Existence of positive ground state solutions to a nonlinear fractional Schrödinger system with linear couplings

Abstract

Abstract In this paper, we investigate a nonlinear fractional Schrödinger system with linear couplings as follows: {(−Δ)αu+(1+a(x))u=Fu(u,v)+λv,in R3,(−Δ)αv+(1+b(x))v=Fv(u,v)+λu,in R3,u,v∈Hα(R3), $$ \textstyle\begin{cases} (-\Delta )^{\alpha }u+(1+a(x))u=F_{u}(u,v)+\lambda v,& \text{in } \mathbb{R}^{3}, \\ (-\Delta )^{\alpha }v+(1+b(x))v=F_{v}(u,v)+\lambda u,& \text{in } \mathbb{R}^{3}, \\ u,v\in H^{\alpha }(\mathbb{R}^{3}), \end{cases} $$ where (−Δ)α,α∈(0,1) $(-\Delta )^{\alpha }, \alpha \in (0,1)$, denotes the fractional Laplacian and λ>0 $\lambda >0$ is the coupling parameter. Under some assumptions, we prove the existence of positive ground state solutions to the above system with the help of the method of Nehari manifold and concentration compactness lemma.

Keywords

• Ground state solution
• Fractional Schrödinger system
• Variational methods
• Nehari manifold