Nonlinear Analysis 75 (2012) 3746–3753

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Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

The spectrum of the p-Laplacian with singular weightMarcelo Montenegro a,∗, Sebastián Lorca b

a Universidade Estadual de Campinas, IMECC, Departamento de Matemática, Rua Sérgio Buarque de Holanda, 651, CEP 13083-859, Campinas, SP, Brazilb Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7 D, Arica, Chile

a r t i c l e i n f o

Article history:Received 4 October 2011Accepted 30 January 2012Communicated by Enzo Mitidieri

MSC:35J2035J7035P0535P30

Keywords:Nonlinear eigenvalue problemp-LaplacianSingular weightIndefinite weight

a b s t r a c t

We use the Hardy–Sobolev inequality to characterize the first eigenvalue λ1 of thep-Laplacian with singular weight. In some cases it is shown that λ1 is positive simple,isolated and has a nonnegative corresponding eigenfunction φ1. Higher eigenvalues, inparticular the second one, are also determined.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

In the present paper we study the weighted eigenvalue problem−∆pu = λm(x)|u|p−2u in Ω

u = 0 on ∂Ω,(1)

where Ω ⊂ RN ,N ≥ 1, is a bounded domain, 1 < p < ∞ and ∆pu = div(|∇u|p−2∇u). This problem was addressed in

[1–9] for domains with boundary at least C2 and bounded weight. These works proved that there exists a first eigenvalueλ1 > 0 (see (6) and (7)), which is simple in the sense that two eigenfunctions corresponding to it are proportional. Moreover,the corresponding first eigenfunction φ1 can be assumed to be positive. Here we will assume that ∂Ω is a piecewise C1

domain. The simplicity of λ1 with m ≡ 1 without any regularity assumption on the boundary was established in [10] withthe aid of special test functions. We follow this procedure here. In [1] the isolation from the left hand side of λ1 was furtherproved. With that aim, a nodal set estimate for eigenfunctions corresponding to eigenvalues greater than λ1 was obtained.We also prove such an estimate in the present paper.

Our main tool is the Hardy–Sobolev inequality proved in [11]. The optimal constant problem for such an inequalityis addressed in [12]. The relationship between the Hardy–Sobolev inequality and the spectrum of a class of nonlineardifferential operators has been addressed in [13]; see also [14] for one-dimensional results dealing with singular eigenvalueproblems. An inspection of the proof in [11] reveals that theHardy–Sobolev inequality holds true in a domainwith piecewiseC1 boundary, on choosing local charts avoiding points where ∂Ω is nondifferentiable.

∗ Correspondence to: Universidade Estadual de Campinas, IMECC, Departamento de Matemática, Rua Sérgio Buarque de Holanda, 651, P.O. Box 6065,CEP 13083-859, Campinas, SP, Brazil.

E-mail addresses:msm@ime.unicamp.br (M. Montenegro), slorca@uta.cl (S. Lorca).

0362-546X/$ – see front matter© 2012 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2012.01.028

M. Montenegro, S. Lorca / Nonlinear Analysis 75 (2012) 3746–3753 3747

Lemma 1.1. If ∂Ω is piecewise C1, then uδτ

Lt (Ω)

≤ C∥∇u∥Lq(Ω) for every u ∈ W 1,q0 (Ω)

for 1t =

1q −

1−τN if q < N and for 1

t =τq if q ≥ N, where δ(x) = dist(x, ∂Ω), τ ∈ [0, 1] and C = C(N, q, τ ) > 0. In the case

t = q = p, no regularity on ∂Ω is needed.

