Abstract
We study the global bifurcation and exact multiplicity of positive solutions of $\\\\\\\\\\\\\\\\displaystyle \\\\\\\\\\\\\\\\left \\\\\\\\\\\\\\\\{ \\\\\\\\\\\\\\\\begin {array}{l} u^{\\\\\\\\\\\\\\\\prime \\\\\\\\\\\\\\\\prime }(x)+\\\\\\\\\\\\\\\\lambda f_{\\\\\\\\\\\\\\\\vare... ...n }(u)=-\\\\\\\\\\\\\\\\varepsilon u^{3}+\\\\\\\\\\\\\\\\sigma u^{2}+\\\\\\\\\\\\\\\\tau u+\\\\\\\\\\\\\\\\rho \\\\\\\\\\\\\\\\text {,}\\\\\\\\\\\\\\\\end{array}\\\\\\\\\\\\\\\\right .$ where $ \\\\\\\\\\\\\\\\lambda ,\\\\\\\\\\\\\\\\varepsilon >0$ are two bifurcation parameters, and $ \\\\\\\\\\\\\\\\sigma ,\\\\\\\\\\\\\\\\rho >0,\\\\\\\\\\\\\\\\tau \\\\\\\\\\\\\\\\geq 0$ are constants. By developing some new time-map techniques, we prove the global bifurcation of bifurcation curves for varying $ \\\\\\\\\\\\\\\\varepsilon >0$. More precisely, we prove that, for any $ \\\\\\\\\\\\\\\\sigma ,\\\\\\\\\\\\\\\\rho >0,\\\\\\\\\\\\\\\\tau \\\\\\\\\\\\\\\\geq 0$, there exists $ \\\\\\\\\\\\\\\\varepsilon ^{\\\\\\\\\\\\\\\\ast }>0$such that, on the $ (\\\\\\\\\\\\\\\\lambda ,\\\\\\\\\\\\\\\\vert\\\\\\\\\\\\\\\\vert u\\\\\\\\\\\\\\\\vert\\\\\\\\\\\\\\\\vert _{\\\\\\\\\\\\\\\\infty })$-plane, the bifurcation curve is S-shaped for $ 0<\\\\\\\\\\\\\\\\varepsilon <\\\\\\\\\\\\\\\\varepsilon ^{\\\\\\\\\\\\\\\\ast }$ and is monotone increasing for $ \\\\\\\\\\\\\\\\varepsilon \\\\\\\\\\\\\\\\geq \\\\\\\\\\\\\\\\varepsilon ^{\\\\\\\\\\\\\\\\ast }$. (We also prove the global bifurcation of bifurcation curves for varying $ \\\\\\\\\\\\\\\\lambda >0$.) Thus we are able to determine the exact number of positive solutions by the values of $ \\\\\\\\\\\\\\\\varepsilon $ and $ \\\\\\\\\\\\\\\\lambda $. We give an application to prove a long-standing conjecture for global bifurcation of positive solutions for the problem $\\\\\\\\\\\\\\\\displaystyle \\\\\\\\\\\\\\\\left \\\\\\\\\\\\\\\\{ \\\\\\\\\\\\\\\\begin {array}{l} u^{\\\\\\\\\\\\\\\\prime \\\\\\\\\\\\\\\\prime }(x)+\\\\\\\\\\\\\\\\lambda (-\\\\\\\\\\\\\\\\varep... ...+1)=0\\\\\\\\\\\\\\\\text {, \\\\\\\\\\\\\\\\ } -1<x<1\\\\\\\\\\\\\\\\text {,} \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ u(-1)=u(1)=0\\\\\\\\\\\\\\\\text {,} \\\\\\\\\\\\\\\\end{array} \\\\\\\\\\\\\\\\right .$ which was studied by Crandall and Rabinowitz (Arch. Rational Mech. Anal. 52 (1973), p. 177). In addition, we give an application to prove a conjecture of Smoller and Wasserman (J. Differential Equations 39 (1981), p. 283, lines 2-3) on the maximum number of positive solutions of a positone problem.