Global Behavior of the Components for the Second Order -Point Boundary Value Problems

Volume 2008, Article ID 254593,10pages doi:10.1155/2008/254593

Research Article

Global Behavior of the Components for the Second

Order

m

-Point Boundary Value Problems

Yulian An1, 2 _{and Ruyun Ma}1

1_{Department of Mathematics, Northwest Normal University, Lanzhou 730070, China} 2_{Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China}

Correspondence should be addressed to Yulian An,an [email protected]

Received 9 October 2007; Accepted 16 December 2007

Recommended by Kanishka Perera

We consider the nonlinear eigenvalue problems u rfu 0, 0 < t < 1, u0 0, u1

m−2

i1 αiuη_i, wherem ≥ 3,η_i ∈ 0,1, andαi > 0 fori 1, . . . , m−2, withim−21 αi < 1;r ∈ R;

f ∈ C1_{R,R. There exist two constantss}

2 < 0 < s1 such that fs1 fs2 f0 0 and

f0 :limu→0fu/u∈0,∞,f∞:lim|u|→∞fu/u∈0,∞. Using the global bifurcation

tech-niques, we study the global behavior of the components of nodal solutions of the above problems.

Copyrightq2008 Y. An and R. Ma. 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.

1. Introduction

In 1, Ma and Thompson were concerned with determining values of real parameter r, for which there exist nodal solutions of the boundary value problems:

uratfu 0, 0< t <1,

u0 u1 0, 1.1

whereaandfsatisfy the following assumptions:

H1 f ∈CR,Rwithsfs>0 fors / 0;

H2 there existf0, f∞∈0,∞such that

f0 lim |s|→0

fs

s , f∞|slim|→∞ fs

s ; 1.2

H3 a:0,1→0,∞is continuous andat/≡0 on any subinterval of0,1.

Theorem 1.1. Let (H1), (H2), and (H3) hold. Assume that for somek∈N, either

λk

f∞ < r < λk

f0 1.3

or

λk

f0

< r < λk

f∞. 1.4

Then1.1have two solutionsu_k andu−_k such thatu_k has exactlyk−1zeros in0,1and is positive near0, andu−_k has exactlyk−1zeros in0,1and is negative near0. In1.3and1.4,λk is thekth eigenvalue of

ϕλatϕ0, 0< t <1, ϕ0 ϕ1 0. 1.5

Recently, Ma2extended this result and studied the global behavior of the components of nodal solutions of1.1under the following conditions:

H1 f ∈CR,Rand there exist two constantss2<0< s1, such thatfs1 fs2 f0

0 andsfs>0 fors∈R\ {0, s1, s2}; H4 f satisfies Lipschitz condition ins2, s1.

Using Rabinowitz global bifurcation theorem, Ma established the following theorem.

Theorem 1.2. LetH1, (H2), (H3), and (H4) hold. Assume that for somek∈N,

λk

f∞ < λk

f0. 1.6

Then

iifr ∈λk/f∞, λk/f0, then1.1have at least two solutionsu±_k,∞, such thatu_k,∞has exactly

k−1 zeros in0,1and is positive near0, andu_k,−_∞has exactlyk−1zeros in0,1and is negative near0,

iiifr ∈λk/f0,∞, then1.1have at least four solutionsu±_k,∞andu±_k,0, such thatu_k,∞(resp., u_k,0) has exactlyk−1zeros in0,1and is positive near0;u−_k,∞(resp.,u−_k,0) has exactlyk−1zeros in

0,1and is negative near0.

Remark 1.3. LetH1,H2,H3, andH4hold. Assume that for somek∈N,λk/f0< λk/f∞.

Similar results toTheorem 1.2have also been obtained.

Making a comparison between the above two theorems, we see that asf has two zeros

s1, s2 : s2 < 0 < s1, the bifurcation structure of the nodal solutions of 1.1 becomes more

complicated: two new nodal solutions are obtained whenr >max{λk/f0, λk/f∞}.