The weightmmay be unbounded and change sign. We assume thatmδτ∈ La(Ω) andm+

≡ 0, where a, τ and p satisfy oneof the following conditions:

∂Ω is piecewise C1, 0 < τ < 1,p

1 − τ≤ a and a ≤

NpN − τp

if N > τp; (2)

∂Ω is piecewise C1, 0 < τ < 1, and p <N

1 − τ< a; (3)

∂Ω is piecewise C1, τ = 1 and a = ∞; (4)Ω is any bounded domain, τ = 0 and a = ∞. (5)

Condition (5) impliesmδτ= m ∈ L∞(Ω), including results of the previously cited papers; see also Example 1.2 below. Here

∂Ω is piecewise C1 except for (5). Conditions (2), (3), (4) or (5) are enough to set up a variational framework, namely theconstrained minimization

λ1 = infA(u) : u ∈ W 1,p0 (Ω) and B(u) = 1, (6)

where A, B : W 1,p0 (Ω) −→ R are the C1 functions defined by

A(u) =1p∥u∥p

W1,p0 (Ω)

and B(u) =1p

Ω

m(x)|u|pdx. (7)

Example 1.2. Wewill exhibit now aweightm such thatmδτ∈ La(Ω)withm+

≡ 0, where a, τ and p satisfy (3). The weightm(x) = δ(x)−β

= (1−|x|)−β is admissible in the open unit ball of RN , i.e.,Ω = B1(0). We need to choose β > 0 adequately.We fix p < N , say, p = 3/2 and N = 3. We also choose τ = 1/2 and b = 7/4; then a = 21/2. Thus, for 1/2 < β < 25/42,we conclude that m ∈ LN/p(Ω) = L2(Ω), but mδ1/2

∈ L21/2(Ω). More generally, given N and 1 < p < N , it is possible tochoose τ , b and β such that m ∈ LN/p(Ω), butmδτ

∈ La(Ω).

Example 1.2 is not included in the eigenvalue papers quoted above, since m is not bounded. This example does not verifythe assumptions of [15], since it is needed there that m ∈ Ls(Ω), if s > N/p and 1 < p ≤ N . In [16,17] it is allowed thatm ∈ LN/p(Ω) if 1 < p ≤ N . In [17] they only show that φ1 ≥ 0 (no strict positiveness). In [16] they show simplicity anduniqueness of φ1, but no isolation of λ1 and λ2 is addressed.

Section 2 is devoted to the study of λ1 and its corresponding eigenfunction φ1. We prove the following results.

Theorem 1.3. If one assumes that ∂Ω is piecewise C1 and that mδτ∈ La(Ω) with m+

≡ 0, where a, τ and p satisfy (2),(3), (4) or (5), then the number λ1 is attained by some φ1 ∈ W 1,p

0 (Ω), where we may assume that φ1 ≥ 0 a.e. in Ω , φ+

1 ≡ 0.Moreover, λ1 is positive and isolated.

Proposition 1.4. If one assumes the same conditions as for Theorem 1.3, for an eigenfunction v corresponding to an eigenvalueλ > λ1 there is a constant C independent of v such that |x ∈ Ω : v(x) < 0 a.e. in Ω| ≥ C.

In some steps of our proofs we will need to use the Harnack inequality. With that aim, according to [18, Section 5] wemake the following definitions involving locally integrable weights. Let ε(ρ) be a smooth function defined for ρ > 0 suchthat

ε(ρ) → 0 as ρ → 0+ and ρ∗

0

ε(ρ)

ρdρ < ∞ (8)

for some ρ∗ > 0. We denote by Kx0(ρ) an N-dimensional cube contained in Ω whose edges are of length ρ and are parallelto the coordinate axes. We define

Ltε(ρ)(Ω) = u ∈ Lt(Ω) : ∥u∥t,ε(ρ),Ω < ∞, (9)

where

∥u∥t,ε(ρ),Ω = sup

∥u∥Lt (Kx0 (ρ)∩Ω)

ε(ρ): x0 ∈ Ω, ρ > 0

. (10)

3748 M. Montenegro, S. Lorca / Nonlinear Analysis 75 (2012) 3746–3753

Remark 1.5. The weight m in Example 1.2 is such thatm ∈ LN/pε(ρ)(Ω), but m ∈ Ls(Ω) for s > N/p if 1 < p ≤ N .

Theorem 1.6. If in addition to the hypotheses of Theorem 1.3 one supposes that either m ∈ LN/pε(ρ)(Ω) for 1 < p ≤ N or

m ∈ L1(Ω) for p > N, then the first eigenvalue is simple and isolated from the right.

