Exact Multiplicity of Positive Solutions for a Class of Second-Order Two-Point Boundary Problems with Weight Function

Volume 2010, Article ID 207649,16pages doi:10.1155/2010/207649

Research Article

Exact Multiplicity of Positive Solutions for

a Class of Second-Order Two-Point Boundary

Problems with Weight Function

Yulian An

_{and Hua Luo}

1_{Department of Mathematics, Shanghai Institute of Technology, Shanghai 200235, China}

2_{School of Mathematics and Quantitative Economics, Dongbei University of Finance and Economics,}

Dalian 116025, China

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

Received 6 March 2010; Revised 18 July 2010; Accepted 11 August 2010

Academic Editor: Raul F. Manasevich

Copyrightq2010 Y. An and H. Luo. 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.

An exact multiplicity result of positive solutions for the boundary value problemsuλatfu

0,t ∈ 0,1,u0 0,u1 0 is achieved, whereλis a positive parameter. Here the function

f :0,∞ → 0,∞isC2_{and satisfiesf0 fs 0,fu >0 foru∈0, s∪s,∞for some}

s ∈ 0,∞. Moreover,fis asymptotically linear andf can change sign only once. The weight functiona:0,1 → 0,∞isC2_{and satisfiesat<0, 3at}2_{<2atatfort∈0,1. Using}

bifurcation techniques, we obtain the exact number of positive solutions of the problem under consideration forλlying in various intervals inR. Moreover, we indicate how to extend the result to the general case.

1. Introduction

Consider the problem

uλatfu 0, t∈0,1,

u0 0, u1 0,

1.1

whereλ >0 is a parameter anda∈C2_{0,1is a weight function.}

in these literatures. In4,5, Ma and Thompson obtained the multiplicity results for a class of second-order two-point boundary value problems depending on a positive parameterλby using bifurcation theory.

Exact multiplicity of positive solutions have been studied by many authors. See, for example, the papers by Korman et al.6, Ouyang and Shi7,8, Shi9, Korman and Ouyang 10,11, Korman12, Rynne13, Bari and Rynne14 for 2mth-order problems, as well as Korman and Li15. In these papers, bifurcation techniques are used. The basic method of proving their results can be divided into three steps: proving positivity of solutions of the linearized problems; studying the direction of bifurcation; showing uniqueness of solution curves.

Ouyang and Shi 7 obtained the curves of positive solutions for the semilinear problem

Δuλfu 0, inBn,

u0, on∂Bn, 1.2

whereBn_{is the unit ball inR}n _{n≥1andf ∈C}2_{R R 0,∞. In7, the following two} cases were considered:

ifdoes not change its sign onR;iifchanges its sign only once onR. Korman and Ouyang10studied the problem

uλft, u 0, t∈−1,1,

u−1 0, u1 0 1.3

under the conditionsf ∈C2_−1,1;Rand

fuut, u>0 for t∈−1,1, u∈0,∞. 1.4

They obtained a full description of the positive solution set of 1.3 and proved that all positive solutions of 1.3 lie on a single smooth solution curve bifurcating from the point 0,0and tending to0,∞in theλ, uplane. Condition1.4is very important to conclude the direction of bifurcation curve.

Of course a natural question is how about the structure of the positive solution set of 1.3whenfuuchanges its sign only once onR?

It is extremely difficult to answer such a question in general. So we shift our study to the problem1.1in this paper. We are interested in discussing the exact multiplicity of positive solutions of1.1with a weight functionawhenfchanges its sign only once onR.

Suppose the following.

H1One hasf ∈ C2_{0,∞withf0 fs 0 for somes ∈ 0,∞andfu > 0 for} u∈0, s∪s,∞.

H2fis concave convexthat is, there existsθ >0 such that

H3The limitsf0limu→0fu/u∈0,∞andf∞limu→0fu/u∈0,∞. H4a∈C2_{0,1satisfiesat>0; at<0 and 3at}2_{<2atat,ift∈0,1.}

In this paper, we obtain exactly two disjoint smooth curves of positive solutions of 1.1under conditionsH1–H4. According to this, we can conclude the existence and exact numbers of positive solutions of1.1forλlying in various intervals inR.

