Extremal Values of Half-Eigenvalues for -Laplacian with Weights in Balls

Volume 2010, Article ID 690342,21pages doi:10.1155/2010/690342

Research Article

Extremal Values of Half-Eigenvalues for

p

-Laplacian with Weights in

L

1

Balls

Ping Yan

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Correspondence should be addressed to Ping Yan,[email protected]

Received 24 May 2010; Accepted 21 October 2010

Academic Editor: V. Shakhmurov

Copyrightq2010 Ping Yan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

For one-dimensionalp-Laplacian with weights inLγ _{: L}γ_{0,1,R 1 ≤ γ ≤ ∞balls, we are} interested in the extremal values of themth positive half-eigenvalues associated with Dirichlet, Neumann, and generalized periodic boundary conditions, respectively. It will be shown that the extremal value problems for half-eigenvalues are equivalent to those for eigenvalues, and all these extremal values are given by some best Sobolev constants.

1. Introduction

Occasionally, we need to solve extremal value problems for eigenvalues. A classical example studied by Krein1is the infimum and the supremum of themth Dirichlet eigenvalues of Hill’s operator with positive weight

infμDmw:w∈Er,h

, supμDmw:w∈Er,h

, 1.1

where 0< r ≤h <∞and

Er,h:

w∈ Lγ: 0≤w≤h,

₁

0

wtdtr

. 1.2

Neumann eigenvalues, which plays a crucial role in population dynamics. Given constants

κ∈0,∞andα∈0,1, denote

Sκ,α:

ω∈ L∞:−1≤ω≤κ, ω0, ₁

0

ωtdt≤ −α

. 1.3

The positive principal eigenvalueμN₀ ωis well-defined for anyω∈Sκ,α, and the minimiza-tion problem in3is to find

infμN₀ w:ω∈Sκ,α

. 1.4

In solving the previous three problems, two crucial steps have been employed. The first step is to prove that the extremal values can be attained by some weights. For regular self-adjoint linear Sturm-Liouville problems the continuous dependence of eigenvalues on weights/potentials in the usual Lγ _{topology is well understood, and so is the Fr´echet} differentiable dependence. Many of these results are summarized in4. It is remarkable that this step cannot be answered immediately by such a continuity results, because the space of weights is infinite-dimensional. The second step is to find the minimizers/maximizers. This step is tricky and it depends on the problem studied. ForL1_{weights the solution is suggested}

by the Pontrjagin’s Maximum Principle5, Sections 48.6–48.8.

For Sturm-Liouville operators and Hill’s operators Zhang6proved that the eigenval-ues are continuous in potentials in the sense of weak topologywγ. Such a stronger continuity result has been generalized to eigenvalues and half-eigenvalues on potentials/weights for scalarp-Laplacian associated with different types of boundary conditionssee7–10.

As an elementary application of such a stronger continuity, the proof of the first step, that is, the existence of minimizers or maximizers, of the extremal value problems as in1–3

was quite simplified in9,10.

Based on the continuity of eigenvalues in weak topology and the Fr´echet diff erentia-bility, some deeper results have also been obtained by Zhang and his coauthors in10–12by using variational method, singular integrals and limiting approach.

The extremal values of eigenvalues for Sturm-Liouville operators with potentials inL1

balls were studied in11,12. Forγ∈1,∞,r ≥0 andm∈Z:{0,1,2, . . .}, denote

LFm,γr:inf

λFm

q:q∈ Lγ, q _γ≤r,

MFm,γr:sup

λFm

q:q∈ Lγ, q _γ≤r,

1.5

where the superscriptFdenotesNorPifm0 andDorNifm >0. By the limiting approach

γ↓1, the most important extremal values inL1balls are proved to be finite real numbers, and they can be evaluated explicitly by using some elementary functionsZ0r,Z1r,Rmr, and

Y1r. None of the extremal valuesLF_m,1 can be attained by any potential ifr > 0, while all

extremal valuesLF

The extremal value of the mth Dirichlet eigenvalue for p-Laplacian with positive weight was studied by Yan and Zhang 10. It was proved for γ ∈ 1,∞, r > 0, and

m∈N:{1,2,3, . . .}that

infμDmw:w∈ Lγ, w≥0, wγ ≤r

infμDmw:w∈ Lγ, w≥0, wγ r

mp·K

pγ∗, p r ,

1.6

whereγ∗:γ/γ−1is the conjugate exponent ofγ, andK·,·is the best Sobolev constant

Kα, p: inf u∈W₀1,p0,1

upp

upα

, ∀α∈1,∞. 1.7

Moreover, the infimum can be attained by some weight if onlyγ ∈ 1,∞. By letting the radiusr↓0one sees that the supremum

supμDmw:w∈ Lγ, w≥0, wγ ≤r

∞, 1.8

so only infimum of weighted eigenvalues is considered.

