Global Structure of Nodal Solutions for

Boundary Value Problems

Volume 2011, Article ID 715836,12pages doi:10.1155/2011/715836

Research Article

Global Structure of Nodal Solutions for

Second-Order m-Point Boundary Value Problems with Superlinear Nonlinearities

Yulian An

Department of Mathematics, Shanghai Institute of Technology, Shanghai 200235, China

Correspondence should be addressed to Yulian An,an [email protected] Received 8 May 2010; Revised 1 August 2010; Accepted 23 September 2010 Academic Editor: Feliz Manuel Minh ´os

Copyrightq2011 Yulian An. 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.

We consider the nonlinear eigenvalue problemsuλfu 0, 0 < t < 1,u0 0,u1 _m−2

i1 αiuηi, wherem ≥ 3,ηi ∈ 0,1,andαi > 0 fori 1, . . . , m−2,with_m−2

i1 αi < 1,and f ∈C¹R\{0},R∩CR,Rsatisfiesfss > 0 fors /0, andf0 ∞, wheref0lim_{|s| →}0fs/s.

We investigate the global structure of nodal solutions by using the Rabinowitz’s global bifurcation theorem.

1. Introduction

We study the global structure of nodal solutions of the problem

uλfu 0, t∈0,1, 1.1

u0 0, u1 ^m−2

i1

αiu ηi

. 1.2

Here m ≥ 3, ηi ∈ 0,1,and αi > 0 fori 1, . . . , m−2 with_m−2

i1 αi < 1; λis a positive parameter, andf∈C¹R\ {0},R∩CR,R.

In the case thatf0∈0,∞,the global structure of nodal solutions of nonlinear second- order m-point eigenvalue problems 1.1, 1.2 have been extensively studied; see 1–5 and the references therein. However, relatively little is known about the global structure of solutions in the case thatf0∞,and few global results were found in the available literature when f0 ∞ f_∞. The likely reason is that the global bifurcation techniques cannot be

used directly in the case. On the other hand, whenm-point boundary value condition1.2 is concerned, the discussion is more difficult since the problem is nonsymmetric and the corresponding operator is disconjugate. In6, we discussed the global structure of positive solutions of1.1,1.2withf0 ∞.However, to the best of our knowledge, there is no paper to discuss the global structure of nodal solutions of1.1,1.2withf0∞.

In this paper, we obtain a complete description of the global structure of nodal solutions of1.1,1.2under the following assumptions:

A1αi>0 fori1, . . . , m−2,with 0<_m−2

i1 αi<1;

A2f ∈C¹R\ {0},R∩CR,Rsatisfiesfss >0 fors /0;

A3f0:lim_{|s| →0}fs/s∞;

A4f∞:lim_{|s| → ∞}fs/s∈0,∞.

LetY C0,1with the norm

u_∞ max

t∈0,1|ut|. 1.3

Let

X

u∈C¹0,1|u0 0, u1 ^m−2

i1

αiu ηi

,

E

u∈C²0,1|u0 0, u1 ^m−2

i1

αiu ηi

1.4

with the norm

u_Xmax

u_∞, u

∞

, umax

u_∞, u

∞, u

∞

, 1.5

respectively. DefineL:E → Y by setting

Lu:−u, u∈E. 1.6

ThenLhas a bounded inverseL⁻¹ :Y → Eand the restriction ofL⁻¹toX,that is,L⁻¹:X → X is a compact and continuous operator; see1,2,6.

For anyC¹ function u, if ux0 0, then x0 is a simple zero of uif ux0/0.For any integerk ≥ 1 and anyν ∈ {,−},define setsS^ν_k, T_k^ν ⊂ C²0,1consisting of functions u∈C²0,1satisfying the following conditions:

S^ν_k:iu0 0, νu0>0,

iiuhas only simple zeros in0,1and has exactlyk−1 zeros in0,1;

T_k^ν:iu0 0, νu0>0 andu1/0,

iiuhas only simple zeros in0,1and has exactlykzeros in0,1, iiiuhas a zero strictly between each two consecutive zeros ofu.

