LOWER BOUNDS FOR EIGENVALUES OF THE ONE-DIMENSIONAL p-LAPLACIAN

We also prove that the lower bound is sharp. Eigenvalue problems for quasilinear operators of p-Laplace type like (1.1) have received considerable attention in the last years (see, e.g., [1, 2, 3, 5, 8, 13]). The asymptotic behavior of eigenvalues was obtained in [6, 7]. Lyapunov inequalities have proved to be useful tools in the study of qualitative nature of solutions of ordinary linear differential equations. We recall the classical Lyapunov’s inequality.


Introduction
In [9], Krein obtained sharp lower bounds for eigenvalues of weighted second-order Sturm-Liouville differential operators with zero Dirichlet boundary conditions.In this paper, we give a new proof of this result and we extend it to the one-dimensional p-Laplacian − u (x) p−2 u (x) = λr(x) u(x) p−2 u(x), x ∈ (a,b), where λ is a real parameter, p > 1, and r is a bounded positive function.The method of proof is based on a suitable generalization of the Lyapunov inequality to the nonlinear case, and on some elementary inequalities.Our main result is the following theorem.
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Theorem 1.1.Let λ n be the nth eigenvalue of problem (1.1).Then, Lyapunov inequalities have proved to be useful tools in the study of qualitative nature of solutions of ordinary linear differential equations.We recall the classical Lyapunov's inequality.
Then, the following inequality holds: For the proof, we refer the interested reader to [10,11,12].We wish to stress the fact that those proofs are based on the linearity of (1.3), by direct integration of the differential equation.Also, in [12], the special role played by the Green function g(s,t) of a linear differential operator L(u) was noted, by reformulating the Lyapunov inequality for We added "by".Please check.
The paper is organized as follows.Section 2 is devoted to the Lyapunov inequality Please check if the slash in the highlighted part denote a fraction.If so, we will change the highlighted part to a stacked fraction.
for the one-dimensional p-Laplace equation.In Section 3, we focus on the eigenvalue problem and we prove Theorem 1.1.

The Lyapunov inequality
We consider the following quasilinear two-point boundary value problem: where r is a bounded positive function and p > 1.By a solution of problem (2.1), we understand a real-valued function The regularity results of [4] imply that the solutions u are at least of class C 1,α loc , and satisfy the differential equation almost everywhere in (a,b).
Our first result provides an estimation of the location of the maxima of a solution in (a,b).We need the following lemma.
where q is the conjugate exponent of p, that is, Proof.Clearly, by using Hölder's inequality, We note that u (c) = 0. So, integrating by parts in (2.1) after multiplying by u gives Then, the first inequality follows after cancelling u(c) in both sides while the second is proved in a similar fashion.
Remark 2.2.The sum of both inequalities shows that c cannot be too close to a or b.We have (2.7) Our next result gives the Lyapunov inequality.

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Theorem 2.3.Let r : [a,b] → R be a bounded positive function, let u be a solution of problem (2.1), and let q be the conjugate exponent of p ∈ (1,+∞).The following inequality holds: (2.8) Proof.For every c ∈ (a,b), we have (2.9) By using Hölder's inequality, (2.10) We now choose c in (a,b) such that |u(x)| is maximized.Then, After cancelling, we obtain and the theorem is proved.
We changed " the classical one in Theorem 1.2" to "inequality (1.4)".for the sake of clarity.

Eigenvalues bounds
In this section, we focus on the following eigenvalue problem: where r ∈ L ∞ (a,b) is a positive function, λ is a real parameter, and p > 1.
Remark 3.1.The eigenvalues could be characterized variationally: where and γ : Σ → N ∪ {∞} is the Krasnoselskii genus, The spectrum of problem (1.1) consists of a countable sequence of nonnegative eigenvalues and coincides with the eigenvalues obtained by Ljusternik-Schnirelmann theory.Now, we prove the lower bound for the eigenvalues of problem (3.1) for every p ∈ (1,+∞).We recall Theorem 1.1.Theorem 3.2.Let λ n be the nth eigenvalue of problem (3.1).Then, Since the style of the journal does not permit repeated statements of theorems, we suggest removing "Theorem 3.2" and replacing it by "we now prove our main result (Theorem 1.1)," as this was already mentioned in the paragraph presenting the structure of the paper.At then, the "Proof " head will be "Proof of Theorem 1.1".Please check.
Proof.Let λ n be the nth eigenvalue of problem (3.1) and let u n be an associate eigenfunction.As in the linear case, u n has n nodal domains in [a,b] (see [2,13]).Applying inequality (2.8) in each nodal domain, we obtain where Now, the sum on the left-hand side is minimized when all the summands are the same, which gives the lower bound ( The theorem is proved.
Finally, we prove that the lower bound is sharp.
Theorem 3.3.Let ε ∈ R be a positive number.There exist a family of weight functions r n,ε such that where λ n,ε is the nth eigenvalue of Proof.We begin with the first eigenvalue λ 1 .We fix b a r(x)dx = M, and let c be the midpoint of the interval (a,b).
Let r 1 be the delta function Mδ c (x).We obtain (3.12) Now, we define the functions r 1,ε : and the result follows by testing, in the variational formulation (3.2), the first Steklov We added the highlighted "by".Please check. eigenfunction Thus, the inequality is sharp for n = 1.We now consider the case n ≥ 2. We divide the interval (a,b) in n subintervals I i of Should we change "(a,b)" to "[a,b]" to be consistent with the rest of the paper?
equal length, and let c i be the midpoint of the ith subinterval.By using a symmetry argument, the nth eigenvalue corresponding to the weight restricted to I i , is the first eigenvalue in this interval, that is, (3.16) The proof is now completed.

Mathematics Subject Classification
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