Our concerns in this paper are the infimum of the mth positive half-eigenvalues

HF

m,γrand the infimum of themth positive eigenvaluesEFm,γrforp-Laplacian with weights inLγ_{γ∈1,∞balls, whereFdenotesD,NorG, whilemis related to the nodal property of} the corresponding half-eigenfunctions or eigenfunction. The detailed definitions ofHF

m,γr andEF

m,γrare given by2.35–2.39and2.44–2.48inSection 2.

Some results on eigenvalues and half-eigenvalues are collected inSection 2. Compared with the results in 8, the characterizations on antiperiodic half-eigenvalues have been improved, see Theorems2.2and2.4. These characterizations make the definition ofHG

m,γr clearer and also easier to evaluate; seeRemark 2.5.

In Section 3, by using 1.6 and the relationship between Dirichlet, Neumann and generalized periodic eigenvaluesseeLemma 3.2, we will show that

EDm,γr ENm,γr EGm,γr mp·

Kpγ∗, p

r 1.9

for anyγ∈1,∞,m∈Nandr >0. It will also be proved that

EN_0,γr EG_0,γr 0, ∀γ∈1,∞, ∀r >0. 1.10

A natural idea to characterize HF

m,γr is to employ analogous method as done for EF

of the jumping terms involved, which is quite different from the eigenvalue case; see

Remark 3.3.

Section 4is devoted toHF

m,γ. It is possible that for some weights inLγ balls themth positive half-eigenvalue does not exist; seeRemark 2.3. So it is impossible to utilize directly the continuous dependence of half-eigenvalues in weights in weak topology or the Fr´echet differential dependence, as done in 10–12. Some more fundamental continuous results in weak topology and differentiable resultsin Lemma 2.1 will be used instead. We will first show two facts. One is the monotonicity of the half-eigenvalues on the weightsa, b. The other is the infimumHF

m,γrcan be attained by some weights for anyγ ∈ 1,∞. As consequence of these two facts, for each minimizeraγ, bγ, one sees thataγ and bγ do not overlap ifγ ∈1,∞. Moreover the extremal problem for half-eigenvalues is reduced to that for eigenvalues. Roughly speaking, for anyγ∈1,∞andr >0 we have

Hm,γF r EFm,γr mp·

Kpγ∗, p

r , ∀m∈N, ∀F∈ {D, N, G}, 1.11 H0F,γr EF0,γr 0, ∀F∈ {N, G}. 1.12

Based on some topological fact onLγ_{balls, the extremal values inL}1_{balls can be obtained by}

the limiting approachγ ↓1. Consequently1.11and1.12also hold forγ1.

2. Preliminary Results and Extremal Value Problems

Denote by φp·the scalar p-Laplacian and let x±· max{±x·,0}. Let us consider the

positive half-eigenvalues of

φp

xλatφpx−λbtφpx− 0 a.e. t∈0,1 2.1

with respect to the boundary value conditions

x0 x1 0, D

x0 x1 0, N

x0±x1 x0±x1 0, G

respectively.

Denote bycospθ,sinpθthe unique solution of the initial value problem

dx dθ −φp∗

y, dy

dθ φpx,

x0, y0 1,0. 2.2

iboth cospθand sinpθare 2πp-periodic, where

πp

2πp−11/p

psinπ/p ; 2.3

iicospθ 0 if and only ifθ πp/2mπp,m ∈ Z, and sinpθ 0 if and only if

θmπp,m∈Z;

iii|cospθ|p p−1|sinpθ|p∗≡1.

By settingφpx −yand introducing the Pr ¨ufer transformationx r2/pcospθ,y

r2/p∗_sin

pθ, the scalar equation

φp

xatφpx−btφpx− 0 a.e. t∈0,1 2.4

is transformed into the following equations forrandθ:

θAt, θ;a, b

:

⎧ ⎨ ⎩

atcospθpp−1sinpθp∗ if cospθ≥0,

btcospθpp−1sinpθp∗ if cospθ <0,

2.5

logrGt, θ;a, b

:

⎧ ⎪ ⎨ ⎪ ⎩

p

2at−1φp

cospθφp∗sinpθ if cospθ≥0,

p

2bt−1φp

cospθ

φp∗

sinpθ

if cospθ <0.

2.6

For anyϑ0 ∈R, denote byθt;ϑ0, a, b, rt;ϑ0, a, b,t∈0,1, the unique solution of2.5 2.6satisfyingθ0;ϑ0, a, b ϑ0andr0;ϑ0, a, b 1. Let

Θϑ0, a, b:θ1;ϑ0, a, b,

Rϑ0, a, b:r1;ϑ0, a, b.

