Exact multiplicity and bifurcation diagrams of positive solutions of a one-dimensional multiparameter prescribed mean curvature problem

We study the exact multiplicity and bifurcation diagrams of positive solutions u∈C2(−L,L)∩C[−L,L]u∈C2(−L,L)∩C[−L,L] of the one-dimensional multiparameter prescribed mean curvature problem{−(u′(x)1+(u′(x))2)′=λ(up+uq),−L0λ>0 is a bifurcation parameter, L>0L>0, the radius of the one-dimensional ball (−L,L)(−L,L), is an evolution parameter, and 0≤p0λ,L>0. In addition, if 0≤p0L>0. For any fixed p≥0p≥0 and q≥q¯(p)=p+2+22p+3, we prove that there exist positive L⁎L⁎L>L⁎, (ii) L=L⁎L=L⁎, (iii) L⁎≤L