Remark 1.7. If p > N andm ∈ L1(Ω), then a weak solution inW 1,p0 (Ω) of the eigenvalue problem belongs to L∞(Ω), by the

Rellich–Kondrachov Theorem. A result due to [19] allows the same conclusion to be reached for 1 < p ≤ N , on assumingm ∈ Ls(Ω) for s > N/p and 1 < p ≤ N . Then the Harnack inequality of [18, Section 5] can be applied to ensure positivenessof φ1.

In Section 3 we construct a Lusternik–Schnirelman sequence λn of higher eigenvalues, as in [2], with the aid of Clarke’sTheorem [20] in a non-Hilbertian setting.We show that there is no eigenvalue of the p-Laplacian on the interval (λ1, λ2). Thisfact was shown in [21] for bounded weights. By means of estimates on the number of nodal regions of the eigenfunctions,the second eigenvalue is shown to be well defined. Monotonicity properties of a sequence of eigenvalues with respect tothe weights are also studied.

2. The first eigenvalue

In the sequel we establish the existence of the first eigenvalue and its isolation from the right.

Proof of Theorem 1.3. Define

b :=

ap

a − pif a < ∞

p if a = ∞.

The function B is well defined. Indeed, for u ∈ W 1,p0 (Ω) we obtain

B(u) =1p

Ω

mδτ |u|δτ

|u|p−1dx ≤1p∥mδτ

∥La(Ω)

uδτ

Lb(Ω)

∥|u|p−1∥L

pp−1 (Ω)

,

by the Hölder inequality with 1a +

1b +

p−1p = 1. We have under condition (2) u

δτ

Lb(Ω)

≤ C∥∇u∥Lτb(Ω) < ∞,

because τb ≤ p. Condition (3) implies uδτ

Lb(Ω)

≤ C∥∇u∥L

bNN+b(1−τ)(Ω)

< ∞,

since bN < p(N + b(1 − τ)). Finally, by virtue of (4),uδ

Lp(Ω)

≤ C∥∇u∥Lp(Ω) < ∞

whereC = C(p,N, τ , a) > 0 is a constant that may differ in each case. The assumption (5) gives ∥u∥Lp(Ω) ≤ C∥∇u∥Lp(Ω) <

∞. If a = ∞, C = C(p,N) > 0. Since A is weakly lower semicontinuous and coercive in W 1,p0 (Ω), so it also is in V =

u ∈ W 1,p0 (Ω) : B(u) = 1. It remains to verify that V is weakly closed for concluding that A is bounded from below in

V and that A(φ1) = infA(u) : u ∈ W 1,p0 (Ω) and B(u) = 1 for some φ1 ∈ W 1,p

0 (Ω). If un → u weakly in W 1,p0 (Ω), up

to a subsequence, un → u in Lp(Ω) and |un|p−1

→ |u|p−1 in Lp

p−1 (Ω), because of the boundedness of ∥un∥W1,p0 (Ω)

and the

compact imbeddingW 1,p0 (Ω) ⊂ Lp(Ω). Hence

Ω

m(x)(|un|p− |u|p)dx

≤ c∥mδτ∥La(Ω)

|un| + |u|δτ

Lb(Ω)

|un|p−1

− |u|p−1Lp−1p (Ω)

→ 0, (11)

because the first two norms in (11) are bounded. The constant c = c(p) > 0 comes from the inequality |ap − bp| ≤

c(a + b)|ap−1− bp−1

| for positive numbers a and b. To be precise, c = 1 if p ≥ 2 and c > p/(p − 1) if 1 < p < 2. Thus Vis weakly closed in W 1,p

0 (Ω). Since B(φ1) = 1, φ1 ≡ 0 in Ω . Clearly, |φ1| also minimizes A over V , so we may assume thatφ1 ≥ 0 a.e. in Ω and φ+

1 ≡ 0. If v is an eigenfunction corresponding to an eigenvalue λ, then λ > λ1.

We prove the nodal set estimate.

Proof of Proposition 1.4. Let v ∈ W 1,p0 (Ω) be an eigenfunction corresponding to an eigenvalue λ > λ1; then v changes

sign, that is, v−≡ 0 and v+

≡ 0 in Ω . Indeed, if v does not change sign, we may assume that v ≥ 0 a.e. in Ω with v+≡ 0.