2.7

For anym∈Z, denote byΣ_ma, bthe set of nonnegative half-eigenvalues of2.1 2.2for which the corresponding half-eigenfunctions have preciselymzeroes in the interval

0,1. Define

Θa, b: max ϑ0∈0,2πp

{Θϑ0, a, b−ϑ0}max

ϑ0∈R

{Θϑ0, a, b−ϑ0}, _2.8

Θa, b: min ϑ0∈0,2πp

{Θϑ0, a, b−ϑ0}min

ϑ0∈R

{Θϑ0, a, b−ϑ0}, 2.9

λL_mλL_ma, b:minλ >0|Θλa, λb mπp

, m∈N, 2.10

λRmλ R

ma, b:max

λ≥0|Θλa, λb mπp

Similar arguments as in the proof of Lemma 3.2 in8show that

λL/R_m a, b, λL/Rm a, b∈Σma, b 2.12

if only these numbers exist.

Lemma 2.1see7,8. Denote bywγthe weak topology inLγ. Then

i Θϑ, a, bis jointly continuous inϑ, a, b∈R×Lγ_{, w} γ2;

ii Θλa, λbandΘλa, λbare jointly continuous inλ, a, b∈R×Lγ_{, w} γ2, and

Θ0,0∈0, πp

, Θ0,0 0; 2.13

iii Θϑ, a, b is continuously differentiable in ϑ, a, b ∈ Lγ_{, ·}

γ2. The derivatives of

Θϑ, a, b atϑ, at a ∈ Lγ _{and atb ∈ L}γ _{(in the Fr´echet sense), denoted, respectively,}

by∂ϑΘ,∂aΘ, and∂bΘ, are

∂ϑΘϑ, a, b 1

R2_{ϑ, a, b},

∂aΘϑ, a, b Xp∈ C0⊂

Lγ_,· γ

∗

,

∂bΘϑ, a, b X−p∈ C0⊂

Lγ_,· γ

∗

,

2.14

whereC0_:C0,1,Rand

XXt Xt;ϑ, a, b: {rt;ϑ, a, b}

2/p_cosp

θt;ϑ, a, b

{r1;ϑ, a, b}2/p 2.15

is a solution of 2.4.

Givena, a1, a2, b1, b2∈ L1, writea0 ifa≥0 and

₁

0atdt >0. Writea1, b1≥a2, b2

ifa1 ≥a2andb1 ≥b2. Writea1, b1a2, b2ifa1, b1≥a2, b2and botha1t> a2tand

b1t> b2thold fortin a commonsubset of0,1of positive measure. Denote

Wγ

:{a, b|a, b∈ Lγ, a, b0,0}. 2.16

Theorem 2.2. Supposea, b∈ W1

. There hold the following results.

iAll positive Dirichlet half-eigenvalues of 2.1consist of two sequences{λD

ma, b}m∈Nand

{λD

mb, a}m∈N, whereλDma, bis the unique solution of

Θ

−πp

2 , λa, λb

−πp

2 mπp, ∀m∈N,

λD₁a, b< λD₂a, b<· · ·< λDma, b<· · ·−→ ∞.

iiAll nonnegative Neumann half-eigenvalues of 2.1 consist of two sequences {λN

ma, b}m∈Zand{λNmb, a}m∈Z, whereλNma, bis determined by

Θ0, λa, λb mπp, ∀m∈Z,

0≤λN₀ a, b< λN₁ a, b< λ₂Na, b<· · ·< λNma, b<· · ·−→ ∞.

2.18

Moreover,

λN₀ a, b>0⇐⇒a0, ₁

0

at<0. 2.19

iiiAll solutions of

Θλa, λb mπp, ∀m∈N,

Θλa, λb mπp, ∀m∈Z

2.20

are contained inΣ_ma, b. DenoteλL₀ :0; then

λL/R_m a, b, λL/Rm a, b

⊂Σ

ma, b⊂

λL_ma, b, λRma, b

, ∀m∈Z. 2.21

There hold the ordering

0<λL₁ ≤λR₁ < λL₂ ≤λR₂ <· · ·< λL_m≤λR_m<· · ·−→ ∞,

0λL0 ≤λ

R

0 < λ

L

1 ≤λ

R

1 <· · ·< λ

L m≤λ

R

m<· · · −→ ∞,

0≤λR0 < λL2 ≤λ

R

2 <· · ·< λL2m≤λ R

2m< λL2m2≤λ

R

2m2· · ·−→ ∞.

2.22

Moreover,

λR0a, b>0⇐⇒

1

0

atdt <0 or

1

0

btdt <0. 2.23

Proof. Compared with results in8, we need only prove

Σ

2m1a, b∈

λL_2m1a, b, λR2m1a, b

∀m∈Z. 2.24

Remark 2.3. The restrictiona, b∈ W1

inTheorem 2.2guarantees the existence of such

half-eigenvalues, to which the corresponding half-eigenfunction have arbitrary many zeros in

0,1. However, it is possible for some weights a, b ∈ L1_{, for example, a 0 and b 0,}

that only finite of these positive half-eigenvalues exists. We refer this to Remark 2.4 in8. In other cases, for example ifa <0 andb <0, there exist no positive half-eigenvalues. Since we are going to study the infimum of positive half-eigenvalues, if one of these half-eigenvalues, sayλD

ma, b, does not exist, we defineλDma, b ∞for simplicity.

