classification and evolution of bifurcation curves for a multiparameter p-laplacian dirichlet problem

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] ∪ [r_n + p − 1,∞) or has exactly one turning point where the curve turns to the right if q ∈ (p−1, r_n+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 ∞.