M. Montenegro, S. Lorca / Nonlinear Analysis 75 (2012) 3746–3753 3749

Let u ≥ 0 be an eigenfunction corresponding to λ1. For uε = u+ε and vε = v+ε with ε > 0, defineUε and Vε byUε =upε−v

pε

up−1ε

and Vε =vpε−upεvp−1ε

, which are genuine test functions in W 1,p0 (Ω); see [10]. Inserting them in the integral relation satisfied by

the weak solutions u and v of (1) and adding the two expressions one gets Jε = Iε with Jε ≥ 0, where

Jε =

Ω

1 + (p − 1)

vε

uε

p|∇u|p +

1 + (p − 1)

uε

vε

p|∇v|

pdx

−

Ω

p

vε

uε

p−1

|∇u|p−2∇u · ∇v + p

uε

vε

p−1

|∇v|p−2

∇v · ∇u

dx

and

Iε =

Ω

m(x)

λ1

uuε

p−1

− λ

v

vε

p−1

(upε − vp

ε )dx.

By the fact that |∇ log uε| =|∇u|uε

, |∇ log vε| =|∇v|

vεand rewriting Jε , we obtain for p ≥ 2

0 ≤1

2p−1 − 1

Ω

1vpε

+1up

ε

|vε∇u − uε∇v|

pdx ≤ Jε. (12)

For 1 < p < 2 we get

0 ≤3p(p − 1)

16

Ω

1up

ε

+1vpε

|vε∇u − uε∇v|

2

(vε|∇u| + uε|∇v|)2−pdx ≤ Jε. (13)

Inequalities (12) and (13) are proved in [10].We may assume that ∥∇u∥Lp(Ω) = ∥∇v∥Lp(Ω) = 1; therefore

Iε →

Ω

m(x)(λ1 − λ)(up− vp)dx = (λ1 − λ)

1λ1

−1λ

≥ 0 as ε → 0+.

But

(λ1 − λ)

1λ1

−1λ

< 0,

which is an absurd, proving the claim.Define Ω−

= x ∈ Ω : v(x) < 0 a.e. in Ω. We will prove that

|Ω−| ≥ C(λ∥mδτ

∥La(Ω))−1γ

for some constant C > 0 depending only on p,N, τ and a. The constant γ > 0 will be appropriately specified according tothe case. Let p∗

= Np/(N − p) if p < N and p∗ > p if p ≥ N . In these two cases ∥v−∥Lp∗(Ω) ≤ C∗

∥∇v−∥Lp(Ω) where C∗ > 0

is a constant depending only on p and N . We distinguish, in analogy with the proof that B is well defined, four steps withrespect to (2), (3), (4) or (5). Define

b :=

ap

a − pif a < ∞

p if a = ∞.

First, since (2) impliesv−

δτ

Lb(Ω)

≤ C∥∇v−∥Lτb(Ω),

we obtain

∥∇v−∥pLp(Ω) ≤ λ∥mδτ

∥La(Ω)

v−

δτ

Lb(Ω)

∥v−∥p−1Lp(Ω) ≤

≤ λ∥mδτ∥La(Ω)

C |Ω−|p−τbτbp ∥∇v−

∥Lp(Ω)∥v−∥p−1Lp(Ω),

and then

∥∇v−∥p−1Lp(Ω) ≤ λ∥mδτ

∥La(Ω)C |Ω−

|p−τbτbp ∥v−

∥p−1Lp(Ω).

3750 M. Montenegro, S. Lorca / Nonlinear Analysis 75 (2012) 3746–3753

The Hölder inequality and Sobolev imbedding furnish1C∗

p−1

∥v−∥p−1Lp∗(Ω) ≤ λ∥mδτ

∥La(Ω)C |Ω−

|p−τbτbp |Ω−

|(1−pp∗ )(

p−1p )

∥v−∥p−1Lp∗(Ω),

which is equivalent to

|Ω−| ≥

1C∗

p−1 1C (λ∥mδτ∥La(Ω))

−1

1γ

,

where

γ =

p − τbτbp

+

1 −

pp∗

p − 1p

.