Theorem 2.4. Supposea, b∈ L1. There hold the following results.

iIfλL_ma, b<∞for somem∈N, then

λ≥λL_ma, b, ∀λ∈Σma, b; 2.25

iiifλRma, b<∞for somem∈Z, then

λ≤λRma, b, ∀λ∈Σma, b. 2.26

Proof. One has the following steps.

Step 1. By checking the proof of Lemma 3.3 in8, results therein still hold for arbitrarya,

b∈ L1_{, that is,}

1IfΘμa, μb mπpfor someμ >0 andm∈N, then there existsδ >0 such that

Θλa, λb> mπp, ∀λ∈

μ, μδ. 2.27

2IfΘμa, μb mπpfor someμ >0 andm∈Z, then there existsδ∈0, μsuch that

Θλa, λb< mπp, ∀λ∈

μ−δ, μ. 2.28

Step 2. It follows fromStep 1that

Θλa, λb

⎧ ⎨ ⎩

< mπp if 0≤λ < λL_m,

≥mπp if λ≥λL_m

∀m∈N, 2.29

ifλL_ma, b<∞, and

Θλa, λb

⎧ ⎪ ⎨ ⎪ ⎩

≤mπp if 0≤λ < λ R m,

> mπp ifλ > λ R m

∀m∈Z 2.30

Step 3. Suppose λL_ma, b < ∞for some m ∈ N. For anyλ ∈ Σma, b, there existsϑ ∈ R

depends onλsuch that

Θϑ, λa, λb ϑmπp. 2.31

Consequently,

Θλa, λb max ϑ0∈R

{Θϑ0, λa, λb−ϑ0} ≥mπp. _2.32

It follows from2.29thatλ≥ λL_ma, b, which completes the proof ofi. Resultsiican be proved analogously by using2.30.

In the product spaceLγ_{× L}γ_{, 1≤γ≤ ∞, one can define the norm| · |} γas

|a, b|_γ :

1

0

|at|γ|bt|γdt

1/γ

, ∀a, b∈ Lγ_{× L}γ_{, γ∈1,∞,}

|a, b|_∞: lim

γ→ ∞|a, b|γ max{a∞,b∞}, ∀a, b∈ L

∞_{× L}∞_.

2.33

Givenγ∈1,∞, andr >0. We take the notations

Bγr:a, b∈ Lγ× Lγ :|a, b|γ≤r

,

B_δγa, b:a1, b1∈ Lγ× Lγ :|a1−a, b1−b|γ ≤δ

,

Sγr:a, b∈ Lγ× Lγ:|a, b|γ r

,

Sγr:

a, b∈Sγr:a≥0, b≥0,

Bγr:a∈ Lγ :aγ ≤r

,

Sγr:a∈ Lγ:a≥0, a_γr.

2.34

Now we can define the infimum of positive half-eigenvalues

Hm,γD r:inf

λDma, b:a, b∈Bγr

, ∀m∈N, 2.35

HN

m,γr:inf

λN

ma, b:a, b∈Bγr

HG

m,γr:inf

λ∈Σma, b:a, b∈Bγr

, ∀m∈N, 2.37

H0N,γr:inf

λNma, b>0 :a, b∈Bγr

, 2.38

H₀G_,γr:inf

λR0a, b>0 :a, b∈Bγr

. 2.39

Remark 2.5. i It follows fromTheorem 2.2that all the extremal values defined by 2.35–

2.39are finite.

ii Although there may exist nonvariational half-eigenvalues inΣ_ma, b cf. 13,

Theorem 2.4shows that

λL_ma, b infΣ_ma, b ∀a, b∈ L1, ∀m∈N. 2.40

Therefore2.37can be rewritten as

HG

m,γr:inf

λL_ma, b:a, b∈Bγ_{r, ∀m∈N. 2.41}

Notice that if a b, then the half-eigenvalue problem of 2.1 is equivalent to the eigenvalue problem of

φp

xλatφpx 0, a.e. t∈0,1. 2.42

Ifa0, thena, a∈ W1.Theorem 2.2shows that all positive Dirichlet eigenvalues of2.42

consist of a sequence{λDma, a}m∈N, all nonnegative Neumann eigenvalues of2.42consist of

{0}∪{λNma, a}m∈Z, while bothλLma, aandλ R

ma, aare periodic or antiperiodic eigenvalues of2.42ifmis even or odd, respectively. We take the notations

λDma:λDma, a, λNma:λNma, a,

λL_ma:λL_ma, a, λRma:λ R ma, a

2.43

andΣ_ma: Σ_ma, a.