Remark 1.1. Obviously, ifu ∈ T_k^ν,then u ∈ S^ν_k oru ∈ S^ν_k1.The setsT_k^ν are open inEand disjoint.

Remark 1.2. The nodal properties of solutions of nonlinear Sturm-Liouville problems with separated boundary conditions are usually described in terms of sets similar to S^ν_k; see 7. However, Rynne1stated thatT_k^ν are more appropriate than S^ν_k when the multipoint boundary condition1.2is considered.

Next, we consider the eigenvalues of the linear problem

Luλu, u∈E. 1.7

We call the set of eigenvalues of1.7the spectrum ofLand denote it byσL. The following lemmas or similar results can be found in1–3.

Lemma 1.3. LetA1hold. The spectrumσLconsists of a strictly increasing positive sequence of eigenvaluesλk, k1,2, . . . ,with corresponding eigenfunctionsϕkx sin

λk x.In addition, ilimk→ ∞λk∞;

iiϕk∈T_k,for eachk≥1,andϕ1is strictly positive on0,1.

We can regard the inverse operator L⁻¹ : Y → E as an operator L⁻¹ : Y → Y. In this setting, eachλk, k 1,2, . . . ,is a characteristic value ofL⁻¹,with algebraic multiplicity defined to be dim_∞

j1NI−λkL^−1j,whereNdenotes null-space andIis the identity on Y.

Lemma 1.4. LetA1hold. For each k ≥ 1, the algebraic multiplicity of the characteristic value λk, k1,2, . . . ,ofL⁻¹:Y → Y is equal to 1.

LetER×Eunder the product topology. As in7, we add the points{λ,∞|λ∈R}

to our spaceE.LetΦ^ν_k R×T_k^ν.LetΣ^ν_kdenote the closure of set of those solutions of1.1, 1.2which belong toΦ^ν_k.The main results of this paper are the following.

Theorem 1.5. Let (A1)–(A4) hold.

aIff∞0,then there exists a subcontinuumC^ν_kofΣ^ν_kwith0,0∈ C^ν_kand

Proj_RC^ν_k 0,∞. 1.8

bIff∞∈0,∞,then there exists a subcontinuumC^ν_kofΣ^ν_kwith 0,0∈ C^ν_k, Proj_RC^ν_k⊆

0, λ1

f_∞

. 1.9

cIff∞ ∞,then there exists a subcontinuumC^ν_kofΣ^ν_k with 0,0 ∈ C^ν_k, Proj_RC^ν_k is a bounded closed interval, andC^ν_kapproaches0,∞asu → ∞.

Theorem 1.6. Let (A1)–(A4) hold.

aIff_∞0, then1.1,1.2has at least one solution inT_k^νfor anyλ∈0,∞.

bIff∞∈0,∞, then1.1,1.2has at least one solution inT_k^νfor anyλ∈0, λ1/f∞. cIff∞ ∞,then there existsλ∗ >0 such that1.1,1.2has at least two solutions inT_k^ν

for anyλ∈0, λ_∗.

We will develop a bifurcation approach to treat the case f0 ∞. Crucial to this approach is to construct a sequence of functions{fⁿ} which is asymptotic linear at 0 and satisfies

fⁿ−→f, fⁿ

0−→ ∞. 1.10

By means of the corresponding auxiliary equations, we obtain a sequence of unbounded components{C_k^νn}via Rabinowitz’s global bifurcation theorem8, and this enables us to find unbounded componentsC^ν_ksatisfying

0,0∈ C^ν_k⊂lim supC^νn_k . 1.11

The rest of the paper is organized as follows. Section 2 contains some preliminary propositions. In Section 3, we use the global bifurcation theorems to analyse the global behavior of the components of nodal solutions of1.1,1.2.

