Bifurcation curve for the Minkowski-curvature equation with concave or geometrically concave nonlinearity

Abstract

We study the bifurcation curve and exact multiplicity of positive solutions in the space C2((−L,L))∩C([−L,L])\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$C^{2}\\left ( (-L,L)\\right ) \\cap C\\left ( [-L,L]\\right ) $\\end{document} for the Minkowski-curvature equation {−(u′(x)1−(u′(x))2)′=λf(u),−L<x<L,u(−L)=u(L)=0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left \\{ \ extstyle\\begin{array}{l} -\\left ( \\dfrac{u^{\\prime }(x)}{\\sqrt{1-\\left ( {u^{\\prime }(x)}\\right ) ^{2}}}\\right ) ^{\\prime }=\\lambda f(u),\ ext{\\ \\ }-L< x< L, \\\\ u(-L)=u(L)=0.\\end{array}\\displaystyle \\right . $$\\end{document}where λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda >0$\\end{document} is a bifurcation parameter, f∈C[0,∞)∩C2(0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f\\in C[0,\\infty )\\cap C^{2}(0,\\infty )$\\end{document} satisfies f(u)>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f(u)>0$\\end{document} for u>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$u>0$\\end{document} and f is either concave or geometrically concave on (0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(0,\\infty )$\\end{document}. If f is a concave function, we prove that the bifurcation curve is monotone increasing on the (λ,∥u∥∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\lambda ,\\left \\Vert u\\right \\Vert _{\\infty })$\\end{document}-plane. If f is a geometrically concave function, we prove that the bifurcation curve is either ⊂-shaped or monotone increasing on the (λ,∥u∥∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\lambda ,\\left \\Vert u\\right \\Vert _{\\infty })$\\end{document}-plane under a mild condition. Some interesting applications are given.