Given γ ∈ 1,∞ and r > 0, now we can define the infimum of positive half-eigenvalues

EDm,γr:inf

λDma:a∈Bγr

, ∀m∈N, 2.44

EN

m,γr:inf

λN

ma:a∈Bγr

EGm,γr:inf{λ∈Σma:a∈Bγr}

infλL_ma:a∈Bγr, ∀m∈N,

2.46

EN_0,γr:infλmNa>0 :a∈Bγr

, 2.47

EG

0,γr:inf

λR0a>0 :a∈Bγr

. 2.48

3. Infimum of Eigenvalues with Weight in

L

_Balls

Theorem 3.1. For anyγ∈1,∞,m∈Nandr >0, one has

EDm,γr mp·

Kpγ∗, p

r . 3.1

Ifγ∈1,∞, thenED

m,γrcan be attained by some weight, called a minimizer, and each minimizer is

contained inSγr. Ifγ1, thenED

m,γrcannot be attained by any weight inBγr.

Proof. Ifa≤ 0, then2.42has no positive Dirichlet eigenvalues, that is,λD

ma ∞by our notations. Ifa 0 anda−0, then|a| aand

λDm|a|< λmDa<∞, 3.2

compare, for example,9, Theorem 3.9, see alsoLemma 4.2i. Consequently one has

ED

m,γr inf

λD

mw:w∈ Lγ, w≥0, wγ≤r

. 3.3

Now the theorem can be completed by the proof of10, Theorem 5.6; see also1.6.

Lemma 3.2. Givena∈ Lγ_{, definea}

st:astfor anys,t∈R. Then

λL_ma min s∈R

λDmas

min

s∈R

λNmas

, ∀m∈N,

λRma max_s∈R

λDmas

max

s∈R

λNmas

, ∀m∈N,

λR0a max_s∈R

λN₀ as

.

3.4

Proof. This lemma can be proved as done in14, where eigenvalues forp-Laplacain with

potentialwere studied by employing rotation number functions.

Remark 3.3. Results inLemma 3.2can be generalized to half-eigenvalues exclusively for even

Notice thata ∈ Bγ_{rif and only if a}

s ∈ Bγrfor any s ∈ R. One can obtain the following theorem immediately fromTheorem 3.1andLemma 3.2.

Theorem 3.4. There holds1.9for anyγ ∈1,∞,m∈Nandr > 0. Ifγ ∈1,∞, any extremal

value involved in1.9can be attained by some weight, and each minimizer is contained inSγr. If

γ1, none of these extremal values can be attained by any weight inBγ_r.

However, we cannot characterizeEN

0,γ andEG0,γ by usingTheorem 3.1andLemma 3.2, becauseλD

0adoes not exist for any weighta∈ Lγ. Theorem 3.5. There holds1.10for anyγ ∈1,∞andr >0.

Proof. Choose a sequence of weights

akt

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

r t∈

0,1

2 − 1

k

,

−r t∈

1 2 −

1

k,1

,

k >2. 3.5

Thenak ∈ Bγr,ak 0 and

₁

0aktdt < 0. It follows fromTheorem 2.2iithat νk :

λN

0 ak λN0 ak, ak>0, andνkis determined by

Θ0, νkak, νkak 0. 3.6

Sinceak1ak, byLemma 4.2iiione hasνk1 < νk. Letk → ∞. Then

ak−→a0 a0t

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

r t∈

0,1

2

,

−r t∈

1 2,1

,

a.e. t∈0,1, 3.7

andνk → ν≥0. ByLemma 2.1the limiting equality of3.6is

Θ0, νa0, νa0 0. 3.8

Since₀1a0tdt0, it follows fromTheorem 2.2iiagain thatν0. HenceEN_0,γr 0.

Notice that2.42has no positiveNeumann or periodiceigenvalues if the weight

a≤0. On the other hand,Theorem 2.2shows that ifa0 then

λR0a>0⇐⇒λN0 a⇐⇒

1

0

atdt <0. 3.9

CombiningLemma 3.2and the definitions in2.47and2.48, one hasEG

4. Infimum of Half-Eigenvalues with Weights in

L

Balls

4.1. Monotonicity Results of Half-Eigenvalues

Applying Fr´echet differentiability of λD

ma, b and λNma, b in weights a, b ∈ L1, some monotonicity results of eigenvalues have been obtained in8.

Lemma 4.1see8. Givenγ∈1,∞andai, bi∈ Wγ,i0,1, ifa0, b0≥a1, b1, then iλDma0, b0<≤λDma1, b1for anym∈N,

iiλN

ma0, b0<≤λNma1, b1for anym∈N,

iiiif moreover₀1a0tdt <0, then0< λN₀ a0, b0<≤λN₀ a1, b1.

By checking the proof in8one sees that the restrictiona, b∈ Wγcan be weakened.

In fact this restriction was used to guarantee the existence of λD

ma, b and λNma, b for arbitrary largem∈N. Employing the boundary value conditions and Fr´echet differentiability ofΘϑ, a, bin weightsLemma 2.1iii, one can prove the following lemma.

Lemma 4.2. Givenai, bi∈ Lγ,i0,1,γ∈1,∞. Supposea0, b0≥a1, b1, then iifλD

ma1, b1<∞for somem∈N, thenλDma0, b0<≤λDma1, b1; iiifλNma1, b1<∞for somem∈N, thenλNma0, b0<≤λNma1, b1; iiiifa10and

1

0a0tdt <0, then0< λN0 a0, b0<≤λN0 a1, b1.