Remark 1.1. Korman and Ouyang10obtained the unique positive solution curve of1.3

under the condition1.4. However they gave no information whenfuu can change sign. In 7, they did not treat the case that the equation contains a weight function.

On the other hand, suppose the following.

H1One hasf ∈C2_{0,∞withf0 0, fu>0, u∈0,∞. There existss >0 such} thatfu< fu/u, u∈0, sandfu> fu/u, u∈s,∞.

Remark 1.2. Ifat > 0, t ∈ 0,1, then we know from the proof in4that the assumptions H1’andH3imply that the component of positive solutions from the trivial solution and the component from infinity are coincident. However, these two components are disjoint under the assumptionsH1and H3 see5. Hence, the essential role is played by the fact of whetherfpossesses zeros inR\ {0}. InSection 3, we prove that1.1has exactly two positive solution curves which are disjoint and have no turning point on themTheorem 3.8

under Conditions H1–H4. And 1.1 has a unique positive solution curve with only one turning point Theorem 3.9if H1 is replaced by H1’. The condition H4 is used to prove the positivity of solutions of the linearized problems of1.1and the direction of bifurcation.

Our main tool is the following bifurcation theorem of Crandall and Rabinowitz.

Theorem 1.3 see 16. Let X and Y be Banach spaces. Let λ, u ∈ R ×X and let F be a

continuously differentiable mapping of an open neighborhood of λ, u into Y. Let the null-space

NFuλ, u span{w}be one dimensional andcodimRFuλ, u 1. LetFλλ, u/∈RFuλ, u.

IfZ is a complement of span{w}inX, then the solution ofFλ, u Fλ, unearλ, uforms a curveλs, us λτs, uswzs, wheres → τs, zs∈R×Zis a continuously

differentiable function nears0andτ0 τ0 z0 z0 0.

2. Notations and Preliminaries

LetY C0,1with the norm

y_∞ max

t∈0,1yt, 2.1

and let

with the norm

y_Emaxy_∞, y_∞. 2.3

Set

X y∈C20,1|y0 y1 0 2.4

equipped with the norm

_y

Xmaxy∞, y∞, y∞

. 2.5

Define the operatorL:X → Y,

Lu−u, u∈X. 2.6

Then,L−1_{:Y → Eis a completely continuous operator.}

Definition 2.1. For a nontrivial solution of1.1,λ, uis degenerateif the linearized problem

wλatfuw0, t∈0,1,

w0 0, w1 0 2.7

has a nontrivial solution; otherwise, it isnondegenerate.

Lemma 2.2. Let (H1) and (H4) hold. For any degenerate positive solution λ, u of 1.1, the nontrivial solutionwof2.7can be chosen as positive.

Proof. The proof is motivated by Lemma 2.6 in11.

Suppose to the contrary thatwhas zeros on0,1. Without loss of generality, suppose thatw0>0. Note thatwandutsatisfy

wλatfuw0, 2.8

u_tλatfuutλatfu 0, 2.9

respectively. We claim thatwhas at most one zero in0,1. Otherwise, let 0< α < β <1 be the first two zeros ofw. Then,

wα wβ 0, wt<0, t∈α, β ,

Multiplying2.9bygtwand 2.8bygtut, subtracting, and integrating overα, β, we have

β

α

u_tgw dt−

β

α

wgutλ

β

α

agfw0 2.11

with gt > 0 to be specified. We denote the left side of 2.11 by I and a constant −wβgβuβ wαgαuαbyA. Integrating by parts,

I A

β

α

ugw dt2

β

α

ugwdtλ

β

α

agfw dt

A−

β

α

ugw dt−2

β

α

wgudtλ

β

α

agfw dt

A−

β

α

ugw dtλ

β

α

2gaag fw dt.

2.12

Let

2gaag 0, g<0 2.13

on0,1.From2.10,2.13, andut<0, t∈0,1, we have

I−wβ gβ uβ wαgαuα−

β

α

ugw dt >0. 2.14

Note that the right side of2.11is zero, which is a contradiction.

Hence,w has at most one zero in0,1. Suppose that there is one pointγ such that wγ 0. Then,

wγ w1 0, wt<0, t∈γ,1 ,

wγ ≤0, w1≥0. 2.15

Repeating the above proof onγ,1, we can get similar contradiction. Finally, integrating the differential equation in2.13, we can choose

gt a−1/2t. 2.16

The following lemma is an important result in this paper.

