Eigenvalues of the -Laplacian and disconjugacy criteria
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We derive oscillation and nonoscillation criteria for the one-dimensional p-Laplacian in terms of an eigenvalue inequality for a mixed problem. We generalize the results obtained in the linear case by Nehari and Willett, and the proof is based on a Picone-type identity. Copyright © 2006 P. L. De Napoli and J. P. Pinasco. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction In this work we study the following equation: p−2 |u | u + q(t)|u| p−2 u = 0.
(1.1)
Here, 1 < p < ∞, t ∈ [a,+∞), and q(t) is a nonnegative continuous function not vanishing in subintervals of the form (b,+∞). The solutions of (1.1) are classified as oscillatory or nonoscillatory. In the first case, a solution has an infinite number of isolated zeros; in the second case, a solution has a finite number of zeros. However, from the Sturm-Liouville theory for the p-laplacian ([11, 16, 22]; see also the recent monograph [10]) if one solution is oscillatory (resp., nonoscillatory), then every solution is oscillatory (resp., nonoscillatory). Hence, we may speak about oscillatory or nonoscillatory equations, instead of solutions. For the p-laplacian operator, there are several criteria for oscillation and nonoscillation in the literature; see for example [6–9]. Among the class of nonoscillatory equations, when any solution has at most one zero in [a,+∞), the equation is called disconjugate on [a,+∞). The disconjugacy phenomenon is considerably more difficult and less understood than nonoscillation; we refer the interested reader to the surveys [3, 5, 23] for the linear case p = 2. Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 37191, Pages 1–8 DOI 10.1155/JIA/2006/37191
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Eigenvalues and disconjugacy
We consider first the disconjugacy problem on [a,+∞). The relationship between disconjugacy and the eigenvalues of a mixed problem −u = λq(t)u,
u(a) = 0 = u (b)
(1.2)
is due to Nehari [17], and was generalized to different equations in [14, 20, 24]. We prove here the following theorems generalizing some of their results for the p-laplacian. Theorem 1.1. Let λ1 be the first eigenvalue of p−2 |u | u + λq(t)|u| p−2 u = 0,
u(a) = 0 = u (b),
a < b,
(1.3)
then (1.1) is disconjugate in [a,+∞) if and only if λ1 > 1 for all b > a. Also, we have the following result for oscillatory equations. Theorem 1.2. Equation (1.1) is oscillatory if and only if there exists a sequence of intervals [an ,bn ] with an < bn , an +∞ as n +∞ such that the first eigenvalue λ(n) 1 of p−2 |u | u + λq(t)|u| p−2 u = 0,
u an = 0 = u bn
(1.4)
satisfies λ(n) 1 ≤ 1 for n ≥ 1. For the linear case p = 2 and more general functions q(t), the proof in [23] follows by analyzing a Lagrange identity formed by a positive solution of (1.1) and an eigenfunction, and by using Riccati equation techniques. Our main tool for th
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