Secondly, by virtue of (3) we get v−

δτ

Lb(Ω)

≤ C∥∇v−∥ bN

LN+b(1−δ)(Ω)

. Since

∥∇v−∥pLp(Ω) ≤ λ∥mδτ

∥La(Ω)C |Ω−

|p(N+b(1−τ))−bN

pbN ∥∇v−∥Lp(Ω)∥v

−∥p−1Lp(Ω)

it follows that

|Ω−| ≥

1C∗

p−1 1C (λ∥mδτ∥La(Ω))

−1

1γ

,

with

γ =p(N + b(1 − τ)) − bN

pbN+

1 −

pp∗

p − 1p

.

In the third situation, we use (4) to obtainv−

δ

Lp(Ω)

≤ C∥∇v−∥Lp(Ω).

Since

∥∇v−∥pLp ≤ λ∥mδ∥L∞(Ω)

C∥∇v−∥Lp(Ω)∥v

−∥p−1Lp(Ω),

we get

|Ω−| ≥

1C∗

p−1 1C (λ∥mδ∥L∞(Ω))−1

1γ

,

where

γ =

1 −

pp∗

p − 1p

.

With the fourth condition (5), we apply the Sobolev inequality ∥v−∥Lp(Ω) ≤ C∥∇v−

∥Lp(Ω) to get

|Ω−| ≥

1C∗

p−1 1C λ∥m∥L∞(Ω)

−1

1γ

with

γ =

1 −

pp∗

p − 1p

.

In what follows we conclude that the first eigenvalue is simple and isolated.

Proof of Theorem 1.6. If u ≥ 0 and v ≥ 0 are eigenfunctions corresponding to λ1 we take Uε =upε−v

pε

up−1ε

and Vε =vpε−upεvp−1ε

as

test functions in W 1,p0 (Ω). Inserting them in the integral relation satisfied by the weak solutions u and v of (1) and adding

M. Montenegro, S. Lorca / Nonlinear Analysis 75 (2012) 3746–3753 3751

the two expressions one gets Jε = Iε with Jε ≥ 0, as in the proof of Proposition 1.4. Notice that now the expression for Iε isdifferent, namely

Iε =

Ω

λ1m(x)

uuε

p−1

−

v

vε

p−1

(upε − vp

ε )dx.

Since Iε → 0 as ε → 0+ it follows that |v∇u− u∇v| = 0 a.e. in Ω irrespective 1 < p < ∞, by Fatou’s Lemma. Hence thereexists a constant k > 0 such that u = kv a.e. in Ω . Hence λ1 is simple.

We proceed to examine the isolation from the right. Supposing the contrary assertion, there exists a sequence ofeigenvalues λn > λ1 such that λn → λ1. Let un be a sequence of eigenfunctions corresponding to λn. We may assumethat ∥∇un∥Lp(Ω) = 1; thus un → u weakly in W 1,p

0 (Ω). For a subsequence we have un → u in Lp(Ω), λn|un|p−2un → λ

|u|p−2u in Lp

p−1 (Ω) and un → u a.e. in Ω . Therefore

C

∥∇un − ∇u∥2

Lp(Ω)(∥∇un∥Lp(Ω) + ∥∇u∥Lp(Ω))p−2 if 1 < p ≤ 2

∥∇un − ∇u∥pLp(Ω) if p ≥ 2

≤

Ω

|∇un|p−2

∇un · (∇un − ∇u) − |∇u|p−2∇u · (∇un − ∇u)dx

=

Ω

m(λn|un|p−2un − λ|u|p−2u)(un − u)dx

≤ ∥mδτ∥La(Ω)

un + uδτ

Lb(Ω)

∥λn|un|p−2un − λ|u|p−2u∥

Lp

p−1 (Ω)→ 0,

where C = C(p) > 0 is a constant, implying un → u inW 1,p0 (Ω); thus u is an eigenfunction corresponding to λ1, and it may

be assumed that u ≥ 0 a.e. in Ω with u+≡ 0. By Egorov’s Theorem un converges uniformly to u except on a subset of Ω

with small measure. But there exists a subset of Ω with small measure such that un ≥ 0 a.e. in Ω and u+n ≡ 0 outside it for

n sufficiently large, contradicting the estimates of |Ω−|.