Lemma 2.3. Let (H1) and (H4) hold. Suppose thatλ∗, u∗is a degenerate positive solution of1.1. Then, the following are considered.

iAll solutions of 1.1nearλ∗, u∗have the formλs, u∗swzsfors∈−δ, δand someδ >0, whereλ0 λ∗, λ0 0, z0 z0 0.

iiOne hasλ0<0,iffis concave convex;λ0>0,iffis convex concave.

Proof. iThe proof is standard. LetF:X → Y be such thatFλ, u uλatfu. We will

show that the conditions ofTheorem 1.3hold.

Sinceλ∗, u∗is a degenerate positive solution of1.1, we denote the corresponding solution of2.7byw. FromLemma 2.2and the theory of compact disturbing of a Fredholm operator,NFuλ∗, u∗ span{w}is one dimensional and codimRFuλ∗, u∗ 1.

Now, we show thatFλλ∗, u∗/∈RFuλ∗, u∗. Suppose to the contrary thatFλλ∗, u∗∈ RFuλ∗, u∗. Then, there is av∈Xsuch that

vλ∗atfu∗vatfu∗, 2.17

v0 v1 0. 2.18

Note thatwsatisfies

wλ∗atfu∗w0, 2.19

w0 w1 0. 2.20

Multiplying2.17bywand2.19byv, subtracting, and integrating on both sides, we obtain

1

0

vw−wv dt

1

0

atfu∗w dt >0. 2.21

However, the left side of2.21is equal to zero according to boundary conditions2.18and 2.20. This implies that Fλλ∗, u∗/∈ImFuλ∗, u∗. According to Theorem 1.3, the resulti holds.

iiSubstitutingλλs, uu∗swzsinto1.1, we obtain

u∗swzsλsatfu∗swzs 0. 2.22

Sincef ∈C2_{0,∞,then, by the implicit function theorem, the solution curve nearλ}∗_{, u}∗_is alsoC2_{.Differentiating2.22twice with respect tos, we have}

u_ssλsatfu 2λsatfuusλsatfuu2sλsatfuuss0. 2.23

Evaluating ats0, we obtain

Multiplying2.24bywand2.19byuss, subtracting, and integrating, we get

λ0 −λ

∗1

0atfu∗w3dt

₁

0atfu∗w dt

. 2.25

According toH1,H4, andLemma 2.2, we see that₀1atfu∗w dt >0. Next, for the sign ofλ0, we consider the sign of₀1atfu∗w3_dt.

We first prove that

1

0

fu∗u_t∗2w dt0. 2.26

Differentiating1.1and2.19with respect tot, we have

u∗_tλatfu∗ λatfu∗u∗_t 0, 2.27

w_t λatfu∗wλatfu∗u∗_twλatfu∗wt0. 2.28

Multiplying,2.27byhtwtand2.28byhtu∗t, subtracting, and integrating over0,1, we get

1

0

u∗_twth−wtu∗th

dtλ

1

0

afu∗wt−fu∗wu∗t

h dtλ

1

0

afu∗u∗_t2wh dt 2.29

withht>0 to be specified. Integrating by parts on the left side of2.29,

₁

0

u∗_twth−wtu∗th

dtλ

₁

0

afu∗wt−fu∗wu∗t

h dt

λ

1

0

ahah fu∗w−fu∗wu∗dt.

2.30

Let

ahah0. 2.31

From2.29, we get

1

0

afu∗u_t∗2wh dt0. 2.32

Solving the equationahah0, we can choose the auxiliary function

ha−1. 2.33

The following proof is motivated by the proof of Theorem 2.2 in8.

Sincew > 0,2.26 implies thatfu∗tmust change sign. If f is concave convex, then there existst0∈0,1such that

fu∗t≥0 in0, t0,

fu∗t≤0 int0,1.

2.34

Next, we claim that there existsk >0, such that

kw≥ −u∗_t in0, t0,

kw≤ −u∗_t int0,1.

