A complete classification of bifurcation diagrams of a p-Laplacian Dirichlet problem

We study the bifurcation diagrams of classical positive solutions u   with ‖u‖∞∈(0,∞)‖u‖∞∈(0,∞) of the p-Laplacian Dirichlet problem{(φp(u′(x)))′+λfq(u(x))=0,−11p>1, φp(y)=|y|p−2yφp(y)=|y|p−2y, (φp(u′))′(φp(u′))′ is the one-dimensional p  -Laplacian, λ>0λ>0 is a bifurcation parameter, and fq(u)=q|1−u|fq(u)=|1−u|q is defined on [0,∞)[0,∞) with q>0q>0. More precisely, for different (p,q)(p,q), we give a complete classification of bifurcation diagrams of classical positive solutions on the (λ,‖u‖∞)(λ,‖u‖∞)-plane. Hence we are able to determine the exact multiplicity of classical positive solutions for each (p,q,λ)(p,q,λ).