2. Preliminaries

Definition 2.1see9. LetW be a Banach space and{Cn |n1,2, . . .}a family of subsets of W.Then the superior limitDof{Cn}is defined by

D:lim sup

n→ ∞ Cn{x∈W| ∃{ni} ⊂Nandxni ∈Cni, such thatxni −→x}. 2.1 Lemma 2.2see9. Each connected subset of metric spaceW is contained in a component, and each connected component ofWis closed.

Lemma 2.3see6. Assume that

ithere existzn∈Cn n1,2, . . .andz^∗∈W,such thatzn → z^∗; iirn ∞, wherernsup{x |x∈Cn};

iiifor allR >0, _∞

n1Cn∩BRis a relative compact set ofW, where

BR{x∈W| x ≤R}. 2.2

Then there exists an unbounded connected componentCinDandz^∗∈ C.

Define the mapTλ:Y → Eby

Tλut λ ₁

0

Ht, sfusds, 2.3

where

Ht, s Gt, s _m−2

i1 αiG ηi, s 1−_m−2

i1 αiηi

t, Gt, s

⎧⎨

⎩

1−ts, 0≤s≤t≤1,

t1−s, 0≤t≤s≤1. 2.4

It is easy to verify that the following lemma holds.

Lemma 2.4. Assume that (A1)-(A2) hold. ThenTλ:Y → Eis completely continuous.

Forr >0, let

Ωr {u∈Y | u_∞< r}. 2.5

Lemma 2.5. Let (A1)-(A2) hold. Ifu∈∂Ωr, r >0, then

Tλu_∞≤λMr

1

_m−2

i1 αi

1−_m−2

i1 αiηi 1 0

Gs, sds, 2.6

whereMr 1max_0≤|s|≤r{|fs|}.

Proof. The proof is similar to that of Lemma 3.5 in6; we omit it.

Lemma 2.6. Let (A1)-(A2) hold, and {μl, yl} ⊂ Φ^ν_k is a sequence of solutions of 1.1,1.2.

Assume thatμl≤C0for some constantC0>0, and lim_{l→ ∞}yl∞.Then

l→ ∞lim yl

∞∞. 2.7

Proof. From the relation ylt μl

₁

0Ht, sfylsds, we conclude that y₁t μl

₁

0 Htt, sfylsds.Then y_l

∞≤C0

1

_m−2

i1 αi

1−_m−2

i1 αiηi 1 0

f

ylsds, 2.8

which implies that{y_l∞}is bounded whenever{yl_∞}is bounded.

3. Proof of the Main Results

For eachn∈N,definefⁿs:R → Rby

fⁿs

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

fs, s∈

1 n,∞

∪

−∞,−1 n

,

nf 1

n

s, s∈

−1 n,1

n

.

3.1

Thenfⁿ∈CR,R∩C¹R\ {±1/n},Rwith

fⁿss >0, ∀s /0, fⁿ

0nf

1 n

. 3.2

ByA3, it follows that

n→ ∞lim

fⁿ

0∞. 3.3

Now let us consider the auxiliary family of the equations

uλfⁿu 0, t∈0,1, 3.4

u0 0, u1 ^m−2

i1

αiu ηi

. 3.5

Lemma 3.1see1, Proposition 4.1. Let (A1), (A2) hold. Ifμ, u∈Eis a nontrivial solution of 3.4,3.5, thenu∈T_k^νfor somek, ν.

Letζⁿ∈CR,Rbe such that

fⁿu

fⁿ

0uζⁿu nf 1

n

uζⁿu. 3.6

Note that

|s| →lim0

ζⁿs

s 0. 3.7

Let us consider

Lu−λ fⁿ

0uλζⁿu 3.8

as a bifurcation problem from the trivial solutionu≡0.

Equation3.8can be converted to the equivalent equation

ut ₁

0

Ht, s λ

fⁿ

0us λζⁿus

ds

:λL⁻¹ fⁿ

0u·

t λL⁻¹

ζⁿu·

t.

3.9

Further we note thatL⁻¹ζⁿuouforunear 0 inE.