2.35

Letxkwu∗_t. Then,x0 kw0 u∗_t0 kw0>0, andx1 kw1 u∗_t1<0. So,x has at least one zero in0,1. Moreover, we can prove thatxhas only one zero in0,1. Note thatkwsatisfies

kwλ∗atfu∗kw 0. 2.36

We get

xλatfu∗x−λatfu∗≥0, 2.37

sinceat <0 andfu∗>0. Suppose thatxhas more than one zero in0, 1. Lett1 < t2be the last two zeros ofx, then we say that

xt1 xt2 0, xt>0, if t∈t1, t2,

xt1≥0, xt2≤0.

2.38

We first prove the above statement. On the contrary, suppose that

xt1 xt2 0, xt<0, ift∈t1, t2∪t2,1. 2.39

Consider the problem

qt λatfu∗qt 0, qt1 q1 0. 2.40

Now let us consider the claim related tok. Multiplying2.36byxand2.37bykw, subtracting, and integrating overt1, t2, we get

−kwt2xt2 kwt1xt1 λ

t2

t1

atfu∗kwdt <0 2.41

sinceat<0. Note that the left side is nonnegative. Such a contradiction implies thatxhas only one zero in0,1. By varyingksuch thatxt0 kwt0 u∗tt0 0,we can conclude the claim.

From the claim andat<0, we have

k2

1

0

atfu∗w3dt

t0

0

afu∗k2w2w dt

1

t0

afu∗k2w2w dt

>

t0

0

afu∗u2_tw dt

1

t0

afu∗u2_tw dt

>

_t0

0

at0fu∗u2tw dt

₁

t0

at0fu∗u2tw dt

at0

₁

0

fu∗u2_tw dt0.

2.42

Hence,λ0<0 from2.25.

Iffis convex concave, thenλ0>0 with a similar proof.

3. The Main Results and the Proofs

In this section we state our main results and proofs.

Definition 3.1. Define

λ0₁ λ1 f0

λ∞₁ λ1

f∞, 3.1

whereλ1is the first eigenvalue of the corresponding linear problem

ϕλatϕ0, t∈0,1,

ϕ0 0, ϕ1 0. 3.2

Remark 3.2. It is well known that the eigenvalues of3.2are given by

0< λ1< λ2<· · ·< λk< λk1<· · ·, lim

For eachk∈N, algebraic multiplicity ofλkis equal to 1, and the corresponding eigenfunction ϕkhas exactlyk−1 simple zeros in0,1.

Definition 3.3see7. Letf ∈C1_{a, b. Thenf is said to be superlinearresp.,sublinearon} a, biffu/u≤ fu resp.,fu/u ≥fuona, b. Andfis said to be sup-subresp.,

sub-supona, bif there existsc ∈ a, bsuch thatfuis superlinearresp., sublinearon

a, c, and superlinearresp., sublinearonc, b.

Lemma 3.4. (i) Letf∈C1_{0,∞, f0 0, f0>0,and (H4) hold. Suppose thatλ}

∗,0is a point

where a bifurcation from the trivial solutions occurs and thatΓ1is the corresponding positive solution

bifurcation curve of 1.1. If there existsδ1>0such thatfis superlinear (resp., sublinear) on0, δ1,

thenΓ1tends to the left (resp., the right) nearλ∗,0.

(ii) Letf ∈C2_{0,∞, f}

∞ ∈ 0,∞,and (H4) hold. Suppose thatλ∗,∞is a point where a

bifurcation from infinity occurs and thatΓ2is the corresponding positive solution bifurcation curve of 1.1. If there existsδ2>0such thatfis superlinear (resp., sublinear) onδ2,∞andfu≥0/≡0

(resp.,fu≤0/≡0) foru > δ2, thenΓ2tends to the right (resp., the left) nearλ∗,∞.

Proof. The proof is similar to that of Proposition 3.4 in7, so we omit it.

Lemma 3.5. Let (H1)–(H4) hold, letI ⊂ Rbe a bounded and closed interval, and letλ0₁, λ∞₁ ∈ I. Suppose thatλn,unare positive solutions of 1.1. Then,

iλn → λ0₁, ifunE → 0, iiλn → λ∞₁ , ifunE → ∞.

