Laplacian Dirichlet problem
ϕp(u′(x))+ λuq(∑k=1nakun)-1=0, -1<x<1,= 0, −1 < x < 1,
q > 0, 0 = r1 < r2 < · · · < rn, n ≥ 2, ak > 0 for k = 1, 2, . . . , n,
u(−1) = u(1) = 0,
where p > 1, ϕp(y) = |y|p−2y, (ϕp(u′))′ is the one-dimensional p-Laplacian, and λ > 0 is a bifurcation parameter, and q > 0 is an evolution parameter. We give a classification of totally five qualitatively different bifurcation curves for different q > 0. More precisely, we prove that, on the (λ, ‖u‖∞)-plane, each bifurcation curve is either a monotone curve if q ∈ (0, p − 1] ∪ [rn + p − 1,∞) or has exactly one turning point where the curve turns to the right if q ∈ (p−1, rn+p−1). Hence the problem has at most two positive solutions for each λ > 0. We also show evolution of five bifurcation curves as q varies from 0+ to ∞.