Abstract
We study the classification and evolution of bifurcation curves for the multiparameter p-Laplacian Dirichlet problem {(φ <sup>p</sup> (u′(x))) ′ +λu <sup>q</sup> (∑ <sub>k=1</sub> <sup>n</sup> a <sub>k</sub> u <sup>rk</sup> ) <sup>-1</sup> =0,-1<x<1,q>0,0=r <sub>1</sub> <r <sub>2</sub> <⋯<r <sub>n</sub> ,n<2, <sup>ak</sup> >0for k=1,2,⋯,n,u(-1)=u(1)=0,where p>1, φ <sub>p</sub> (y)=|y <sup>|p-2</sup> y, (φ <sub>p</sub> (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|| <sub>∞</sub> )-plane, each bifurcation curve is either a monotone curve if q∈(0,p-1]∪[r <sub>n</sub> +p-1,∞) or has exactly one turning point where the curve turns to the right if q∈(p-1,r <sub>n</sub> +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 <sup>+</sup> to ∞. © 2011 Elsevier Ltd. All rights reserved.