Proof. Letζ, ξ∈CR,Rbe such that

fu f0uζu, fu f∞uξu. 3.4

Clearly,

lim

|u| →0 ζu

u 0, |ulim| → ∞

ξu

u 0. 3.5

Let us consider

Lu−λatf0uλatζu 3.6

as a bifurcation problem fromu≡0. Note that3.6is the same as to1.1. FromRemark 3.2

and the standard bifurcation theorem from simple eigenvalues17, we havei. Let us consider

Lu−λatf∞uλatξu 3.7

Lemma 3.6. Let (H1), (H4) hold. Suppose thatλ, uis a positive solution of 1.1. Then,

max{ut|t∈0,1}/s. 3.8

Proof. Suppose to the contrary that

max{ut|t∈0,1}s. 3.9

By1.1andH1, we haveu0 s. Note thatfs 0. By the uniqueness of solutions of initial value problem, the problem

uλatfu 0, t∈0,1,

u0 0, u0 s

3.10

has a unique solutionut≡s. This contradictsu1 0.

The following Lemma is an interesting and important result.

Lemma 3.7. Let (H1)–(H4) hold. Suppose thatλ∗, u∗is a positive solution of1.1, thenλ∗, u∗is

nondegenerate.

Proof. From conditionsH1–H3, we can check easily that

fuu < fu, ∀u∈0, s; fuu > fu, ∀u∈s,∞. 3.11

In fact, letGu fu−ufu, then

G0 Gs 0, 3.12

sincef0 fs fs 0. Note thatGu −fuu > 0, ifu∈0, θ,andGu< 0, if u∈θ,∞. This together with3.12implies thatθ < sand3.11.

Now, we give the proof in two cases.

Case Imax{u∗t | t ∈ 0, 1} < s. On the contrary, suppose thatλ∗, u∗is a degenerate

solution with max{u_∗t|t∈0,1}< s, thenu_∗t< s,for allt∈0,1. By3.11, we get

u∗tfu∗t< fu∗t, ∀t∈0,1. 3.13

Multiplying1.1byw∗and2.7byu∗, subtracting, and integrating, we have

0

1

0

u₀w∗−w∗u0 dτλ

1

0

aτfu∗u∗−fu∗w∗dτ. 3.14

Case IImax{u∗t | t ∈ 0,1} > s. On the contrary, suppose thatλ∗, u∗is a degenerate

solution with max{u_∗t|t∈0,1}> s. According to Lemmas2.2and2.3, we know that all solutions of1.1nearλ∗, u∗satisfyλs, u∗sw∗zsfors∈ −δ, δand someδ > 0,

whereλ0 λ∗, λ0 0, z0 z0 0. It follows that for λclose toλ∗ we have two

solutionsu−t, λand ut, λwith u−t, λ strictly increasing inλand ut, λwith strictly decreasing inλ. We will show that the lower branchu−t, λis strictly increasing for allλ < λ_∗. Note that u−_λt, λ > 0 for λ close to λ_∗ and all t ∈ 0,1. Let λ∗ be the largest λ where this inequality is violated; that is,u−_λt, λ∗≥ 0 andu−_λt0, λ∗ 0 for somet0 ∈0,1. Differentiating1.1with respect toλ,

u_λλ∗atfuuλ−atfu≤0, uλ0 uλ1 0. 3.15

We can extend evenlya, u, anduλon−1,1, then we obtain

u_λλ∗atfuuλ−atfu≤0, uλ−1 uλ1 0. 3.16

By the strong maximum principle, we conclude thatu−_λt, λ∗>0 for all0,1. This contradicts thatu−_λt0, λ∗ 0.

ByLemma 2.3, we getλ0<0 at every degenerate positive solution. Hence, there is

no degenerate positive solution on the lower branchu−t, λ. However, the lower branch has no place to go. In fact, there must exist some positive constantα≥ssuch that max{ut|t∈

0,1} > αfor anyλ, ulying onu−t, λ. Hence, the lower branch cannot go to theλaxis. And it also cannot go to theuaxis, since1.1has only the trivial solution atλ0.

So,λ∗, u∗is nondegenerate.

Our main result is the following.

