Abstract
We study the exact multiplicity and bifurcation diagrams of positive solutions u∈C <sup>2</sup> (-L, L)∩C[-L, L] of the one-dimensional multiparameter prescribed mean curvature problem{-(u'(x)1+(u'(x))2)'=λ(up+uq),-L<x<L,u(-L)=u(L)=0, where λ>0 is a bifurcation parameter, L>0, the radius of the one-dimensional ball (-L, L), is an evolution parameter, and 0≤p<q<∞ are two constants. We prove that the problem has at most two positive solutions for any 0≤p<q<∞ and λ, L>0. In addition, if 0≤p<q≤q~(p)=p+1+2p+1, we give a classification of totally three qualitatively different bifurcation diagrams on the (λ, {norm of matrix}u{norm of matrix} <sub>∞</sub> )-plane for any L>0. For any fixed p≥0 and q≥q-(p)=p+2+22p+3, we prove that there exist positive L <sub>*</sub> <L <sup>*</sup> such that the bifurcation diagrams on the (λ, {norm of matrix}u{norm of matrix} <sub>∞</sub> )-plane are individually qualitatively different for the cases (i) 0<L<L <sub>*</sub> or L>L <sup>*</sup> , (ii) L=L <sup>*</sup> , (iii) L <sub>*</sub> ≤L<L <sup>*</sup> .