Due to the so-called parametric resonance15or the so-called coexistence of periodic and antiperiodic eigenvalues 16, half-eigenvalues λL_ma, b and λRma, b, m ∈ N, are not continuously differentiable in a, b in general. This add difficulty to the study of monotonicity ofλL_ma, band λRma, bina, b. Even if we go back to 2.10and 2.11by which λL_ma, b and λRma, b are determined, we find that Θa, b and Θa, b are not differentiable. Finally, we have to resort to the comparison result on Θϑ, a, b. It can be proved that

a0, b0≥a1, b1 ⇒Θϑ, a0, b0<≤Θϑ, a1, b1, ∀ϑ∈R. 4.1

New difficulty occurs since the weights are sign-changing, that is, we cannot conclude from

a0, b0≥a1, b1that

Θϑ, λa0, λb0<≤ Θϑ, λa1, λb1, ∀ϑ∈R, ∀λ >0. 4.2

So we can only obtain some weaker monotonicity results for generalized periodic half-eigenvalues.

Lemma 4.3. Givena, b, ai, bi∈ Lγ,i0,1,γ∈1,∞. There hold the following results.

iIfλL_ma, b<∞for somem∈N, thenλL_ma, b≤λLma, b.

iiiIfa0, b0≥ a1, b1≥0,0andλL_ma1, b1<∞for somem∈N, thenλL_ma0, b0< ≤λL_ma1, b1.

ivIfa0, b0≥ a1, b1≥0,0andλ

R

ma1, b1<∞for somem∈N, thenλ

R

ma0, b0< ≤λRma1, b1.

Proof. Givena, b∈ Lγ_{. For anyλ≥0, one hasλa}

, λb≥λa, λb. It follows from4.1that

Θϑ, λa, λb≥Θϑ, λa, λb, ∀ϑ∈R, ∀λ≥0. 4.3

Notice thatΘϑ, a, b−ϑis 2πp-periodic inϑ∈R. Combining the definition ofΘa, bin2.8, one has

Θλa, λb≥Θλa, λb, ∀λ≥0. 4.4

ByLemma 2.1ii,Θ0·a,0·a∈0, πpandΘλa, λbis continuous inλ∈R. As functions ofλ∈0,∞, the smooth curveΘλa, λblies aboveΘλa, λb. By the definition ofλLma, b in2.10, ifλL_ma, b<∞for somem∈NthenλL_ma, b≤λLma, b. Thus the proof ofiis completed.

Resultsii,iii, andivcan be proved analogously.

4.2. The Infimum in

L

_γ

_∈

₁

_,

_∞

_{Balls Can Be Attained}

Givena, b∈ Lγ_{,γ∈1,∞,m∈N, andτ >0, one has}

λDmτa, τb

λD ma, b

τ , λ

N mτa, τb

λN ma, b

τ , λ

L mτa, τb

λL_ma, b

τ . 4.5

Hence

HF m,γr1

HF m,γr2

r2

r1, ∀r1, r2∈

0,∞, ∀γ∈1,∞, ∀m∈N, 4.6

whereFdenotesD,NorG.

Theorem 4.4. Givenγ∈1,∞,r >0,m∈NandF∈ {D, N, G}. ThenHF

m,γr>0and it can be

attained by some weights. Moreover, any minimizeraF, bF∈Sγr.

Proof. We only prove for the caseF G, other cases can be proved analogously. There exists

a sequence of weightsan, bn∈Bγr,n∈N, such that

By the definition ofλL_min2.10, there existϑn∈0,2πp,n∈N, such that

Θϑn, νnan, νnbn−ϑnmπp,

Θϑ, νnan, νnbn−ϑ≤mπp, ∀ϑ∈

0,2πp

.

4.8

Notice thatBγ_r⊂Lγ_{× L}γ_{,| · |}

γ,γ∈1,∞, is sequentially compact inLγ, wγ2. Passing to a subsequence, we may assumeϑn → ϑ0and

an, bn−→a0, b0∈Bγr, in

Lγ_{, w} γ

2

. 4.9

Letn → ∞in4.8. ByLemma 2.1i, one has

Θϑ0, ν0a0, ν0b0−ϑ0mπp,

Θϑ, ν0a0, ν0b0−ϑ≤mπp, ∀ϑ∈

0,2πp

. 4.10

ThusΘν0a0, ν0b0 mπp. It follows from2.10and2.13thatr:|a0, b0|γ >0 and

ν0≥λL_ma0, b0>0. 4.11

On the other hand, sincea0, b0∈Bγr, one has

λL_ma0, b0≥inf

λL_ma, b:a, b∈Bγrν0 Hm,γG r. 4.12

To complete the proof of the lemma, it suffices to showrr. If this is false, then 0<r < r and

Hm,γG r≤λLma0, b0 Hm,γG r, 4.13

which contradicts4.6.

