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 $$ \left\{% \begin{array}{ll} (-Δ)^s u +u=μ_1 |u|^{2p-2}u+β|v|^p|u|^{p-2}u,~~x\in \R^N,\vspace{2mm}\\ (-Δ)^s v +v=μ_2 |v|^{2p-2}v+β|u|^p|v|^{p-2}v,~~x\in \R^N, \end{array}% \right. $$ where $0<s<1,{\mu }_1>0,{\mu }_2>0,1<p<2_s^{\ast }/2,2_s^{\ast }=+\infty$ for $N\le 2s$ and $2_s^{\ast }=2N/(N-2s)$ 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.