Existence of normalized solutions for Schrödinger systems with linear and nonlinear couplings

Abstract

In this paper we study the nonlinear Bose–Einstein condensates Schrödinger system {−Δu1−λ1u1=μ1u13+βu1u22+κ(x)u2in R3,−Δu2−λ2u2=μ2u23+βu12u2+κ(x)u1in R3,∫R3u12=a12,∫R3u22=a22,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extstyle\\begin{cases} -\\Delta u_{1}-\\lambda _{1} u_{1}=\\mu _{1} u_{1}^{3}+\\beta u_{1}u_{2}^{2}+ \\kappa (x) u_{2}\\quad\ ext{in }\\mathbb{R}^{3}, \\\\ -\\Delta u_{2}-\\lambda _{2} u_{2}=\\mu _{2} u_{2}^{3}+\\beta u_{1}^{2}u_{2}+ \\kappa (x) u_{1}\\quad\ ext{in }\\mathbb{R}^{3}, \\\\ \\int _{\\mathbb{R}^{3}} u_{1}^{2}=a_{1}^{2},\\qquad \\int _{\\mathbb{R}^{3}} u_{2}^{2}=a_{2}^{2}, \\end{cases} $$\\end{document} where a1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$a_{1}$\\end{document}, a2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$a_{2}$\\end{document}, μ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mu _{1}$\\end{document}, μ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mu _{2}$\\end{document}, κ=κ(x)>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\kappa =\\kappa (x)>0$\\end{document}, β<0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\beta <0$\\end{document}, and λ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda _{1}$\\end{document}, λ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda _{2}$\\end{document} are Lagrangian multipliers. We use the Ekeland variational principle and the minimax method on manifold to prove that this system has a solution that is radially symmetric and positive.