3. Higher eigenvalues

The method of the Ljusternik–Schnirelmann theory used in [2] transforms the eigenvalue problem into that of findingmultiple nonconstrained critical values of the functional Φ : W 1,p

0 (Ω) → R given by Φ(u) = A(u)2 − B(u). The elementsof Kc = u ∈ W 1,p

0 (Ω) : Φ(u) = c and Φ ′(u) = 0 are related by λ = 2(−c)−12 . Recall that

γ : F ⊂ W 1,p0 (Ω) : F is closed and F = −F → N ∪ +∞

is the genus function defined byγ (F) = mink ∈ N : there exists ϕ ∈ C0(F , Rk

− 0) such that ϕ(x) = −ϕ(−x),and otherwise γ (F) = +∞. If mδτ

∈ La(Ω) and m+≡ 0, with a fulfilling one of the four conditions (2), (3), (4) or (5), we

also define

b :=

ap

a − pif a < ∞

p if a = ∞.

The functional Φ has a sequence of critical values given bycn = inf

K∈Γnmaxu∈K

Φ(u),

whereΓn = K ⊂ W 1,p

0 (Ω) : K is compact , K = −K and γ (K) ≥ n.Furthermore, −∞ < infW1,p

0 (Ω)Φ = c1 ≤ c2 ≤ · · · ≤ cn ≤ · · · < 0 = Φ(0) and cn → 0. Note that if cj = ck = c for j < k,

then γ (Kc) ≥ j − k + 1 ≥ 2. Therefore, Kc has infinitely many critical points. A result in [20] allows us to verify that cn is amonotone sequence of critical values. Indeed, note that Φ is bounded from below since

Φ(u) =1p∥u∥2p

W1,p0 (Ω)

−1p

Ω

m(x)|u|pdx

≥1p∥u∥2p

W1,p0 (Ω)

−1p∥mδτ

∥La(Ω)

uδτ

Lb(Ω)

∥u∥p−1Lp(Ω)

≥1p∥u∥2p

W1,p0 (Ω)

− ∥mδτ∥La(Ω)C∥u∥p

W1,p0 (Ω)

,

3752 M. Montenegro, S. Lorca / Nonlinear Analysis 75 (2012) 3746–3753

for every u ∈ W 1,p0 (Ω). The last inequality follows along the same lines as the proof of B beingwell defined; see the beginning

of the proof of Theorem 1.6. Notice that τb ≤ p and bN/(N + b(1 − τ)) < p; hence C = C(p,N, τ , a, |Ω|) > 0. Theabove relation involving Φ also provides its coercivity. Taking this into account we verify that Φ satisfies the so-calledPalais–Smale condition. Let un ∈ W 1,p

0 (Ω) be a sequence such that Φ(un) is bounded and Φ ′(un) → 0 in the dual space

W−1, pp−1 (Ω). By virtue of the coerciveness of Φ we conclude that un is bounded in W 1,p

0 (Ω). A relabeled subsequence un

satisfies un → u weakly in W 1,p0 (Ω), un → u in Lp(Ω) and |un|

p−2un → |u|p−2u in Lp

p−1 (Ω). Clearly, Φ ′(un) = (−∆pun)

2∥un∥p

W1,p0 (Ω)

− m|un|p−2un. Notice thatm|un|

p−2un converges in W−1, pp−1 (Ω). Indeed, for every v ∈ W 1,p

0 (Ω) we obtainΩ

m(x)|un|p−2unvdx ≤ ∥mδτ

∥La(Ω)

v

δτ

Lb(Ω)

∥un∥p−1Lp(Ω)

≤ ∥mδτ∥La(Ω)C∥v∥W1,p

0 (Ω)∥un∥

p−1Lp(Ω) < ∞,

where C = C(p,N, τ , a, |Ω|) > 0 is a constant. Hence m|un|p−2un defines a sequence of continuous linear functionals in

W−1, pp−1 (Ω) that converges tom|u|p−2u in W−1, p

p−1 (Ω). Actually, for ∥v∥W1,p0 (Ω)

≤ 1 we obtainΩ

m(|un|p−2un − |u|p−2u)vdx ≤ ∥mδτ

∥La(Ω)∥v

δτ∥Lb(Ω)∥|un|

p−2un − |u|p−2u∥L

pp−1 (Ω)

→ 0.

