Eigenvalues and the One-Dimensional p-Laplacian

Abstract

We consider the boundary value problem ϕ p u + λF t u = 0, with p > 1, t ∈ 0 1 , u 0 = u 1 = 0, and with λ > 0. The value of λ is chosen so that the boundary value problem has a positive solution. In addition, we derive an explicit interval for λ such that, for any λ in this interval, the existence of a positive solution to the boundary value problem is guaranteed. In addition, the existence of two positive solutions for λ in an appropriate interval is also discussed.  2002 Elsevier Science (USA)

EIGENVALUE PROBLEMS FOR p-LAPLACIAN DYNAMIC EQUATIONS ON TIME SCALES

“…But very little work has been done to the existence of positive solutions for p-Laplacian boundary value problem on time scales [3,17,18]. In particular, we would like to mention some results of Anderson, Avery, and Henderson [3], Sun and Li [18], Agarwal and Lü [1], which motivate us to consider our problem.…”

“…In [1], Agarwal and Lü considered the following problem (ϕ p (u (t))) + λF (t, u(t)) = 0, t ∈ (0, 1),…”

“…They obtained series results of the existence and nonexistence of positive solution. It is noted that in the paper [1], the authors assumed that f 0 = lim u→0 + f (u)/ϕ p (u) and f ∞ = lim x→∞ f (u)/ϕ p (u) exist. However, there is little work has been done to the eigenvalue problem of multi-point boundary value problems for p-Laplacian dynamic equation on time scales.…”

Existence and iteration of positive solutions for a singular two-point boundary value problem with a p-Laplacian operator

“…10371006. , and the coefficient q(t) may be singular at t = 0, 1. Several papers have been devoted in the recent years to the study of (1) subject to different linear or nonlinear boundary conditions, see [1], [2], [4], [6], [7] and their references. Here, only positive solutions are meaningful.…”

“…By using the fixed point theorem in cones due to Krasnoselskii [3], Wang [1] and Kong and Wang [2] established the existence of one positive solution for (1) subject to one of the following nonlinear boundary conditions: By applying a new twin fixed point theorem due to Avery and Henderson [5], He and Ge [4] obtained the existence of two positive solutions for (1) subject to (w1), (w2), (w3). By using the fixed point theorem in cones due to Krasnoselskii [3], R. P. Agarwal, Haishen Lü and D. O'Regan [6] studied the problem of eigenvalues of (1) subject to (BCa) when B 0 = B 1 = 0, they also obtained the existence of two positive solutions. By using an extension of the Leggett-Williams theorem, i.e., the fixed point theorem of five functionals, Guo and Ge [7] got the existence of three positive solutions for (1) subject to (w1), (w2), (w3), and…”