Classification and evolution of bifurcation curves for a multiparameter pp-Laplacian Dirichlet problem

We study the classification and evolution of bifurcation curves for the multiparameter pp-Laplacian Dirichlet problem {(φp(u′(x)))′+λuq(∑k=1nakurk)−1=0,−10,0=r10for k=1,2,…,n,u(−1)=u(1)=0,where p>1p>1, φp(y)=|y|p−2yφp(y)=|y|p−2y, (φp(u′))′(φp(u′))′ is the one-dimensional pp-Laplacian, and λ>0λ>0 is a bifurcation parameter, and q>0q>0 is an evolution parameter. We give a classification of totally five   qualitatively different bifurcation curves for different q>0q>0. More precisely, we prove that, on the (λ,‖u‖∞)(λ,‖u‖∞)-plane, each bifurcation curve is either a monotone curve if q∈(0,p−1]∪[rn+p−1,∞)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)q∈(p−1,rn+p−1). Hence the problem has at most two positive solutions for each λ>0λ>0. We also show evolution of five bifurcation curves as qq varies from 0+0+ to ∞∞.