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  • Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case
    Zhen, Maoding ; Zhang, Binlin ; Rǎdulescu, Vicenţiu, 1958-
    In this paper, we study the following coupled nonlocal system ▫$$\begin{cases} (-\Delta)^{s}u-\lambda_{1}u = \mu_{1}|u|^{\alpha}u+\beta|u|^{\frac{\alpha-2}{2}}u|v|^{\frac{\alpha+2}{2}} & \text{in} \ ... \ \mathbb{R}^{N},\\ (-\Delta)^{s}v-\lambda_{2}v = \mu_{2}|v|^{\alpha}v+\beta|u|^{\frac{\alpha+2}{2}}|v|^{\frac{\alpha-2}{2}}v& \text{in} \ \ \mathbb{R}^{N}, \end{cases}$$▫ satisfying the additional conditions ▫$$\int_{\mathbb{R}^{N}}u^{2}dx = b^{2}_{1}\quad \text{and} \;\; \int_{\mathbb{R}^{N}}v^{2}dx = b^{2}_{2},$$▫ 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 [T. Bartsch, L. Jeanjean and N. Soave, Normalized solutions for a system of coupled cubic Schrödinger equations on ▫$\mathbb{R}^{3}$▫, J. Math. Pures Appl., 106 (2016), 583-614] for the particular case ▫$N = 3$▫, ▫$s = 1$▫, ▫$\alpha = 2$▫.
    Vir: Discrete and continuous dynamical systems. - ISSN 1078-0947 (Vol. 41, no. 6, June 2021, str. 2653-2676)
    Vrsta gradiva - članek, sestavni del
    Leto - 2021
    Jezik - angleški
    COBISS.SI-ID - 66189571

    Povezava(-e):

    https://www.aimsciences.org/article/doi/10.3934/dcds.2020379
    DOI

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