In 3, Ma and O’Regan established some existence results which are similar to Theorem 1.1of the nodal solutions of them-point boundary value problems

ufu 0, 0< t <1,

u0 0, u1

m−2

i1 αiu

ηi

under the following condition:

H1 f ∈C1_{R,Rwithsfs>0 fors /0.}

Remark 1.4. For other results about the existence of nodal solution of multipoint boundary value problems, we can see4–7.

Of course an interesting question is, as for m-point boundary value problems, when

f possesses zeros in R\ {0}, whether we can obtain some new results which are similar to Theorem 1.2.

We consider the eigenvalue problems

urfu 0, 0< t <1, 1.8

u0 0, u1 m−2

i1 αiu

ηi

, 1.9

where m ≥ 3, ηi ∈ 0,1, andαi > 0 for i 1, . . . , m−2. Also using the global bifurcation techniques, we study the global behavior of the components of nodal solutions of1.8,1.9 and give a positive answer to the above question. However, when m-point boundary value condition1.9is concerned, the discussion is more difficult since the problem is nonsymmetric and the corresponding operator is disconjugate.

In the following paper, we assume that

H0 αi>0 fori1, . . . , m−2,with 0<mi−12αi<1;

H1 f ∈ C1_{R,R and there exist two constantss}

2 < 0 < s1, such that fs1 fs2

f0 0;

H2 there existf0, f∞∈0,∞such that

f0 lim |s|→0

fs

s , f∞|slim|→∞ fs

s . 1.10

The rest of the paper is organized as follows.Section 2contains preliminary definitions and some eigenvalue results of corresponding linear problems of 1.8, 1.9. In Section 3, we give two Rabinowize-type global bifurcation theorems. Finally, in Section 4, we consider two bifurcation problems related to1.8,1.9, and use the global bifurcation theorems from Section 3to analyze the global behavior of the components of nodal solutions of1.8,1.9.

2. Preliminary definitions and eigenvalues of corresponding linear problems

LetY C0,1with the norm

u∞ max

t∈0,1

_{ut. 2.1}

Let

X

u∈C1_{0,1|u0 0, u1}

m−2

i1 αiu

ηi

,

E

u∈C2_{0,1|u0 0, u1} m−2

i1 αiu

ηi

with the norm

uX max

u∞,u∞}, uEmax{u∞,u∞,u∞

, 2.3

respectively. DefineL:E→Y by setting

Lu:−u, u∈E. 2.4

ThenLhas a bounded inverseL−1_{:Y → Eand the restriction ofL}−1_{toX, that is,L}−1 _:X→X

is a compact and continuous operator, see3,4,8.

LetER×Eunder the product topology. As in9, we add the points{λ,∞|λ∈R}

to our space E. For anyC1 functionu, ifux0 0, thenx0is a simple zero of uifux0/0.

For any integer k ≥ 1 and any ν ∈ {±}, define sets Sν_k, T_kν ⊂ C20,1 consisting of functions

u∈C20,1satisfying the following conditions:

Sν_k:

iu0 0, νu0>0;

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

T_kν:

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 2.1. Obviously, ifu∈T_kν, thenu∈Sν_k oru∈Sν_k1. The setsT_kνare open inEand disjoint.

Remark 2.2. The nodal properties of solutions of nonlinear Sturm-Liouville problems with sep-arated boundary conditions are usually described in terms of sets similar toSν

k, see1,2,5,9– 11. However, Rynne 4 stated that Tν

k are more appropriate than Sνk when the multipoint boundary condition1.9is considered.

Next, we consider the eigenvalues of the linear problem

Luλu, u∈E. 2.5

We call the set of eigenvalues of2.5the spectrum ofL, and denote it byσL. The following lemmas can be found in3,4,12.

