EIGENVALUES OF THE p-LAPLACIAN AND DISCONJUGACY CRITERIA

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.

Here, 1 < p < ∞, t ∈ [a, +∞), and q(t) is a nonnegative continuous function not vanishing in subintervals of the form (b, +∞). The solutions of equation (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 [1,12,16], 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 [5,6,7,8]. Among the class of nonoscillatory equations, when any solution has a unique zero in [a, +∞), the equation is called disconjugate.
The disconjugacy phenomena is considerably more difficult and less understood than nonoscillation, we refer the interested reader to the surveys [3,4,17] for the linear case p = 2.
In order to fix ideas, we may consider it in a closed interval [a, b] with b < ∞. Clearly, the existence of two zeros in [a, b] of a solution u is related with the Dirichlet eigenvalue problem and the equation (1.1) is disconjugate if the first eigenvalue satisfies λ 1 < 1. This observation gives a condition for disconjugacy on finite intervals, namely, the equation is disconjugate if and only if the first Dirichlet eigenvalue is greater than one. Therefore, the problem of find disconjugacy conditions on finite intervals is equivalent to find lower bounds for eigenvalues. This problem has a long history, which can be traced back to [11], similar results for the p-laplacian were obtained in [14].
We consider now the disconjugacy problem on [a, +∞). The relationship between disconjugacy and the eigenvalues of a mixed problem is due to Nehari [13], and was generalized to different equations in [9,15,18]. We prove here the following theorems generalizing their results for the p-laplacian: Theorem 1.1. Let q(t) ≥ 0, not vanishing in subintervals of the form (b, +∞) and let λ 1 be the first eigenvalue of Also, we have the following result for oscillatory equations: Let q(t) ≥ 0, not vanishing in subintervals of the form (b, +∞). Then, equation (1.1) is oscillatory if and only if there exists a sequence of intervals [a n , b n ] whit a n +∞ as n +∞ such that the first eigenvalue λ 1 ≤ 1 for n ≥ 1. The proof for the linear case p = 2 in [17] follows by analyzing a Lagrange identity formed by a positive solution of equation (1.1) and an eigenfunction, and by using Riccati equation techniques. Our main tool for the proof of both theorems is a Picone type identity as in [1], and the variational characterization of the first eigenvalue, which can be obtained from the Rayleigh quotient, Let us note that the eigenvalue problem for the p-laplacian has been widely studied in recent years, see for example [2,16] among several others, and the references therein. Hence, a characterization for disconjugacy in terms of eigenvalues could be an useful tool.
As an application, we consider the so called Roundabout Theorem [7,8] for bounded intervals, adding another equivalent criterium for disconjugacy. Also, we analyze in this way Hille and Leighton type criteria (see [5,7,8]) for oscillation and nonoscillation in the half line.
The paper is organized as follows: in Section 2 we prove the main Theorems 1.1 and 1.2; in section 3 we discuss other oscillation theorems.

Main Theorems
Our main tool is the following Picone type identity which can be found in [1]: Then, We are ready to proof Theorems 1.1 and 1.2.
Proof of Theorem 1.1. Let us assume that the first eigenvalue λ 1 of is greater than 1. If equation (1.1) is not disconjugate in (a, +∞), there exists a solution v with two zeros t 1 , t 2 ∈ [a, +∞). Now, we choose v as a test function for the eigenvalue problem in [a, t 2 ], extending it by zero in [a, t 1 ). Clearly, the Rayleigh quotient gives Here, we use the weak formulation of Problems (1.1) and (1.2), multiplying the first by (u p /v −1 ) and the second by u.
Since R(u, v) = L(u, v) ≥ 0, we have λ = 1, which gives R(u, v) = L(u, v) = 0. Hence, u ≡ kv for some constant k ∈ R, and we can choose an eigenfunction such that k = 1 due to the homogeneity of the p-laplacian.
We extend the eigenfunction u as u(b) in (b, c), and let us call itũ. Sinceũ is an admissible function in the variational characterization of the first eigenvalue λ 1 in (a, c), we obtain Proof of Theorem 1.2. Let us assume first that equation (1.1) is oscillatory. Therefore, there exists a solution v with infinitely many zeros a < t 1 < t 2 < . . .
+∞. Let us choose a n = t n , b n = t n+1 . The first Dirichlet eigenfunction in [a n , b n ] coincides with u up a multiplicative constant, with eigenvalue equal to 1. The eigenvalue λ (n) 1 of (|u | p−2 u ) + λq(t)|u| p−2 u = 0, u(a n ) = 0 = u (b n ) satisfies λ Suppose now that the eigenvalue condition is satisfied for a family of intervals [a n , b n ]. Let us suppose that there exist a nonoscillatory solution u, and let us take one of the intervals with a N greater than the last zero of u. Therefore, equation (1.1) is disconjugate in (a N , +∞) (if not, there exist a solution with two zeros, and the Sturmian theory implies that u must have a zero between them). Hence, from Theorem 1.1 we get λ (N ) 1 > 1, a contradiction.

Final Remarks
3.1. Disconjugacy on bounded intervals. The arguments of the previous section can be added to the so called Roundabout Theorem (see [7,8]) as a disconjugacy criteria in an interval [a, b]. We state it here with a modification of Theorem 1.1 as condition (v): (iv) The p-degree functional is positive for every u ∈ W 1,p 0 (a, b), u not identically zero on I. (v) The first eigenvalue λ 1 of (|u | p−2 u ) + λq(t)|u| p−2 u = 0, u(a) = 0 = u(b) Proof. It is easy to see that (iv) is equivalent to (v). From the variational characterization of the first eigenvalue Remark 3.2. The eigenvalue problem in unbounded intervals was studied in [10]. However, it is not know if the eigenvalues can be characterized variationally.

Oscillation Criteria in the Half Line.
Here we consider some oscillation criteria which can be obtained from Theorems 1.1 and 1.2. We begin with a classical oscillation result: Proof. The proof follows from Theorem 1.2. For any a n ≥ a, we choose b n such that bn an+1 q(t)dt ≥ 1 and we compute the Rayleigh quotient for the first eigenvalue λ (n) 1 of the mixed problem (|u | p−2 u ) + λq(t)|u| p−2 u = 0, u(a n ) = 0 = u (b n ) with the test function v = t − a n , if t ∈ [a n , a n + 1) 1, if t ∈ [a n + 1, b n ] Hence, The following Hille type theorem can be found in [7].
It is convenient to introduce the notion of strongly oscillatory and strongly nonoscillatory equations since the results of the inequalities of the previous criteria could be changed by introducing a parameter λ in equation (1.1).
Remark 3.5. Strongly nonoscillatory equations were considered in [10] and the references therein.
We give a characterization of strongly oscillation (resp., nonoscillation) following the ideas of Nehari [13].  , which is valid if and only if q * = 0, and the upper limit coincides with the limit since q(t) ≥ 0.