Global bifurcation phenomena for singular one-dimensional p-Laplacian

Abstract

In this paper, we present global existence results for the following problemwhere ϕ p (x) = |x| p−2 x, p > 1, λ a positive parameter and h a nonnegative measurable function on (0, 1) which may be singular at t = 0 and/or t = 1, and f ∈ C(R + , R + ) with R + = [0, ∞). By applying the global bifurcation theorem and figuring the shape of unbounded subcontinua of solutions, we obtain many different types of global existence results of positive solutions. We also obtain existence results of signchanging solutions for (P λ ) when f is an odd symmetric function.

One-dimensional p-Laplacian with a strong singular indefinite weight, I. Eigenvalue

“…In the case of h ∈ L ∞ (0, 1) with meas{t ∈ (0, 1), h(t) > 0} = 0, Anane, Chakrone and Moussa [2] obtained a sequence of eigenvalues and proved that its corresponding eigenfunctions have properties (i), (ii) and (iii) given below, employing the Ljusternik-Schnirelmann critical theory. In the case of h ∈ L 1 (0, 1) with h 0, using Sturm type comparison, Zhang [12] showed the existence of a sequence of eigenvalues and their corresponding eigenfunctions for (E λ ) which hold the properties (i) and (iii), and using Picone identity, Lee and Sim [10] proved the property (ii):…”

Positive convex solutions of boundary value problems arising from Monge–Ampère equations

“…Since T p k is completely continuous in E, the Leray-Schauder degree d LS ðI À T p k ; B r ð0Þ; 0Þ is well defined for arbitrary r-ball B r ð0Þ and k -l k ðpÞ; k 2 N. Applying the similar argument as in the proof of [[19], Lemma 2.8], we may get that.…”

C1-regularity of solutions forp-Laplacian problems

“…Let us consider the following one-dimensional singular p-Laplacian problem [1,3,4,9]: ϕ p (u (t)) + f (t, u(t)) = 0, t ∈ (0, 1), u(0) = u(1) = 0, (P) where ϕ p (x) = |x| p−2 x, p > 1. We assume that f ∈ C ((0, 1) × R, R) satisfies (F ) f (t, u)u > 0, for a.e.…”