Spectrum of one dimensional p-laplacian Operator with indefinite weight

Spectrum of one dimensional p-laplacian Operator with indefinite weight A. Anane, O. Chakrone and M. Moussa 2 Département de mathématiques, Faculté des Sciences, Université Mohamed I er, Oujda. Maroc. e-mail: anane@sciences.univ-oujda.ac.ma chakrone@sciences.univ-oujda.ac.ma 2 Départment de mathématiques, Faculté des Sciences, Université Ibn Tofail, Kénitra. Maroc. e-mail: moussa@univ-ibntofail.ac.ma Abstract This paper is concerned with the nonlinear boundary eigenvalue problem ( u p 2 u ) = λm u p 2 u u I =]a, b[, u(a) = u(b) = 0, where p >, λ is a real parameter, m is an indefinite weight, and a, b are real numbers. We prove there exists a unique sequence of eigenvalues for this problem. Each eigenvalue is simple and verifies 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 zeros in (a, b). At the end, we give a simple variational formulation of eigenvalues. MSC 2000 35P30. Key words and phrases: p-laplacian spectrum, simplicity, isolation, strict monotonicity property, zeros of eigenfunctions, variational formulation. 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,p 0 (Ω) of problem { p u = λm u (V.P (m,ω) ) p 2 u in Ω, u = 0 on Ω, where p >, p : is the p-laplacian operator, defined by p u := div( u p 2 u), in a bounded domain Ω IR N, and m M(Ω) is an indefinite weight, with M(Ω) := {m L (Ω)/meas{x Ω, m(x) > 0} 0}. EJQTDE, 2002 No. 7, p.

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 σp ( p, m) the set of negative eigenvalues. For p = 2 ( p = Laplacian operator) it is well known (cf [4, 7, 8]) that, σ + 2 (, m) = {µ k (m, Ω), k =, 2, }, with 0 < µ (m, Ω) < µ 2 (m, Ω) µ 3 (m, Ω) +, µ k (m, Ω) repeated according to its multiplicity. The k-th eigenfunction corresponding to µ k (m, Ω), has at most k nodal domains. The eigenvalues µ k (m, Ω), k, verify the strict monotonicity property (SMP in brief), i.e if m, m M(Ω), m(x) m (x) a.e in Ω and m(x) < m (x) in some subset of nonzero measure, then µ k (m, Ω) > µ k (m, Ω). 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 [5]), 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, m, and p 2. The principal results for the problem seems to be given in (cf [, 2, 3, 5, 6, 9, 0,, 2, 3]), where is shown that there exists a sequence of eigenvalues of (V.P (m,ω) ) given by, λ n (m, Ω) = inf max Ω v p dx () K B n v K Ω m v p dx B n = {K, symmetrical compact, 0 K, and γ(k) n }, γ is the genus function, or equivalently, λ n (m, Ω) = sup Ω min m v p dx K B n v K Ω v p dx which can be written simply, λ n (m, Ω) = sup min m v p dx (3) K A n v K Ω A n = {K S, K B n }. S is the unit sphere of W,p 0 (Ω) endowed with the usual norm ( v p,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 < λ (m, Ω) < λ 2 (m, Ω) λ 3 (m, Ω) +. The first eigenvalue λ (m, Ω) is of special importance. We give some of its properties which will be of interest for us (cf []). First, λ (m, Ω) is given by, λ (m, Ω) = sup m v p dx = m φ p dx (4) v S Ω Ω φ S is any eigenfunction corresponding to λ (m, Ω), for this reason λ (m, Ω) is called the principal eigenvalue, also we know that λ (m, Ω) > 0, simple (i.e if v and u are two (2) EJQTDE, 2002 No. 7, p. 2

