A complete classification of bifurcation diagrams of classes of multiparameter p-Laplacian boundary value problems

Abstract

We study the bifurcation diagrams of positive solutions of the multiparameter p-Laplacian problemwhere p > 1, ϕ p (y) = |y| p−2 y, (ϕ p (u )) is the one-dimensional p-Laplacian, f λ,μ (u) = g(u, λ) + h(u, μ), and λ > λ 0 and μ > μ 0 are two bifurcation parameters, λ 0 and μ 0 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 0 (resp. λ > λ 0 ), we give a classification of totally eight qualitatively different bifurcation diagrams. We prove that, on the (λ, u ∞ )-plane (resp. (μ, u ∞ )-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 0 (resp. μ > μ 0 ). 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.

Bifurcation diagrams of a p-Laplacian Dirichlet problem with Allee effect and an application to a diffusive logistic equation with predation

“…16) In addition, by Lemma 4.4(ii), we obtain thatλ(k) = C p /(k − 1) andλ(k) = (T k,m (D(k, m))) p are both continuous, strictly decreasing functions of k on (1, ∞), …”

Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros

“…Problems with linear/superlinear nonlinearities at infinity have been extensively studied. For the Laplacian, see for example [3,5,10,12,21,25,26,28,29,30,31,32,35,37,44,45,55]. For the p-Laplacian, see for example [6,14,46].…”

A complete classification of bifurcation diagrams of classes of a multiparameter Dirichlet problem with concave-convex nonlinearities

“…This research is motivated by papers by Ambrosetti et al [1], Caldwell et al [3], Caldwell et al [2], Wang and Yeh [8], and Hung and Wang [4]. Ambrosetti et al [1] studied the combined effects of concave and convex nonlinearities on the exact structure of solutions for the elliptic Dirichlet problem Δu + λu q + μu r = 0 in Ω, (1.2) where Ω is a general bounded domain in R N (N 1) with smooth boundary ∂Ω, λ, μ > 0, and 0 < q < 1 < r. For μ > 0, they obtained the existence of two positive solutions of (1.2) for small λ > 0 by using sub-and supersolutions and variational arguments.…”

“…Very recently, Hung and Wang [4] studied positive solutions of the p-Laplacian multiparameter Dirichlet problem ϕ p u (x) + g(u, λ) + h(u, μ) = 0, −1 < x < 1,…”