In the case of low perturbations of the coupling parameter, by using two-dimensional linking arguments, we show that there exists $ \beta_{1}>0 $ such that when $ 0<\beta<\beta_{1} $, then the system has a positive radial solution. Next, in the case of high perturbations of the coupling parameter, we prove that there exists $ \beta_{2}>0 $ such that the system has a mountain-pass type solution for all $ \beta>\beta_{2} $. These results correspond to low and high perturbations with respect to the values of the coupling parameter $ \beta $. This paper extends and complements the main results established in [2] for the particular case $ N = 3 $, $ s = 1 $, $ \alpha = 2 $.