eigenfunctions corresponding to λ (m, Ω) then v = αu for some α IR), isolate (i.e there is no eigenvalue in ]0, a[ for some a > λ (m, Ω), finally it is the unique eigenvalue which has an eigenfunction with constant sign. We denote φ (x) the positive eigenfunction corresponding to λ (m, Ω), φ (x) verifies the strong maximum principle (cf [7]), φ (x) < 0, n for x in Ω satisfying the interior ball condition. In [4] Otani considers the case N =, m(x), and proves that, σ p ( p, ) = {µ k (m, I), k =, 2, }, the sequence can be ordered as 0 < µ (m, Ω) < µ 2 (m, Ω) < µ 3 (m, Ω) < +, the k-th eigenfunction has exactly k zeros in I = (a, b). In [0] Elbert proved the same results in the case N =, m(x) 0 and continuous, the author gives an asymptotic relation of eigenvalues. In this paper we consider the general case, N = and m(x) can change sign and is not necessarily continuous. We prove that σ + p ( p, m) = {λ k (m, I), k =, 2, }, the sequence can be ordered as 0 < λ (m, Ω) < λ 2 (m, Ω) < λ 3 (m, Ω) < λ k (m, I) + as k +, the k-th eigenfunction has exactly k zeros in I = (a, b). The eigenvalues verify the SMP with respect to the weight m and the domain I. 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) ). u /ω is the extension, by zero, on I of u /ω 2 Results and technical Lemmas We first state our main results Theorem Assume that N= (Ω =]a, b[= I), m M(I) such that m and p 2, we have. Every eigenfunction corresponding to the k-th eigenvalue λ k (m, I), has exactly k- zeros. 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. 3. σ + p ( p, m) = {λ k (m, I), k =, 2, }, for any m M(I). The eigenvalues are ordered as 0 < λ (m, Ω) < λ 2 (m, Ω) < λ 3 (m, Ω) < λ k (m, I) + as k +. Corollary For any integer n, we have the simple variational formulation, F n ={F / F is a n dimensional subspace of W,p 0 (I)}. b λ n (m, I) = sup min m v p dx (5) F F n F S a EJQTDE, 2002 No. 7, p. 3

For the proof of Theorem we need some technical Lemmas. Lemma Let m, m M(I), m(x) m (x), then for any n, λ n (m, I) λ n (m, I) Proof Making use of equation (2), we obtain immediately λ n (m, I) λ n (m, I). Lemma 2 Let (u, λ(m, I)) be a solution of (V.P (m,i) ), m M(I), then m /ω M(ω) for any nodal domain ω of u. Proof Let ω be a nodal domain of u and multiply (V.P (m,i) ) by u /ω so that we obtain 0 < u p dx = λ(m, I) m u p dx. (6) This completes the proof. ω Lemma 3 The restriction of a solution (u, λ(m, I)) of problem (V.P (m,i) ), on a nodal domain ω, is an eigenfunction of problem (V.P (m/ω,ω)), and we have λ(m, I) = λ (m/ ω, ω). Proof Let v W,p 0 (ω) and let ṽ be the extension by zero of v on Ω. It is obvious that ṽ W,p 0 (Ω). Multiply (V.P (m,ω) ) by ṽ u p 2 u v dx = λ(m, I) m u p 2 uv dx (7) ω for all v W,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 (m /ω, ω), ω), which completes the proof. ω ω Lemma 4 Each solution (u, λ(m, I)) of the problem (V.P (m,i) ) has a finite number of zeros. 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, of nodal domains (intervals), I i I j = for i j. We deduce by Lemmas 3 and, respectively, that λ(m, I) = λ (m, I k ) λ (C, I k ) = λ (, I k ) C where C = m. From equation (8) we deduce (meas(i k )) ( λ (,]0,[) λ(m,i)c ) p, for all k, so meas(i) = k (meas(i k )) = +. = λ (, ]0, [) C(meas(I k )) p, (8) This yields a contradiction. Let {I, I 2, I k } be the nodal domains of u. Put I i =]a i, b i [, where a a < b a 2 < b 2 a k < b k b. It is clear that the restriction of u on ]a, b [ is a nontrivial eigenfunction with constant sign corresponding to λ(m, I). The maximum principle (cf [7]) yields u(t) > 0 for all t ]a, b [, so a = a. By a similar argument we prove that b = a 2, b 2 = a 3, b k = b, which completes the proof. Lemma 5 (cf [6]) Let u be a solution of problem (V.P (m,ω) ) and u W,p (Ω) L (Ω) then u C,α (Ω) C ( Ω) for some α (0, ). EJQTDE, 2002 No. 7, p. 4

