Ground state normalized solutions to the Kirchhoff equation with general nonlinearities: mass supercritical case

Abstract

We study the following nonlinear mass supercritical Kirchhoff equation: −(a+b∫RN|∇u|2)△u+μu=f(u)in RN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ - \\biggl(a+b \\int _{\\mathbb{R}^{N}} \\vert \ abla u \\vert ^{2} \\biggr) \ riangle u+ \\mu u=f(u) \\quad \ ext{in } {\\mathbb{R}^{N}}, $$\\end{document} where a,b,m>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$a ,b,m>0$\\end{document} are prescribed, and the normalized constrain ∫RN|u|2dx=m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\int _{\\mathbb{R}^{N}}|u|^{2}\\,dx =m$\\end{document} is satisfied in the case 1≤N≤3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$1\\leq N\\leq 3$\\end{document}. The nonlinearity f is more general and satisfies weak mass supercritical conditions. Under some mild assumptions, we establish the existence of ground state when 1≤N≤3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$1\\leq N\\leq 3$\\end{document} and obtain infinitely many radial solutions when 2≤N≤3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$2\\leq N\\leq 3$\\end{document} by constructing a particular bounded Palais–Smale sequence.