Classification of bifurcation diagrams of a p-Laplacian Dirichlet problem with examples

We study bifurcation diagrams of positive solutions of the p-Laplacian Dirichlet problem { ( φ p ( u ′ ( x ) ) ) ′ + f λ ( u ( x ) ) = 0 , − 1 < x < 1 , u ( − 1 ) = u ( 1 ) = 0 , where φ p ( y ) = | y | p − 2 y , ( φ p ( u ′ ) ) ′ is the one-dimensional p-Laplacian, and p > 1 and λ > 0 are two bifurcation parameters. Assume that f λ ( u ) = λ g ( u ) − h ( u ) where g , h ∈ C [ 0 , ∞ ) ∩ C 2 ( 0 , ∞ ) satisfy hypotheses (H1)–(H5) presented herein. For different values p with 1 < p ⩽ 2 and with p > 2 , we give a classification of totally six different bifurcation diagrams. We prove that, on the ( λ , ‖ u ‖ ∞ ) -plane, each possible bifurcation diagram consists of exactly one curve with exactly one turning point where the curve turns to the right. Hence we are able to determine the exact multiplicity of positive solutions. In addition, for 1 < p ⩽ 2 and for p > 2 , we give interesting examples f λ ( u ) = λ ( k u p − 1 + u q ) − u r satisfying r > q > p − 1 and k ⩾ 0 , and show complete evolution of bifurcation diagrams as evolution parameter k varies from 0 to ∞.