Spectrum of one dimensional p-Laplacian Operator with indefinite weight

This paper is concerned with the nonlinear boundary eigenvalue problem (ju 0 j p 2 u 0 ) 0 = m juj p 2 u u 2 I =]a; b[; u(a) = u(b) = 0; where p > 1, is a real parameter, m is an indenite weight, and a, b are real numbers. We prove there exists a unique sequence of eigenvalues for this problem. Each eigenvalue is simple and veries the strict monotonicity property with respect to the weight m and the domain I, the k-th eigenfunction, corresponding to the k-th eigenvalue, has exactly k 1 zeros in (a; b). At the end, we give a simple variational formulation of eigenvalues.


Introduction
The spectrum of the p-Laplacian operator with indefinite weight is defined as the set σ p (−∆ p , m) of λ := λ(m, I) for which there exists a nontrivial (weak) solution u ∈ W 1,p 0 (Ω) of problem The values λ(m, Ω) for which there exists a nontrivial solution of (V.P (m,Ω) ) are called eigenvalues and the corresponding solutions are the eigenfunctions.We will denote σ + p (−∆ p , m) the set of all positive eigenvalues, and by σ − p (−∆ p , m) the set of negative eigenvalues.
• The k-th eigenfunction corresponding to µ k (m, Ω), has at most k nodal domains.
• Equivalence between the SMP and the unique continuation one.
For p = 2 (nonlinear problem), it is well known that the critical point theory of Ljusternik-Schnirelmann (cf [15]), provides a sequence of eigenvalues for those problems.Whether or not this sequence, denoted λ k (m, Ω), constitutes the set of all eigenvalues is an open question when N ≥ 1, m ≡ 1, and p = 2.The principal results for the problem seems to be given in (cf [1,2,3,5,6,9,10,11,12,13]), where is shown that there exists a sequence of eigenvalues of (V.P (m,Ω) ) given by, B n = {K, symmetrical compact, 0 ∈ K, and γ(K) ≥ n }, γ is the genus function, or equivalently, which can be written simply, A n = {K ∩ S, K ∈ B n }. S is the unit sphere of W 1,p 0 (Ω) endowed with the usual norm ( v p 1,p = Ω |∇v| p dx), the equation ( 2) is the generalized Rayleigh quotient for the problem (V.P (m,Ω) ).The sequence is ordered as 0 ) is of special importance.We give some of its properties which will be of interest for us (cf [1]).First, λ 1 (m, Ω) is given by, φ 1 ∈ S is any eigenfunction corresponding to λ 1 (m, Ω), for this reason λ 1 (m, Ω) is called the principal eigenvalue, also we know that λ 1 (m, Ω) > 0, simple (i.e if v and u are two EJQTDE, 2002 No. 17, p. 2 eigenfunctions corresponding to λ 1 (m, Ω) then v = αu for some α ∈ IR), isolate (i.e there is no eigenvalue in ]0, a[ for some a > λ 1 (m, Ω), finally it is the unique eigenvalue which has an eigenfunction with constant sign.We denote φ 1 (x) the positive eigenfunction corresponding to λ 1 (m, Ω), φ 1 (x) verifies the strong maximum principle (cf [17]), ∂φ 1 ∂n (x) < 0, for x in ∂Ω satisfying the interior ball condition.
In this paper we consider the general case, N = 1 and m(x) can change sign and is not necessarily continuous.We prove that σ In the next section we denote by: M (I) := {m ∈ L ∞ (I)/meas{x ∈ I, m(x) > 0} = 0}, m /J the restriction of m on J for a subset J of I, Z(u) = {t ∈ I/ u(t) = 0}, a nodal domain ω of u is a component of I\Z(u), where (u, λ(m, I)) is a solution of (V.P (m,I) ).
ũ/ω is the extension, by zero, on I of u /ω

Results and technical Lemmas
We first state our main results 2. For every k, λ k (m, I) is simple and verifies the strict monotonicity property with respect to the weight m and the domain I.
Corollary 1 For any integer n, we have the simple variational formulation, For the proof of Theorem 1 we need some technical Lemmas.
Proof Let ω be a nodal domain of u and multiply (V.P (m,I) ) by ũ/ω so that we obtain This completes the proof.
Proof Let v ∈ W 1,p 0 (ω) and let ṽ be the extension by zero of v on Ω.It is obvious that ṽ ∈ W 1,p 0 (Ω).Multiply (V.P (m,Ω) ) by ṽ for all v ∈ W 1,p 0 (ω).Hence the restriction of u in ω is a solution of problem (V.P (m /ω ,ω) ) with constant sign.We then have λ(m, Ω) = λ 1 (m /ω , ω), ω), which completes the proof.Proof This Lemma plays an essential role in our work.We start by showing that u has a finite number of nodal domains.Assume that there exists a sequence I k , k ≥ 1, of nodal domains (intervals), I i ∩ I j = ∅ for i = j.We deduce by Lemmas 3 and 1, respectively, that where C = m ∞ .
From equation (8) we deduce (meas p , for all k, so This yields a contradiction. is a nontrivial eigenfunction with constant sign corresponding to λ(m, I).The maximum principle (cf [17]) yields u(t) > 0 for all t ∈]a, b 1 [, so a = a 1 .By a similar argument we prove that b Lemma 5 (cf [16]) Let u be a solution of problem (V.P (m,Ω) ) and u

