Nehari Manifold for Fractional Kirchhoff Systems with Critical Nonlinearity

摘要

In this paper, we show the existence and multiplicity of positive solutions of the fractional Kirchhoff systemLM(u)=lambda f(x)|u|q-2u+2 alpha alpha+beta|u|alpha-2u|v|beta in omega,LM(v)=mu g(x)|v|q-2v+2 beta alpha+beta|u|alpha|v|beta-2vin omega,u=v=0on partial differential omega,documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$egin{aligned} {left{ egin{array}{ll}mathcal {L}_{M}(u) = {lambda }f(x)|u|<^>{q-2}{u} + rac{2{lpha }}{{lpha }+{eta }} |u|<^>{lpha -2} ,u|v|<^>{eta} &{}quad mathrm{in}, Omega , mathcal {L}_M(v) = {mu }g(x)|v|<^>{q-2}v + rac{2{eta }}{{lpha }+{eta }}|u|<^>{lpha} ,|v|<^>{eta -2}v &{}quad mathrm{in}, Omega , quad ;;; u = v = 0 &{}quad mathrm{on}, {partial }{Omega }, end{array}ight. } end{aligned}$$end{document} where LM(u)=M(integral omega|(-Delta)s2u|2dx)(-Delta)sudocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$mathcal{L}_{M}(u) = M ig(int_{Omega} |(-{Delta})<^>rac{s}{2}u|<^>{2}dxig) (-Delta)<^>{s}u$$end{document} is a double non-local operator due to a Kirchhoff term M(t)=a+btdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$M(t) = a + bt$$end{document} with a, b > 0 and the fractional Laplacian (-Delta)s,s is an element of(0,1)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$(-Delta)<^>{s}, s in (0,1)$$end{document}.