3 Proof of main results Proof of Theorem For n =, we know that λ (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 λ (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, λ (m, I) < λ (m, I) (9) and, if J is a strict sub interval of I such that m /J M(J) then, λ (m, I) < λ (m /J, J). (0) Proof Let m, m M(I) as in Proposition and recall that the principal eigenfunction φ S corresponding to λ (m, I) has no zero in I; i.e φ (t) 0 for all t I. By (3), we get λ (m, I) I = m φ p dx < m φ p dx sup m v p dx = I v S I λ (m, I). () Then inequality (9) is proved. To prove inequality (0), let J be a strict sub interval of I and m /J M(J). Let u S be the (principal) positive eigenfunction of (V.P (m,j) ) corresponding to λ (m /J, J), and denote by ũ the extension by zero on I. Then λ (m /J, J) J = m u p dx = I m ũ p dx < sup m v p dx = v S I λ (m, I). (2) The last strict inequality holds from the fact that ũ vanishes in I/J so can t be an eigenfunction corresponding to the principal eigenvalue λ (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, I, for which we have Z(u) = {c 2, } for any eigenfunction u corresponding to λ 2 (m, I). For this reason, we will say that c 2, 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 I =]a, c[ and I 2 =]c, b[ two nodal domains of u, by Lemma 3, λ (m /I, I ) = λ 2 (m, I) = λ (m /I2, I 2 ). Assume that c < c, choose d ]c, c [ and put J =]a, d[, J 2 =]d, b[; hence J J 2 =, and for i =, 2, I i J i strictly, and m /Ji M(J i ). Making use of Lemma 3, by (0), we get λ (m /J, J ) < λ (m /I, I ) = λ 2 (m, I) (3) and λ (m /J2, J 2 ) < λ (m /I2, I 2 ) = λ 2 (m, I). (4) Let φ i S be an eigenfunction corresponding to λ (m /Ji, J i ), by (4) we have for i =, 2 λ (m, J i ) = J i m φ i p dx (5) EJQTDE, 2002 No. 7, p. 5

φ i is the extension by zero of φ i on I. Consider the two dimensional subspace F = φ, φ 2 and put K 2 = F S W,p 0 (I). Obviously γ(k 2 ) = 2 and we remark that for v = α φ + β φ 2, v,p = α p + β p =. Hence by (3), (3), (4) and (5) we obtain, λ 2 min m v p dx (m,i) v K 2 I = min ( α p m φ p + β p m φ 2 p ) v=α φ +β φ 2 K 2 J J 2 = α 0 p J m φ p + β 0 p J 2 m φ 2 p ) α = 0 p + λ (m /J,J ) > α 0 p + β 0 p λ 2 (m,i) =, λ 2 (m,i) α 0 p λ (m /J2,J 2 ) 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 (0), we get λ 2 (m, I) = λ (m /]a,d[, ]a, d[) < λ (m /]a,c[, ]a, c[) = λ 2 (m, I). (6) 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. Lemma 6 λ 2 (m, I) is simple, hence λ 2 (m, I) < λ 3 (m, I). Proof Let u and v be two eigenfunctions corresponding to λ 2 (m, I). By Lemma 3 the restrictions of u and v on ]a, c 2, [ and ]c 2,, b[ are eigenfunctions corresponding to λ (m /]a,c2, [, ]a, c 2, [) and λ (m /]c2,,b[, ]c 2,, b[), respectively. Making use of the simplicity of the first eigenvalue, we get u = αv in ]a, c 2, [ and u = βv in ]c 2,, b[. But both of u and v are eigenfunctions, so then by Lemma 5, there are C (I). The maximum principle (cf [7]) tell us that u (c 2, ) 0, so α = β. Finally, by the simplicity of λ 2 (m, I) and the theorem of multiplicity (cf [5]) 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, and c 2, be the zeros of λ 2(m, I) and λ 2 (m, I) respectively. We distinguish three cases :. c 2, = c 2, = c, then meas({x I/ m(x) < m (x) ]a, c[}) 0, or meas({x I/ m(x) < m (x) ]c, b[}) 0, by Lemma 3 and (9) we obtain λ 2 (m, I) = λ (m /]a,c[, ]a, c[) < λ (m /]a,c[, ]a, c[) = λ 2 (m, I), (7) or λ 2 (m, I) = λ (m /]c,b[, ]c, b[) < λ (m /]c,b[, ]c, b[) = λ 2 (m, I). (8) EJQTDE, 2002 No. 7, p. 6