Proof of Theorem 1
For n = 1, we know that λ 1 (m, I) is simple, isolate and the corresponding eigenfunction has constant sign.Hence it has no zero in (a, b) and it remains to prove the SMP.
Proposition 1 λ 1 (m, I) verifies the strict monotonicity property with respect to weight m and the domain I. i.e If m, m ∈ M (I), m(x) ≤ m (x) and m(x) < m (x) in some subset of I of nonzero measure then, and, if J is a strict sub interval of I such that m /J ∈ M (J) then, Proof Let m, m ∈ M (I) as in Proposition 1 and recall that the principal eigenfunction φ 1 ∈ S corresponding to λ 1 (m, I) has no zero in I; i.e φ 1 (t) = 0 for all t ∈ I.By (3), we get 1 Then inequality ( 9) is proved.To prove inequality (10), let J be a strict sub interval of I and m /J ∈ M (J).Let u 1 ∈ S be the (principal) positive eigenfunction of (V.P (m,J) ) corresponding to λ 1 (m /J , J), and denote by ũ1 the extension by zero on I. Then The last strict inequality holds from the fact that ũ1 vanishes in I/J so can't be an eigenfunction corresponding to the principal eigenvalue λ 1 (m, I).
For n = 2 we start by proving that λ 2 (m, I) has a unique zero in (a, b).
Proposition 2 There exists a unique real c 2,1 ∈ I, for which we have Z(u) = {c 2,1 } for any eigenfunction u corresponding to λ 2 (m, I).For this reason, we will say that c 2,1 is the zero of λ 2 (m, I).
Proof Let u be an eigenfunction corresponding to λ 2 (m, I).u changes sign in I. Consider , and for i = 1, 2, I i ⊂ J i strictly, and m /J i ∈ M (J i ).Making use of Lemma 3, by (10), we get and Let φ i ∈ S be an eigenfunction corresponding to λ 1 (m /J i , J i ), by (4) we have for i = 1, 2 EJQTDE, 2002 No. 17, p. 5 φi is the extension by zero of φ i on I. Consider the two dimensional subspace F = φ1 , φ2 and put K 2 = F ∩ S ⊂ W 1,p 0 (I).Obviously γ(K 2 ) = 2 and we remark that for v = α φ1 + β φ2 , v 1,p = 1 ⇐⇒ |α| p + |β| p = 1.Hence by (3), ( 13), ( 14) and ( 15) we obtain, a contradiction; hence c = c .On the other hand, let v be another eigenfunction corresponding to λ 2 (m, I).Denote by d its unique zero in (a, b).Assume, for example, that c < d.By Lemma 3 and relation (10), we get This is a contradiction so c = d.We have proved that every eigenfunction corresponding to λ 2 (m, I) has one, and only one, zero in (a, b), and that the zero is the same for all eigenfunctions, which completes the proof of the Proposition.The maximum principle (cf [17]) tell us that u (c 2,1 ) = 0, so α = β.Finally, by the simplicity of λ 2 (m, I) and the theorem of multiplicity (cf [15]) we conclude that λ 2 (m, I) < λ 3 (m, I).
Proposition 3 λ 2 (m, I) verifies the SMP with respect to the weight m and the domain I.
Proof Let m, m ∈ M (I) such that, m(x) ≤ m (x) a.e in I and m(x) < m (x) in some subset of nonzero measure.Let c 2,1 and c 2,1 be the zeros of λ 2 (m, I) and λ 2 (m , I) respectively.We distinguish three cases : Lemma 3 and (9) we obtain or 3. c 2,1 < c 2,1 , as before, by Lemmas 1, 3 and (10), we have For the SMP with respect to the domain, put J =]c, d[ a strict sub interval of I with m /J ∈ M (J), and denote c 2,1 the zero of λ 2 (m /J , J).As in the SMP with respect to the weight, three cases are distinguished: Lemma 3 and (10), we get 2. c 2,1 < c 2,1 , again by Lemma 3 and ( 10), we obtain 3. c 2,1 < c 2,1 , for the same reason as in the last case, we get The proof is complete.
Lemma 7 If any eigenfunction u corresponding to some eigenvalue λ(m, I) is such that Z(u) = {c} for some real number c, then λ(m, I) = λ 2 (m, I).
For n > 2, we use a recurrence argument.Assume that, for any k, 1 ≤ k ≤ n, that the following hypothesis: 1. H.R.1 For any eigenfunction u corresponding to the k-th eigenvalue λ k (m, I), there exists a unique c k,i , 1 5. H.R.5 λ k (m, I) verifies the SMP with respect to the weight m and the domain I.
Holds, and prove them for n + 1.
We obtain by (3) and the same proof as in Proposition 2 a contradiction, so c = c n .On the other hand, let v be an eigenfunction corresponding to λ n+1 (m, I).Denote by 2. c n+1,1 < c n+1,1 , by Lemmas 1, 3 and (10) we have 3. c n,1 < c n,1 , from the same reason as before, we get By similar argument as in proof of Proposition 3, we prove the SMP with respect to the domain I.

Remark
The spectrum of p-Laplacian, with indefinite weight, in one dimension, is entirely determined by the sequence (λ n (m, I)) n≥1 if m(x) ≥ 0 a.e in I. Therefore, if m(x) < 0 in some

Lemma 4
Each solution (u, λ(m, I)) of the problem (V.P (m,I) ) has a finite number of zeros.