Thus, −∆pun converges in W−1, pp−1 (Ω). Since −∆p : W 1,p

0 (Ω) → W−1, pp−1 (Ω) is a homeomorphism, we conclude that

un → u inW 1,p0 (Ω). It remains to show that for everyn ∈ N there exists a setK ∈ Γn such thatmaxu∈K Φ(u) < 0 = Φ(0). Let

w1, . . . , wn ∈ C∞c (Ω) be functions with mutually disjoint supports and B(wi) = 1 for i = 1, . . . , n. LetWn = ⟨w1, . . . , wn⟩

be the finite dimensional subspace of W 1,p0 (Ω) generated by w1, . . . , wn. There exists a constant c > 0 such that cA(w) ≤

B(w) ≤1c A(w), for every w ∈ Wn. The set K =

w ∈ Wn :

c24 ≤ B(w) ≤

c23

is such that maxu∈K Φ(u) < 0 and γ (K) = n.

Finally, the convergence cn → 0 can be found in [2].A calculation similar to that in the proof of Theorem 1.6 permits us to affirm that the set of critical values cn is closed.

The procedure can be performed by taking the limits in sequences of eigenvalues and the corresponding eigenfunctions,respectively, given by λn = 2(−cn)−

12 and un. Therefore, µ = infλ > λ1 : λ is an eigenvalue is an eigenvalue different

from λ1. Since λn = 2(−cn)−12 we have

λn = infK∈Γn

maxu∈K

B(u)=1

A(u).

Thus λ1 corresponds to c1 and λ2 to c2. The same analysis on the number of nodal sets as was employed in [21] can beapplied here in order to characterize λ2 above as λ2 = µ. For completeness we include it here. Let (u, λ) be an eigenpair;define N(u) = the number of connected components of Ω − x ∈ Ω : u(x) = 0 a.e. in Ω and set N(λ) = maxN(u) :

(u, λ) is an eigenpair. If λ is an eigenvalue, then λN(λ) ≤ λ and if λn < λn+1, then N(λn) ≤ n. The correspondingeigenfunction of µ changes sign twice; then N(µ) ≥ 2 and hence λ2 ≤ λN(µ). Since λN(µ) ≤ µ, we have λ2 = µ. Therefore,in the class of singular weights m that we are considering, there exists also a second eigenvalue of the p-Laplacian. It is notknown yet whether all eigenvalues of the p-Laplacian are those generated by the Ljusternik–Schnirelmann sequence. This istrue only for p = 2, when the p-Laplacian is the classical Laplace operator and more information is obtained from a Hilbertspace approach.

We accomplish our taskwith somemonotone properties of the eigenvalueswith respect to theweights. Denote by λn(m)and Bm, respectively, the n-eigenvalue and the function B with respect to the eigenvalue problem with weightm; see (7).

Proposition 3.1. Let m and h be weights such that m+≡ 0, h+

≡ 0 and mδτ , hδτ∈ La(Ω) with a satisfying one of the

properties (2), (3), (4) or (5). If m ≤ h a.e. in Ω , then λn(h) ≤ λn(m).

Proof. By definition

λn(h) = infK∈Γn

maxu∈K

Bh(u)=1

A(u).

Hence

λn(h) ≤ infK∈Γn

maxu∈K

Bm(u)=1

A(u)Bh(u)

≤ inf

K∈Γnmaxu∈K

Bm(u)=1

A(u) = λn(m).

It is clear that if m ≤ h a.e. in Ω and m ≡ h, then λ1(h) < λ1(m). It follows from [21] that, if m < h a.e. in the setx ∈ Ω : m+(x) = 0 and if this set has positive measure, then λ2(h) < λ2(m). This strict inequality also holds for our classof singular weights. A result similar to that for higher eigenvalues λn with n ≥ 3 is an open question.

M. Montenegro, S. Lorca / Nonlinear Analysis 75 (2012) 3746–3753 3753

Acknowledgment

M. Montenegro was partially supported by CNPq and FAPESP.

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