Theorem 3.8. Let (H1)–(H4) hold. Then the following are considered.

iAll positive solutions of 1.1lie on two continuous curvesΣ1andΣ2without intersection. Σ1bifurcates fromλ0₁, 0to infinity andProj_RΣ1 λ0₁,∞;Σ2bifurcates fromλ∞₁, ∞

to infinity andProj_RΣ2 λ∞₁ ,∞. There is no degenerate positive solution on these curves.

For anyλ, u∈Σ1,u∞< s, and for anyλ, u∈Σ2,u∞> s.

iiEquation1.1has no positive solution for0 ≤λ≤min{λ0

1, λ∞1}has exactly one positive

solution formin{λ0₁, λ∞₁}< λ≤max{λ0₁, λ∞₁ }but and has exactly two positive solutions formax{λ0₁, λ∞₁ }< λ <∞(seeFigure 1).

Proof. iSincef0 0 andf0>0, thenλ01>0. FromLemma 3.5iand the standard Crandall and Rabinowitz theorem on local bifurcation from simple eigenvalues17,λ0₁is the unique point where a bifurcation from the trivial solution occurs. Moreover, byLemma 3.4, the curve bifurcates to the right. We denote this local curve byΣ1and continueΣ1 to the right as long as it is possible. Meanwhile, byLemma 3.6, there is no positive solution of1.1which has the maximum valueson0,1. So, ifλ, u∈Σ1, thenu∞< s. From1.1, we have

ut

_t

0

λaτfuτdτ≤M

_t

0

λaτdτ, 3.17

u f

0 θ s

a

E

λ0

1 λ∞1 λ

0

b Figure1

On the other hand,Lemma 3.7and the implicit function theorem ensure thatΣ1cannot stop at a finite pointλ, u.

From the above discussion, we see thatΣ1 can be extended continuously to infinity and Proj_RΣ1 λ0₁,∞. Meanwhile, the maximum values of all positive solutions of1.1are less thans.

Now, we consider positive solutions of1.1, for which the maximum value on0,1 is greater thans.

Let us return to consider3.6as the bifurcation problem from infinity. Note that3.6

is also the same as to1.1. Since lim_|u| → ∞ξu/u 0,by Theorem 1.6 and Corollary 1.8 in

18, there exists a subcontinuumD ⊂R×Eof positive solutions of3.6which meetsλ∞₁ ,∞. TakeΛ⊂Ras an interval such thatΛ∩ {λj/f∞ |j ∈N}{λ∞1}andUas a neighborhood of λ∞₁ ,∞whose projection onRlies inΛand whose projection onEis bounded away from 0. Then, there exists a neighborhoodO ⊂ Usuch that any positive solutionλ, u ∈ Oof1.1

satisfiesλ, u λα, αϕ1zαforα∈δ,∞and someδ >0 and|λ−λ∞₁|o1,zX oαatα∞, whereϕ1denotes the normalized eigenvector of3.2corresponding toλ1. So, ϕ11.

Hence,D ∩ Ois a continuous curve, and we denote it byΣ2. It tends to the right from

Lemma 3.4ii. FromLemma 3.7and the implicit function theorem,Σ2can be continued to a

maximal interval of definition over theλaxis. We claim thatΣ2\ {λ∞1 ,∞}cannot blow up ifλis bounded. In fact, suppose that there exists a positive solutions sequence{λn, un}of 1.1andλ <∞such thatunE → ∞asλn → λ. Then, byLemma 3.5ii,λλ∞₁ . This is a contradiction. On the other hand, the implicit function theorem implies thatΣ2cannot stop at a finite pointλ, u. Thus, Proj_RΣ2 λ∞₁,∞andu∞> sifλ, u∈Σ2.

Finally, we show that both curvesΣ1andΣ2are the only two positive solutions curves of1.1. On the contrary, suppose thatλ, uis a positive solution of1.1withλ, u/∈Σ1∪Σ2. Without loss of generality, assume thatu∞> s. Note thatλ, uis nondegenerate, so we can

extend it to form a curve. We denote this curve byΣand the corresponding maximal interval of definition byI e1, e2. Since all positive solutions of1.1are nondegenerate, according to the implicit function theorem, we must have that

lim λ→e1

ut_E∞. _3.18

u f

0

a

E

λ0 1

λ∞

1 λ∗ λ

0

b Figure2

Similarly, we can show that every positive solution of1.1, the maximum value on 0,1of which is less thans,lies onΣ1.

iiThe resultiiis a corollary ofi.