2. c 2, < c 2,, by Lemmas, 3 and (0), we get λ 2 (m, I) = λ (m /]a,c 2, [, ]a, c 2,[) λ (m /]a,c 2, [, ]a, c 2,[) < λ (m /]a,c2, [, ]a, c 2, [) = λ 2 (m, I). (9) 3. c 2, < c 2,, as before, by Lemmas, 3 and (0), we have λ 2 (m, I) = λ (m /]c 2,,b[, ]c 2,, b[) λ (m /]c 2,,b[, ]c 2,, b[) < λ (m /]c2,,b[, ]c 2,, b[) = λ 2 (m, I). (20) 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, the zero of λ 2 (m /J, J). As in the SMP with respect to the weight, three cases are distinguished:. c 2, = c 2, = l, then ]c, l[ is a strict sub interval of ]a, l[ or ]l, d[ is a strict sub interval of ]l, b[. By Lemma 3 and (0), we get λ 2 (m, I) = λ (m /]a,l[, ]a, l[) < λ (m /]c,l[, ]c, l[) = λ 2 (m /J, J) (2) or λ 2 (m, I) = λ (m /]l,b[, ]l, b[) < λ (m /]l,d[, ]l, d[) = λ 2 (m /J, J). (22) 2. c 2, < c 2,, again by Lemma 3 and (0), we obtain λ 2 (m, I) = λ (m /]c2,,b[, ]c 2,, b[) < λ (m /]c 2,,d[, ]c 2,, b[) = λ 2(m /J, J). (23) 3. c 2, < c 2,, for the same reason as in the last case, we get The proof is complete. λ 2 (m, I) = λ (m /]a,c2, [, ]a, c 2, [) < λ (m /]c,c 2, [, ]c, c 2, [) = λ 2 (m /J, J). (24) 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). Proof We shall prove that c = c 2,. Assume, for example, that c < c 2,. By Lemma and (0) we get On the other hand, λ(m, I) = λ (m /]c,b[, ]c, b[) < λ (m /]c2,,b[, ]c 2,, b[) = λ 2 (m, I). (25) λ 2 (m, I) = λ (m /]a,c2, [, ]a, c 2, [) < λ (m /]a,c[, ]a, c[) = λ(m, I), (26) a contradiction. Hence, c = c 2, and λ(m, I) = λ 2 (m, I). The proof is complete. For n > 2, we use a recurrence argument. Assume that, for any k, k n, that the following hypothesis: EJQTDE, 2002 No. 7, p. 7

. H.R. For any eigenfunction u corresponding to the k-th eigenvalue λ k (m, I), there exists a unique c k,i, i k, such that Z(u) = {c k,i, i k }. 2. H.R.2 λ k (m, I) is simple. 3. H.R.3 λ (m, I) < λ 2 (m, I) < < λ n+ (m, I). 4. H.R.4 If (u, λ(m, I)) is a solution of (V.P (m,i) ) such that Z(u) = {c i, i k }, then λ(m, I) = λ k (m, I). 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 +. Proposition 4 There exists a unique family {c n+,i, i n} such that Z(u) = {c n+,i, i n}, for any eigenfunction u corresponding to λ n+ (m, I). Proof Let u be an eigenfunction corresponding to λ n+ (m, I). By H.R.3 and H.R.4, u has at least n zeros. According to Lemma 4, we can consider the n + nodal domains of u, I =]a, c [, I 2 =]c, c 2 [,..., I n =]c n, c n [, I n+ =]c, b[. We shall prove that c = c n. Remark that the restrictions of u on ]a, c i+ [, i n, are eigenfunctions with i zeros, by H.R.4 λ n+ (m, I) = λ i (m /]a,ci+ [, ]a, c i+ [). Assume that c n < c, choose d in ]c n, c[ and put, J =]a, d[, J 2 =]d, b[. Remark that J J 2 =, ]a, c n [ is a strict sub interval of J I, and ]c, b[ is a strict sub interval of J 2 I. It is clear that m /Ji M(J i ) for i =, 2, by H.R.4 and H.R.5. We have and λ n (m /J, J ) < λ n (m /]a,cn[, ]a, c n [) = λ n+ (m, I), (27) λ (m /J2, J 2 ) < λ (m /]c,b[, ]c, b[) = λ n+ (m, I). (28) Denote by (φ n+, λ (m /J2, J 2 )) a solution of (V.P (m,j2 )), (v, λ n (m /J, J )) a solution of (V.P (m,j )), φ i, i n the restrictions of v on I i, and φ i, i n, their extensions, by zero, on I. Let F n+ = φ, φ 2,, φ n+ and K n+ = F n+ S, then γ(k n+ ) = n +. We obtain by (3) and the same proof as in Proposition 2 λ n+ (m, I) min m v p dx > K n+ I λ n+ (m, I), (29) a contradiction, so c = c n. On the other hand, let v be an eigenfunction corresponding to λ n+ (m, I). Denote by d, d 2,, d n the zeros of v. If d c, then λ n+ (m, I) = λ (m /]a,d [, ]a, d [) λ (m /]a,c [, ]a, c [) = λ n+ (m, I), so d = c, by the same argument we conclude that d i = c i for all i n. Lemma 8 λ n+ (m, I) is simple, hence. λ n+ (m, I) < λ n+2 (m, I). EJQTDE, 2002 No. 7, p. 8