The results of Rabinowitz8for3.8can be stated as follows. For each integerk ≥ 1, ν ∈ {,−}, there exists a continuum{C^νn_k }of solutions of3.8joiningλk/fⁿ₀,0to infinity inE.Moreover,{C^νn_k } \ {λk/fⁿ ₀,0} ⊂Φ^ν_k.

Proof of Theorem1.5. Let us verify that{C^νn_k }satisfies all of the conditions of Lemma2.3.

Since

nlim→ ∞

λk

fⁿ

0

lim

n→ ∞

λk

nf1/n0, 3.10

conditioniin Lemma2.3is satisfied withz^∗ 0,0. Obviously

rnsup

λ y | λ, y

∈ C^νn_k ∞, 3.11

and accordingly,iiholds.iiican be deduced directly from the Arzela-Ascoli Theorem and the definition of fⁿ. Therefore, the superior limit of {C^νn_k }, D^ν_k, contains an unbounded connected componentC^ν_kwith0,0∈ C^ν_k.

From the conditionA2, applying Lemma2.2withp2 in10, we can show that the initial value problem

vλfv 0, t∈0,1,

vt0 0, vt0 β 3.12

has a unique solution on 0,1 for everyt0 ∈ 0,1 and β ∈ R. Therefore, any nontrivial solutionuof1.1,1.2has only simple zeros in0,1andu0/0. Meanwhile,A1implies thatu1/0 1, proposition 4.1. SinceC^νn_k ⊂ Φ^ν_k, we conclude thatC^ν_k ⊂ Φ^ν_k. Moreover, C^ν_k⊂Σ^ν_kby1.1and1.2.

We divide the proof into three cases.

Case 1f_∞0. In this case, we show that Proj_RC^ν_k 0,∞.

Assume on the contrary that sup

λ|λ, u∈ C^ν_k

<∞, 3.13

then there exists a sequence{μl, yl} ⊂ C^ν_ksuch that

llim→ ∞ yl ∞, μl≤C0, 3.14

for some positive constantC0depending not onl. From Lemma2.6, we have

l→ ∞lim yl _∞∞. 3.15

Setvlt ylt/yl_∞.Thenvl_∞1.Now, choosing a subsequence and relabelling if necessary, it follows that there existsμ∗, v∗∈0, C0×Ewith

v∗_∞1, 3.16

such that

llim→ ∞

μl, vl

μ∗, v∗

, inR×E. 3.17

Since lim_{|u| → ∞}fu/u0, we can show that

llim→ ∞

f ylt yl

∞

0. 3.18

The proof is similar to that of the step 1 of Theorem 1 in7; we omit it. So, we obtain

v_∗t μ∗·00, t∈0,1, 3.19

v_∗0 0, v_∗1 ^m−2

i1

αiv_∗ ηi

, 3.20

and subsequently,v_∗t≡0 fort∈0,1. This contradicts3.16. Therefore sup

λ| λ, y

∈ C^ν_k

∞. 3.21 Case 2f∞ ∈ 0,∞. In this case, we can show easily thatCjoins0,0withλk/f∞,∞by using the same method used to prove Theorem 5.1 in2.

Case 3f∞∞. In this case, we show thatC^ν_kjoins0,0with0,∞.

Let{μl, yl} ⊂ C^ν_kbe such that

μl yl −→ ∞, l−→ ∞. 3.22

If{yl}is bounded, say,yl ≤ M1, for someM1 depending not onl, then we may assume that

llim→ ∞μl∞. 3.23

Taking subsequences again if necessary, we still denote{μl, yl}such that{yl} ⊂T_k^ν∩S^ν_k. If {yl} ⊂T_k^ν∩S^ν_k1, all the following proofs are similar.

Let

0τ_l⁰< τ_l¹ <· · ·< τ_l^k−1 3.24 denote the zeros ofyl in0,1. Then, after taking a subsequence if necessary, liml→ ∞τ_l^j : τ_∞^j , j ∈ {0,1, . . . , k−1}. Clearly,τ_∞⁰ 0. Setτ_∞^k 1. We can choose at least one subinterval τ∞^j , τ∞^j1 I∞^j which is of length at least 1/kfor somej ∈ {0,1, . . . , k−1}. Then, for this j, τ_l^j1−τ_l^j>3/4kiflis large enough. Putτ_l^j, τ_l^j1I_l^j.

