Abstract
We study the bifurcation diagrams of positive solutions of the multiparameter p-Laplacian problem{((φ <sub>p</sub> (u <sup>′</sup> (x))) <sup>′</sup> + f <sub>λ, μ</sub> (u (x)) = 0, - 1 < x < 1,; u (- 1) = u (1) = 0,) where p > 1, φ <sub>p</sub> (y) = | y | <sup>p - 2</sup> y, (φ <sub>p</sub> (u <sup>′</sup> )) <sup>′</sup> is the one-dimensional p-Laplacian, f <sub>λ, μ</sub> (u) = g (u, λ) + h (u, μ), and λ > λ <sub>0</sub> and μ > μ <sub>0</sub> are two bifurcation parameters, λ <sub>0</sub> and μ <sub>0</sub> are two given real numbers. Assuming that functions g and h satisfy hypotheses (H1)-(H3) and (H4a) (resp. (H1)-(H3) and (H4b)), for fixed μ > μ <sub>0</sub> (resp. λ > λ <sub>0</sub> ), we give a classification of totally eight qualitatively different bifurcation diagrams. We prove that, on the (λ, {norm of matrix} u {norm of matrix} <sub>∞</sub> )-plane (resp. (μ, {norm of matrix} u {norm of matrix} <sub>∞</sub> )-plane), each bifurcation diagram consists of exactly one curve which is either a monotone curve or has exactly one turning point where the curve turns to the right. Hence the problem has at most two positive solutions for each λ > λ <sub>0</sub> (resp. μ > μ <sub>0</sub> ). More precisely, we prove the exact multiplicity of positive solutions. In addition, for all p > 1, we give interesting examples which show complete evolution of bifurcation diagrams as μ (resp. λ) varies. © 2008 Elsevier Inc. All rights reserved.