Sign-changing solutions for coupled Schrödinger system

Abstract

In this paper we study the following nonlinear Schrödinger system: {−Δu+αu=|u|p−1u+2q+1λ|u|p−32u|v|q+12,x∈R3,−Δv+βv=|v|q−1v+2p+1λ|u|p+12|v|q−32v,x∈R3,u(x)→0,v(x)→0,as |x|→∞,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extstyle\\begin{cases} -\\Delta u+\\alpha u = \\vert u \\vert ^{p-1}u+\\frac{2}{q+1} \\lambda \\vert u \\vert ^{ \\frac{p-3}{2}}u \\vert v \\vert ^{\\frac{q+1}{2}},\\quad x \\in \\mathbb{R}^{3}, \\\\ -\\Delta v+\\beta v = \\vert v \\vert ^{q-1}v+\\frac{2}{p+1} \\lambda \\vert u \\vert ^{ \\frac{p+1}{2}} \\vert v \\vert ^{\\frac{q-3}{2}}v ,\\quad x \\in \\mathbb{R}^{3}, \\\\ u(x)\\rightarrow 0,\\qquad v(x)\\rightarrow 0,\\quad \ ext{as } \\vert x \\vert \\rightarrow \\infty , \\end{cases} $$\\end{document} where 3≤p,q<5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$3\\leq p, q<5$\\end{document}, α, β are positive parameters. We show that there exists λk>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda _{k}>0$\\end{document} such that the equation has at least k radially symmetric sign-changing solutions and at least k seminodal solutions for each k∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$k\\in \\mathbb{N}$\\end{document} and λ∈(0,λk)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda \\in (0, \\lambda _{k})$\\end{document}. Moreover, we show the existence of a least energy radially symmetric sign-changing solution for each λ∈(0,λ0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda \\in (0, \\lambda _{0})$\\end{document} where λ0∈(0,λ1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda _{0}\\in (0, \\lambda _{1}]$\\end{document}.