Obviously, for the above given k, νand j, ylthave the same sign on I_l^j for alll.

Without loss of generality, we assume that

ylt>0, t∈I_l^j. 3.25

Moreover, we have

max

t∈I_l^j |ult| ≤M1. 3.26

Combining this with the fact f

ylt ylt ≥inf

!fs

s |0< s≤M1

"

>0, t∈

τ_l^j, τ_l^j1

, 3.27

and using the relation

y_lt μl

f ylt

ylt ylt 0, t∈

τ_l^j, τ_l^j1

, 3.28

we deduce thatylmust change its sign onτ_l^j, τ_l^j1iflis large enough. This is a contradiction.

Hence{yl}is unbounded. From Lemma2.6, we have that

l→ ∞lim yl

∞∞. 3.29

Note that{μl, yl}satisfies the autonomous equation y_lμlf

yl

0, t∈0,1. 3.30

We see that yl consists of a sequence of positive and negative bumps, together with a truncated bump at the right end of the interval0,1,with the following propertiesignoring the truncated bump see,1:

iall the positive resp., negative bumps have the same shape the shapes of the positive and negative bumps may be different;

iieach bump contains a single zero ofy_l, and there is exactly one zero ofylbetween consecutive zeros ofy_l;

iiiall the positivenegativebumps attain the same maximumminimumvalue.

Armed with this information on the shape ofyl,it is easy to show that for the above givenI_l^j, {yl_I^j

l,∞:max_I^j

lylt}^∞_l1is an unbounded sequence. That is

llim→ ∞ yl

I_l^j,∞∞. 3.31

Sinceylis concave onI_l^j, for anyσ >0 small enough, ylt≥σ yl

I^j_l,∞, ∀t∈

τ_l^jσ, τ_l^j1−σ

. 3.32

This together with3.31implies that there exist constantsα, βwithα, β ⊂I_∞^j , such that

llim→ ∞ylt ∞, uniformly fort∈# α, β$

. 3.33

Hence, we have

llim→ ∞

f ylt

ylt ∞, uniformly fort∈# α, β$

. 3.34

Now, we show that liml→ ∞μl0.

Suppose on the contrary that, choosing a subsequence and relabeling if necessary,μl≥ b0for some constantb0>0. This implies that

llim→ ∞μl

f ylt

ylt ∞, uniformly fort∈# α, β$

. 3.35

From 3.28we obtain that yl must change its sign onα, βif l is large enough. This is a contradiction. Therefore liml→ ∞μl0.

Proof of Theorem1.6. a and b are immediate consequence of Theorem 1.5a and b, respectively.

To provec, we rewrite1.1,1.2to

uλ ₁

0

Ht, sfusdsTλut. 3.36

By Lemma2.5, for everyr >0 andu∈∂Ωr,

Tλu_∞≤λMr

1

_m−2

i1 αi

1−_m−2

i1 αiηi 1 0

Gs, sds, 3.37

whereMr 1max_0≤|s|≤r{|fs|}.

Letλr >0 be such that

λrMr

1

_m−2

i1 αi

1−_m−2

i1 αiηi 1 0

Gs, sdsr. 3.38

Then forλ∈0, λrandu∈∂Ωr,

Tλu_∞<u_∞. 3.39

This means that

Σ^ν_k∩ {λ, u∈0,∞×E|0< λ < λr, u∈E:u_∞r}∅. 3.40

By Lemma2.6and Theorem1.5, it follows thatC^ν_kis also an unbounded component joining 0,0and0,∞in0,∞×Y. Thus,3.40implies that forλ∈0, λr,1.1,1.2has at least two solutions inT_k^ν.

Acknowledgments

The author is very grateful to the anonymous referees for their valuable suggestions. This paper was supported by NSFCno.10671158, 11YZ225, YJ2009-16no.A06/1020K096019.

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