4.3. Minimizers and Infimum in

L

γ

∈

1

,

∞

Balls

We have proved that for anym∈Nthe infimumHm,γF rcan be obtained if onlyγ ∈1,∞. In the following we will study the property of the minimizers.

Theorem 4.5. Givenγ ∈1,∞,r > 0,m ∈ N, andF ∈ {D, N, G}, ifa, bis the minimizer of

HF

mr, thena, b∈S γ

r. Moreover,aandbdo not overlap, that is,

at 0 a.e. t∈Jb :{t|bt>0},

bt 0 a.e. t∈Ja:{t|at>0}.

Proof. We only prove for the caseFG, other cases can be proved analogously.

Step 1Nonnegative. Supposeat< 0 a.e.t ∈J0 ⊂0,1, whereJ0 is of positive measure.

Let

a1t

⎧ ⎪ ⎨ ⎪ ⎩

|at|

2 , ift∈J0,

at, otherwise,

b1t

⎧ ⎨ ⎩

bt ε, ift∈J0,

bt, otherwise,

4.15

whereεεγ>0 can be chosen arbitrary small such that|a1, b1|γ ≤r. Thena1, b1a, b

and it follows fromLemma 4.3iiithat

λL_ma1, b1< λL_ma, b Hm,γF r, 4.16

which is in contradiction to the definition ofHm,γF r. Thusais nonnegative. Analogouslyb is also nonnegative. Then it follows fromTheorem 4.4thata, b∈Sγr.

Step 2Nonoverlap. Ifaandboverlap, thena, b0,0, that is, there existsJ0⊂0,1with

positive measure such that

at>0, bt> a.e. t∈J0⊂0,1. 4.17

LetXtbe the half-eigenfunction corresponding toν :λL_ma, b. Without loss of generality, we may assume that

Xt>0 a.e. t∈J0⊂J0 4.18

for someJ0with positive measure. Let

a1t at, b1t

⎧ ⎨ ⎩

0 ift∈J0,

bt, otherwise. 4.19

Thenr:|a1, b1|γ <|a, b|γ rand

φp

ThereforeλL_ma1, b1≤νHm,γG r. It follows that

Hm,γG r≤λLma1, b1≤Hm,γG r, 4.21

which contradicts4.6. Thusaandbdo not overlap.

Corollary 4.6. Givenγ ∈1,∞,r > 0,m∈ N, andF ∈ {D, N, G}, ifa, bis the minimizer of

HF

mrandXis the corresponding half-eigenfunction, then

Xt>0 a.e. t∈Ja:{at>0}, 4.22

Xt<0 a.e. t∈Jb :{bt>0}. 4.23

Proof. If4.23does not hold. Then there existJ0⊂Jbsuch thatJ0is of positive measure and

Xt>0 a.e. t∈J0. 4.24

Definea1andb1as in4.19. A contradiction can be obtained by similar arguments as in the

proof ofTheorem 4.5. Thus4.23holds. One can prove4.22analogously.

Theorem 4.7. Givenr >0, then1.11holds for anyγ∈1,∞and1.12holds for anyγ∈1,∞.

Proof. By the monotonicity results in Lemmas 4.2and 4.3, HF

m,∞rcan be attained by the

minimizera, b r, rfor anyF∈ {D, N, G}andm∈N. Thus1.11holds forγ∞. Now we prove1.11forγ ∈ 1,∞. Supposea0, b0is the minimizer ofν :HmFr andX is the corresponding half-eigenfunction. Letw0 a0b0. ByTheorem 4.5,a0, b0 ∈

Sγranda0andb0do not overlap, thus

w0γ|a0, b0|γr. 4.25

CombiningCorollary 4.6, one has

φp

Xνw0tφpX 0. 4.26

Hence

Hm,γF r ν≥EFm,γr. 4.27

On the other hand, for anyw∈Bγ_{randλ∈R, one has|w}

, w−|γ wγand

φp

xλwtφpx 0

⇐⇒φp

xλwtφpx λw−tφpx− 0.

Take the notationsλG

ma, b:λLma, bandλGma:λLmafor anya, b ∈ Lγ. ThenλFmw

λF

mw, w−for anyF∈ {D, N, G}and

EFm,γr inf

λFmw:w∈Bγr

infλFmw, w−:w∈Bγr

≥infλF

ma, b:a, bγ∈Bγr

HF

m,γr.

4.29

Therefore1.11is proved forγ∈1,∞.

One can obtain 1.12 for any γ ∈ 1,∞ by the fact that the half-eigenfunction corresponding toλN

0 a, borλ

R

0a, bdoes not change its sign.