Proof Let u and v be two eigenfunctions corresponding to λ n+ (m, I). The restrictions of u and v on ]a, c n+, [ and ]c n+,, b[ respectively, are eigenfunctions corresponding to λ (m /]a,cn+, [, ]a, c n+, [) and λ n (m /]cn+,,b[, ]c n+,, b[). By H.R.2 and H.R.4 we have u = αv in ]a, c n+, [ and u = βv in ]c n+,, b[. On the other hand, u and v are C (I) and u (c n+, ) 0, so α = β. From the simplicity of λ n+ (m, I) and theorem of multiplicity we conclude that λ n+ (m, I) < λ n+2 (m, I). Proposition 5 λ n+ (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) with m(x) < m (x) in some subset of nonzero measure and (c n+,i) i n the zeros of λ n+ (m ) three cases are distinguished,. c n+, = c n+, = c, one of the subsets is of nonzero measure, {x I/ m(x) < m (x)} ]a, c[ and {x I/ m(x) < m (x)} ]c, b[. By Lemma 3 and (9), we have λ n+ (m, I) = λ (m /]a,c[, ]a, c[) < λ (m /]a,c[, ]a, c[) = λ n+ (m, I) (30) or λ n+ (m, I) = λ n (m /]c,b[, ]c, b[) < λ n(m /]c,b[, ]c, b[) = λ n+ (m, I). (3) 2. c n+, < c n+,, by Lemmas, 3 and (0) we have λ n+ (m, I) = λ (m /]a,c n+, [, ]a, c n+, [) λ (m /]a,c n+, [, ]a, c n+, [) < λ (m /]a,cn+, [, ]a, c n+, [) = λ n+ (m, I). (32) 3. c n, < c n,, from the same reason as before, we get λ n+ (m, I) = λ n (m /]c n+,,b[, ]c n+,, b[) λ n (m /]c n+,,b[, ]c n+, [, b) < λ n (m /]cn+,,b[, ]c n+,, b[) = λ n+ (m, I). (33) By similar argument as in proof of Proposition 3, we prove the SMP with respect to the domain I. Lemma 9 If (u, λ(m, I)) is a solution of (V.P (m,i) ) such that Z(u) = {d, d 2, d n }, then λ(m, I) = λ n+ (m, I). Proof It is sufficient to prove that d i = c n+,i for all i n. If c n+, < d then, by Lemma, (0), H.R.4 and H.R.5, λ(m, I) = λ (m /]a,d [, ]a, d [) < λ (m /]a,cn+, [, ]a, c n+, [) = λ n+ (m, I) = λ n (m /]cn+,,b[, ]c n+,, b[) < λ n (m /]d,b[, ]d, b[) = λ(m, I), (34) EJQTDE, 2002 No. 7, p. 9

a contradiction. If d < c n+, then, by Lemma, (0), H.R.4 and H.R.5, λ n+ (m, I) = λ (m /]a,cn+ [, ]a, c n+ [) < λ (m /]a,d [, ]a, d [) = λ(m, I) = λ n (m /]d,b[, ]d, b[) < λ n (m /]cn+,,b[, ]c n+, b[) = λ n+ (m, I), (35) a contradiction. The proof is then complete, which completes the proof of Theorem. have: Proof of Corollary. Since for F F n, the compact F S A n, by (3) we sup F F n min m v p dx v F S Ω λ n (m, Ω). (36) On the other hand, for a n dimensional subspace F of W,p 0 (I), the compact set K = F S A n. Let u be an eigenfunction corresponding to λ n (m, I) and put F = φ (]a, c n, [), φ (]c n,, c n,2 [),, φ (]c n,n, b[), to conclude F S A n. By an elementary computation as in Proposition 2, one can show that λ n (m, I) = min m v p dx. (37) F S I Then combine (36) with (37) to get (5). Which completes the proof. 3. Remark The spectrum of p-laplacian, with indefinite weight, in one dimension, is entirely determined by the sequence (λ n (m, I)) n if m(x) 0 a.e in I. Therefore, if m(x) < 0 in some subset J I of nonzero measure, replace m by m; by Theorem, since m M(I) we conclude that, the negative spectrum σ p ( p, m) = σ + p ( p, m) of this operator is constituted by a sequence of eigenvalues λ n (m, I) = λ n ( m, I). Thus the spectrum of the operator is, σ p ( p, m) = σ + p ( p, m) σ p ( p, m). References [] A. Anane, Simplicité et isolation de la première valeur propre du p-laplacien avec poids, C. R. Acad. Sci. Paris, t.305, Série, (987), pp. 725-728. [2] A. Anane and O. Chakrone, Sur un théorème de point critique et application à un problème de non résonance entre deux valeurs propres du p-laplacien, Annales de la Faculté des Sciences de Toulouse, Vol. IX, No., (2000), pp. 5-30. EJQTDE, 2002 No. 7, p. 0

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