Next, we will give directly other theorems. Their proofs are similar to that of

Theorem 3.8. So, we omit them.

Theorem 3.9. LetH1and (H2)–(H4) hold. Then, the following are considered.

iAll positive solutions of 1.1lie on a single continuous curveΣ. AndΣbifurcates from

λ0

1, 0to the right to a unique degenerate positive solutionλ∗, u∗of1.1, then it tends

to the left toλ∞₁ ,∞.

iiEquation1.1has no positive solution forλ∈0,min{λ0

1, λ∞1}∪λ∗,∞, and has exactly

one positive solution forλ ∈min{λ0

1, λ∞1},max{λ01, λ∞1 }∪ {λ∗}, and has exactly two

positive solutions forλ∈max{λ0₁, λ∞₁ }, λ∗(seeFigure 2).

Remark 3.10. In fact, if we reverse the inequalities in H1, H1’, H2, we will obtain

corresponding results similar to Theorems3.8and3.9.

Also using the method in this paper, we can obtain the exact numbers of positive solutions for the Dirichlet problem

uλatfu 0, t∈−1,1,

u−1 u1 0, 3.19

whereλ >0 is a parameter. We assume that

H4a∈C1−1,1withat>0, a−t at, andat<0,for allt∈0,1.

Definition 3.11. Define

λ0₁ λ1 f0

, λ∞₁ λ1

f_∞, 3.20

Theorem 3.12. Let (H1’), (H2), (H3), and (H4’) hold. Then, the following are considered.

iAll positive solutions of 3.19lie on a single continuous curveΣ. AndΣ bifurcates from

λ0₁, 0to the right to a unique degenerate positive solutionλ∗,u∗of3.19, then it tends to the left toλ∞₁ ,∞.

iiEquation1.1has no positive solution forλ∈0,min{λ0

1, λ∞1 }∪λ∗,∞but has exactly

one positive solution forλ ∈ min{λ0₁, λ∞₁},max{λ0₁, λ∞₁ }∪ {λ∗}and has exactly two positive solutions forλ∈max{λ0

1, λ∞1 },λ∗. Theorem 3.13. Let (H1), (H2), (H3), (H4’) hold. Then

iAll positive solutions of 3.19 lie on two continuous curves Σ1 and Σ2 without

intersection.Σ1bifurcates fromλ0₁, 0to infinity andProj_RΣ1 λ0₁,∞;Σ2bifurcates

fromλ∞₁ , ∞to infinity andProj_RΣ2 λ∞₁,∞. There is no degenerate positive solution

on these curves. For anyλ, u∈Σ1,u∞< s, and for anyλ, u∈Σ2,u∞> s.

iiEquation3.19has no positive solution for0 ≤ λ ≤ min{λ0

1, λ∞1 }, and has exactly one

positive solution for min{λ0₁, λ∞₁} < λ ≤ max{λ0₁, λ∞₁}, and has exactly two positive solutions formax{λ0

1, λ∞1}< λ <∞.

Remark 3.14. Theorems3.12and3.13extend the main result Theorem 1 in10, wheref >0

foru >0.

4. Examples

In this section, we give some examples.

Example 4.1. Let

fu

⎧ ⎨ ⎩

uu−12, foru∈0,2,

−32 lnu21u32 ln 2−40, for u∈2,∞. 4.1

Then,fsatisfiesH1,H2, andH3. Moreover,s1,θ2/3,f01, andf∞21.

Example 4.2. Let

fu

⎧ ⎨ ⎩

uu2_{−u2 , for u∈0,1,}

−4 lnu7u−5, for u∈1,∞. 4.2

Then,fsatisfiesH1,H2, andH3. Moreover,s1/2, θ1/3,andf02, f∞7.

Example 4.3. Letat βt c. Here,β∈C2_{0,1,βt>0, βt<0,for allt∈0,1, andc}

Acknowledgments

The authors are very grateful to the anonymous referees for their valuable suggestions. An is supported by SRFDP no. 20060736001, YJ2009-16 A06/1020K096019, 11YZ225. Luo is supported by grant no. L09DJY065.

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