Lemma 2.3. Let (H0) hold. The spectrum σL consists of a strictly increasing positive sequence of eigenvaluesλk,k1,2, . . . ,with corresponding eigenfunctionsϕkx sin

λkx. 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 operatorL}−1 _{: Y → Y. In this}

setting, eachλk,k1,2, . . . ,is a characteristic value ofL−1, with algebraic multiplicity defined to be dim∞_j1NI−λkL−1j, whereNdenotes null-space andIis the identity onY.

Lemma 2.4. Let (H0) hold. For eachk ≥ 1, the algebraic multiplicity of the characteristic valueλk,

3. Global bifurcation

Letg∈C1_{R,Rand satisfy}

g0 g0 0. 3.1

Consider the following bifurcation problem:

Luμugu, μ, u∈R×X. 3.2

Obviously,u≡0 is a trivial solution of3.2for anyμ∈R. About nontrivial solutions of3.2, we have the following.

Lemma 3.1see4, Proposition 4.1. Let (H0) hold. Ifμ, u∈ Eis a nontrivial solution of 3.2, thenu∈T_kνfor somek, ν.

Remark 3.2. From Lemmas 2.3and 3.1, we can see that T_kν are more effectual than the setSν_k

when the multipoint boundary condition1.9is considered. In fact, eigenfunctionsϕkx sinλkx, k 1,2, . . . , of 2.5do not necessarily belong to S_k. In 3, 4, there were some special examples to show this problem.

Also, in4, Rynne obtained the following Rabinowitz-type global bifurcation result for

3.2.

Lemma 3.3 see4, Theorem 4.2. Let (H0) hold. For eachk ≥ 1andν, there exists a continuum Cν

k ⊂Eof solution of 3.2with the following properties:

1o λk,0∈ Cν_k;

2o Cν_k\ {λk,0} ⊂R×T_kν;

3o Cν

k is unbounded inE.

Now, we consider another bifurcation problem

Luμuhu, μ, u∈R×X, 3.3

where we suppose thath∈C1R,Rand satisfy

lim

|x|→∞

hx

x 0. 3.4

TakeΛ⊂Ras an interval such thatΛ∩ {λj |j ∈N}{λk}andMas a neighborhood of

λk,∞whose projection onRlies inΛand whose projection onEis bounded away from 0.

Lemma 3.4. Let (H0) and3.4hold. For eachk≥1andν, there exists a continuumDν_k ⊂Eof solution of 3.3which meetsλk,∞and either

1o Dν

k\ Mis bounded inEin which caseDνk\ Mmeets{λ,0|λ∈R}or

Moreover, if2ooccurs andDν

k\ Mhas a bounded projection onR, thenDνk\ Mmeetsμ, ∞, whereμ∈ {λj |j∈N}withμ /λk.

In every case, there exists a neighborhoodO ⊂ M ofλk,∞such thatμ, u ∈ Dν_k ∩ O and

μ, u/ λk,∞impliesμ, u∈R×T_kν.

Remark 3.5. A continuum D_kν ⊂ E of solution of 3.3meets λk,∞ which means that there exists a sequence{λn, un} ⊂ D_kνsuch thatunE→ ∞andλn→λk.

Proof. Obviously,3.3is equivalent to the problem

uμL−1uL−1hu, μ, u∈R×X. 3.5

Note thatL−1 :X →Xis a compact and continuous linear operator. In addition, the mapping

u→ L−1_{huis continuous and compact, and satisfiesL}−1_{hu ou}

Xatu∞; moreover, u2

XL−1hu/u2Xis compactsimilar proofs can be found in9. Hence, the problem3.3is of the form considered in9, and satisfies the general hypotheses imposed in that paper. Then by 9, Theorem 1.6 and Corollary 1.8 together with Lemmas 2.3and 2.4 in Section 2, there exists a continuumDν_k ⊂R×Xof solutions of3.3which meetsλk,∞and either

1o Dν_k\ Mis bounded inR×Xin which caseDν_k\ Mmeets{λ,0|λ∈R}or

2o Dν

k\ Mis unbounded inR×X.