4.4. The Infimum in

L

_Balls

We cannot handle extremal problem inL1_{balls in the same way as done forL}γ_{γ > 1case,} becauseL1_{balls are not sequentially compact even in the sense of weak topology.}

Lemma 4.8. Givenγ∈1,∞,r >0, andm∈N, there hold the following properties.

iIfλL_ma0, b0<∞, then there existsδ >0such that

λ_mLa, b<∞, ∀a, b∈Bγ_δa0, b0. 4.30

iiIfλD/Nm a0, b0<∞, then there existsδ >0such that

λD/N

m a, b<∞, ∀a, b∈B γ

δa0, b0. 4.31

Proof. iSupposeλL_ma0, b0<∞. ByTheorem 2.4ithere existε >0 andν > λL_ma0, b0such

that

Θνa0, νb0> mπp2ε. 4.32

By the definition ofΘin2.8, there isϑ0∈Rsuch that

Θϑ0, νa0, νb0−ϑ0> mπp2ε. 4.33

ByLemma 2.1i, that is, the continuous dependence ofΘϑ, a, bin the weightsa, b, there existsδ >0 such that

Therefore,

Θνa, νb> mπpε, ∀a, b∈Bγ_δa0, b0. 4.35

We conclude from2.29that

λ_mLa, b< ν <∞, ∀a, b∈Bγ_δa0, b0, 4.36

completing the proof ofi.

iiSupposeμ:λD/Nm a0, b0<∞. LetX be the half-eigenfunction corresponding to

μ. ThenXsatisfies Dirichlet or Neumann boundary value conditions and

φp

Xμa0tφpX−μb0φpX− 0. 4.37

Multiplying4.37byXand integrating over0,1, one has

₁

0

a0Xpb0X−p

dt 1 μ

₁

0

_Xp

dt >0. 4.38

LetϑD_−π

p/2 andϑN0. ByLemma 2.1iii, one has

d dλΘ

ϑD/N, λa, λb

λμ

1

0

a0Xpb0Xp−

dt >0. 4.39

Notice thatΘϑD/N_{, μa}

0, μb0 ϑD/N mπp. Then there existε >0 andν > μλD/Nm a0, b0

such that

ΘϑD/N, νa0, νb0

> ϑD/Nmπp2ε. 4.40

ByLemma 2.1i, there existsδ >0 such that for anya, b∈Bγ_δa0, b0, one has

ΘϑD/N, νa, νb> ϑD/Nmπpε, 4.41

and henceλD/Nm a, b<∞, completing the proof ofii.

As a function ofα, Kα, pis continuous inα∈1,∞. Explicit formula of Kα, pcan be found in17, Theorem 4.1. For instance, Kp, p πppand K∞, p 2p.

Theorem 4.9. For anyr >0,1.11holds forγ1, that is,

H_m,F₁r E_m,F₁r 2m

p

Proof. By Theorem 4.7, 1.11 holds for any γ ∈ 1,∞. As the Sobolev constant Kα, p is continuous inα∈1,∞, one has limγ↓1Hm,γF r 2mp/r.

Our first aim is to prove

H_m,F₁r≥lim γ↓1 H

F m,γr

2mp

r . 4.43

Anya0, b0∈B1rcan be approximated by elements inBγr,γ >1, in the sense that there

existsaγ, bγ∈Bγrsuch that

lim γ↓1

_{aγ, bγ}

−a0, b0₁0. _4.44

For instance, one can choose

aγr1/γ

∗

|a0t|1/γ·signa0t, bγ r1/γ

∗

|b0t|1/γ·signb0t, 4.45

compare, for example,11, Lemma 2.1. For simplicity, we take the notation

λGma, b:λLma, b. 4.46

GivenmandF. SupposeλF

ma0, b0<∞. ByLemma 4.8, there existsδ >0 such thatλFma, b exists for anya, b ∈ Bγ_δa0, b0. We can assume that λFmaγ, bγexists for any γ ∈ 1,∞ due to4.44. Furthermore, byLemma 2.1iii, one can prove thatλF

ma, bis continuously differentiable in a, b ∈ B_δγa0, b0in | · |1 topology. In particular,λFma, bis continuous in

a, b∈B_δγa0, b0in| · |1topology. Thus we obtain

λFma0, b0 lim

γ↓1 λ

F m

aγ, bγ

≥lim

γ↓1H

F

m,γr, 4.47

and therefore,

H_m,F₁r infλFma0, b0|a0, b0∈B1r

≥lim

γ↓1 Hm,γr

2mp

r . 4.48

On the other hand, we prove

H_m,F₁r≤ 2m

p

r . 4.49

Notice thatBγ₂1/γ−1_r⊂B1_{rfor allγ >1 and allr >0, because}

|a, b|₁a₁b₁≤ a_γb_γ ≤21−1/γ_aγ γbγγ

1/γ

for anya, b∈ Lγ_{× L}γ_{. Thus we obtain}

H_m,F₁r≤Hm,γF

21−1/γrmp·K

pγ∗, p

21−1/γ_r . 4.51

Inequality 4.49 follows immediately by letting γ ↓ 1. The desired result is proved by combining4.43and4.49.

Acknowledgment

This work was supported by the National Natural Science Foundation of ChinaGrant no. 10901089.

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