Existence of ground state solutions to a class of fractional Schrödinger system with linear and nonlinear couplings - Xinsheng Du (Boundary Value Problems, 2019)
Abstract In this paper, we study the existence of ground state solutions to the following fractional Schrödinger system with linear and nonlinear couplings: {(−△)su+(λ1+V(x))u+kv=μ1u3+βuv2,in R3,(−△)sv+(λ2+V(x))v+ku=μ2v3+βu2v,in R3,u,v∈Hs(R3), $$ \textstyle\begin{cases} (-\triangle )^{s}u+(\lambda _{1}+V(x))u+kv=\mu _{1}u^{3}+\beta uv^{2}, \quad \text{in } R^{3},\\ (-\triangle )^{s}v+(\lambda _{2}+V(x))v+ku=\mu _{2}v^{3}+ \beta u^{2}v, \quad \text{in } R^{3},\\ u, v\in H^{s}(R^{3}), \end{cases} $$ where (−△)s $(-\triangle )^{s}$ denotes the fractional Laplacian of order s∈(34,1) $s\in (\frac{3}{4},1)$. Under some assumptions of the potential V(x) $V(x)$ and the linear and nonlinear coupling constants k, β, we prove some results for the existence of ground state solutions for the fractional Laplacian systems by using variational methods.