Moreover, if(2o_{)occurs andD}ν

k \ Mhas a bounded projection onR, thenD ν

k \ Mmeets

μ, ∞whereμ∈ {λj |j ∈N}withμ /λk.

In every case, there exists a neighborhoodO ⊂ Mof λk,∞such that μ, u ∈ Dν_k ∩ O andμ, u/ λk,∞impliesμ, u∈R×T_kν.

On the other hand, by3.5and the continuity of the operatorL−1:Y →E, the setDν_klies inEand the injectionDν_k →Eis continuous. Thus,Dν_k is also a continuum inEand the above properties hold inE.

Now, we assume that

h0 0. 3.6

Lemma 3.6. Let (H0) and3.6hold. Ifμ, u ∈ Eis a nontrivial solution of 3.3, then u ∈Tν k for somek, ν.

Proof. The proof ofLemma 3.6is similar to the proof of Lemma 3.14, Proposition 4.1; we omit it.

Remark 3.7. If3.6holds, Lemma 3.6guarantees thatD_kν inLemma 3.4is a component of so-lutions of 3.3 in T_kν which meets λk,∞. Otherwise, if there exist η1, y1 ∈ Dν_k ∩ T_kν and η2, y2∈ Dν_k∩T_hνfor somek /h∈N, then by the connectivity ofDν_k, there existsη∗, y∗∈ Dν_k

4. Statement of main results

We return to the problem1.8,1.9. LetH1,H2hold and letζ, ξ∈C1_{R,Rbe such that}

fu f0uζu, fu f∞uξu. 4.1

Clearly

ζ0 0, ξ0 0,

lim

|u|→0 ζu

u ζ

_{0 0, lim} |u|→∞

ξu

u 0.

4.2

Let us consider

Lu−rf0urζu 4.3

as a bifurcation problem from the trivial solutionu≡0, and

Lu−rf∞urξu 4.4

as a bifurcation problem from infinity. We note that4.3and4.4are the same, and each of them is equivalent to1.8,1.9.

The results of Lemma 3.3for4.3can be stated as follows: for each integerk ≥ 1,ν ∈

{, −}, there exists a continuumCν_k,0of solutions of4.3joiningλk/f0,0to infinity, andCν_k,0\

{λk/f0,0} ⊂R×T_kν.

The results of Lemma 3.4for4.4can be stated as follows: for each integerk ≥ 1,ν ∈

{, −}, there exists a continuumDν_k,∞of solutions of4.4meetingλk/f∞,∞.

Theorem 4.1. Let (H0),H1,and (H2) hold. Then

iforr, u∈ C_k,0∪ C−_k,0,

s2< ut< s1, t∈0,1; 4.5

iiforr, u∈ D_k,∞∪ D−_k,∞,

max

t∈0,1ut> s1, or tmin∈0,1ut< s2. 4.6

Proof ofTheorem 4.1. Suppose on the contrary that there existsr, u∈ C_k,0∪ C−_k,0∪ D_k,∞∪ D_k,−_∞

such that either

maxut|t∈0,1s1 4.7

or

minut|t∈0, 1s2. 4.8

Sinceu∈Tν

k, byRemark 2.1,u∈Sνk oru∈ Sνk1. We assumeu∈Sνk. Whenu∈Sνk1, we

can prove all the following results with small modifications. Let

0τ0< τ1<· · ·< τk−1<1 4.9

Case 1max{ut|t∈0, 1}s1. In this case, there existsj ∈ {0, . . . , k−2}such that

maxut|t∈τj, τj1

s1 or max{ut|t∈τk−1, 1

s1,

0≤ut≤s1, t∈

τj, τj1

, or t∈τk−1, 1

.

4.10

Sinceu1 m_i−₁2αiuηiandH0, we claimu1< s1.

