文摘
We study exact multiplicity and bifurcation curves of positive solutions for a multiparameter Dirichlet problem{u″(x)+λ[a−bu+up1+up]=0,−1<x<1,u(−1)=u(1)=0, where p>1p>1, a,ba,b are positive dimensionless parameters, and λ>0λ>0 is a bifurcation parameter. For fixed p>1p>1, assume that either 0<a≤a1,p0<a≤a1,p, b>0b>0 and (a,b)(a,b) lies below the curveΓ1={(a,b):a(t)=tp[p−1−tp](tp+1)2,b(t)=ptp−1(tp+1)2,0<t<p−1p+1p},or 0<a≤a2,p0<a≤a2,p and (a,b)(a,b) lies on or above the curve Γ1Γ1 for some positive constants a1,pa1,p and a2,pa2,p. Then on the (λ,‖u‖∞)(λ,‖u‖∞)-plane, we give a classification of three qualitatively different bifurcation curves: an S-shaped curve, a broken S-shaped curve, and a monotone increasing curve. Hence we are able to determine the exact multiplicity of positive solutions by the values of a, b and λ.