Existence of ground state solutions to a class of fractional Schr\xf6dinger system with linear and nonlinear couplings

Abstract

In this paper, we study the existence of ground state solutions to the following fractional Schrödinger system with linear and nonlinear couplings: \t\t\t{(−△)su+(λ1+V(x))u+kv=μ1u3+βuv2,in R3,(−△)sv+(λ2+V(x))v+ku=μ2v3+βu2v,in R3,u,v∈Hs(R3),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} (-\\triangle )^{s}u+(\\lambda _{1}+V(x))u+kv=\\mu _{1}u^{3}+\\beta uv^{2}, \\quad \\text{in } R^{3},\\\\ (-\\triangle )^{s}v+(\\lambda _{2}+V(x))v+ku=\\mu _{2}v^{3}+ \\beta u^{2}v, \\quad \\text{in } R^{3},\\\\ u, v\\in H^{s}(R^{3}), \\end{cases} $$\\end{document} where (-triangle )^{s} denotes the fractional Laplacian of order sin (frac{3}{4},1). Under some assumptions of the potential 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.