Lett0∈τj, τj1 ort0∈τk−1, 1such thatut0 s1, thenut0 0. Note that

fut0fs1 0. 4.11

By the uniqueness of solutions of 1.8subject to initial conditions, we see that ut ≡ s1 on 0,1. This contradicts1.9andH0.

Therefore,

maxut|t∈0, 1/s1. 4.12

Case 2min{ut|t∈0,1}s2. In this case, the proof is similar toCase 1, we omit it.

Consequently, we obtain the resultsiandii.

Theorem 4.2. Let (H0),H1,and (H2) hold. Assume that for somek∈N,

λk

f∞ < λk

f0

resp., λk

f0 < λk

f∞

. 4.13

Then

iif r ∈ λk/f∞, λk/f0 (resp., r ∈ λk/f0, λk/f∞, then 1.8, 1.9 have at least two solutionsu±_k,∞(resp.,u±_k,0), such thatu_k,∞ ∈ T_k andu−_k,∞ ∈ T_k− (resp.,u_k,0 ∈T_kandu−_k,0 ∈ T_k−),

iiifr∈λk/f0,∞(resp., r ∈λk/f∞,∞, then1.8,1.9have at least four solutionsu±_k,∞ andu±_k,0, such thatu_k,∞,u_k,0∈T_k, andu−_k,∞,u−_k,0∈T_k−.

Remark 4.3. Making a comparison between results in3and the above theorem, we see that as f has two zeros s1, s2 : s2 < 0 < s1, the bifurcation structure of the nodal solutions of 1.8,1.9becomes more complicated: the component of the solutions of1.8,1.9from the trivial solution atλk/f0,0and the component of the solutions of1.8,1.9from infinity at λk/f∞,∞are disjoint; two new nodal solutions are born whenr >max{λk/f0, λk/f∞}.

Proof ofTheorem 4.2. Since1.8,1.9have a unique solutionu≡0, we get

C

k,0∪ C−k,0∪ Dk,∞∪ D−k,∞

⊂μ, z∈E|μ≥0. 4.14

TakeΛ ⊂ Ras an interval such thatΛ∩ {λj/f∞ | j ∈ N} {λk/f∞}andMas a

neigh-borhood ofλk/f∞,∞whose projection onRlies inΛand whose projection onEis bounded

Dν

k,∞\ Msatisfies one of the following:

1o Dν

k,∞\ Mis bounded inEin which caseDνk,∞\ Mmeets{λ,0|λ∈R};

2o Dν

k,∞\ Mis unbounded inEin which case ProjRDk,∞\ Mis unbounded.

Obviously,Theorem 4.1iiimplies that1odoes not occur. SoD_k,∞\ Mis unbounded inE. Thus

Proj_RD_k,∞⊃

λk

f∞,∞

,

Proj_RD−_k,∞⊃

λk

f∞,∞

.

4.15

ByTheorem 4.1, for anyr, u∈C_k,0∪ C−_k,0,

u∞<maxs1,s2:s∗. 4.16

Equations4.16,1.8, and1.9imply that

uE <max

rmax

|s|≤s∗

_{fs, s}∗_{, 4.17}

which means that the sets{μ, z∈ C_k,0 | μ∈0, d}and{μ, z∈ C−_k,0 | μ∈0, d}are bound-ed for any fixbound-edd∈0,∞. This, together with the fact thatC_k,0resp.,C−_k,0joinsλk/f0,0to

infinity, yields that

Proj_RC_k,0⊃

λk

f0,∞

,

Proj_RC−_k,0⊃

λk

f0,∞

.

4.18

Combining4.15with4.18, we conclude the desired results.

Acknowledgments

This paper is supported by the NSFCno. 10671158, the NSF of Gansu Provinceno. 3ZS051-A25-016, NWNU-KJCXGC-03-17, the Spring-sun program no. Z2004-1-62033, SRFDP no. 20060736001, the SRF for ROCS, SEM2006311, and LZJTU-ZXKT-40728.

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