Existence, non-degeneracy of proportional positive solutions and least energy solutions for a fractional elliptic system

Abstract

In this paper, we study the following fractional nonlinear Schrödinger system $$ \left\{ \begin{array}{ll} (-\Delta)^s u +u=\mu_1 |u|^{2p-2}u+\beta |v|^p|u|^{p-2}u, ~~x\in \mathbb R^N, \\ (-\Delta)^s v +v=\mu_2 |v|^{2p-2}v+\beta |u|^p|v|^{p-2}v, ~~x\in \mathbb R^N, \end{array} \right. $$ where $0 < s < 1, \mu_1 > 0, \mu_2 > 0, 1 < p < 2_s^*/2, 2_s^*=+\infty$ for $N\le 2s$ and $2_s^*=2N/(N-2s)$ for $N > 2s$, and $\beta \in \mathbb R$ is a coupling constant. We investigate the existence and non-degeneracy of proportional positive vector solutions for the above system in some ranges of $\mu_1,\mu_2, p, \beta$. We also prove that the least energy vector solutions must be proportional and unique under some additional assumptions.