Bifurcation curves of positive solutions for the

Bifurcation curves of positive solutions for the

Minkowski-curvature problem with cubic nonlinearity

Shao-Yuan Huang

and Min-Shu Hwang

1Department of Mathematics and Information Education National Taipei University of Education, Taipei 106, Taiwan

2Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan Received 25 February 2021, appeared 22 May 2021

Communicated by Paul Eloe

Abstract. In this paper, we study the shape of bifurcation curveS_Lof positive solutions for the Minkowski-curvature problem











−





u⁰(x) q

1−(u⁰(x))²





0

=λ

−εu³+u²+u+1

, −L<x<L, u(−L) =u(L) =0,

where λ,ε > 0 are bifurcation parameters and L > 0 is an evolution parameter. We prove that there existsε₀>0 such that the bifurcation curveS_L is monotone increasing for all L > 0 ifε ≥ ε0, and the bifurcation curve SL is from monotone increasing to S-shaped for varyingL>0 if 0<ε<ε0.

Keywords: bifurcation curve, positive solution, Minkowski-curvature problem.

2020 Mathematics Subject Classification: 34B15, 34B18, 34C23, 74G35.

1 Introduction and main result

In this paper, we study the shapes of bifurcation curves of positive solutions u∈C²(−L,L)∩ C[−L,L]for the one-dimensional Minkowski-curvature problem











−





u⁰(x) q

1−(u⁰(x))²





0

=λf(u), −L< x< L, u(−L) =u(L) =0,

(1.1)

whereλ>0 is a bifurcation parameter, L>0 is an evolution parameter and the nonlinearity f(u)≡ −εu³+u²+u+1, ε>0. (1.2)

BCorresponding author. Email: [email protected]

It is well-known that studying the multiplicity of positive solutions of problem (1.1) is equiv- alent to studying the shape of bifurcation curveSL of (1.1) where

S_L ≡ {(λ,ku_λk_∞):λ>0 andu_λ is a positive solution of (1.1)} for L>0. (1.3) Thus this investigation is essential.

Before going into further discussions on problems (1.1), we give some terminologies in this paper for the shape of bifurcation curveS_Lon the(λ, kuk_∞)-plane.

Definition 1.1. LetS_L be the bifurcation curve of (1.1) on the(λ,kuk_∞)-plane.

(i) S-like shaped: The curve S_L is said to be S-like shaped if S_L has at least two turning points at some points λ₁,ku_λ1k_∞ and λ₂,ku_λ2k_∞ where λ₁ < λ₂ are two positive numbers such that:

(a) at λ₁,ku_λ1k_∞the bifurcation curveS_Lturns to the right, (b) ku_λ2k_∞< ku_λ1k_∞,

(c) at λ₂,ku_λ2k_∞the bifurcation curveS_Lturns to the left.

(ii) S-shaped: The curve S_L is said to be S-shaped if S_L is S-like shaped, has exactly two turning points, and has at most three intersection points with any vertical line on the (λ,kuk_∞)-plane.

(iii) Monotone increasing: The curveSL is said to bemonotone increasingif λ₁ < λ₂ for any two points λ_i,ku_λik_∞,i=1, 2, lying inS_Lwithku_λ1k_∞ ≤ ku_λ2k_∞.

Crandall and Rabinowitz [2, p. 177] first considered shape of bifurcation curve of positive solutions for then-dimensionalsemilinearproblem

(−_∆u(x) =λ −εu³+u²+u+1

inΩ,

u(x) =0 on ∂Ω, (1.4)

whereΩis a general bounded domain inRⁿ(n≥1) with smooth boundary∂Ω. They applied the implicit function theorem and perturbation arguments to prove that the bifurcation curve of positive solutions of (1.4) is S-like shaped on the(λ,ku_λk_∞)-plane whenε>0 is sufficiently small. Shi [17, Theorem 4.1] proved that the bifurcation curve of positive solutions of (1.4) is S-shaped when ε > 0 is small and Ω is a ball in Rⁿ with 1 ≤ n ≤ 6. Hung and Wang [6]

consider the one-dimensional case

(−u⁰⁰(x) =λ −εu³+u²+u+_1, −₁< x<_1,

u(−1) =u(1) =0. (1.5)

Then they provided the complete variational process of shape of bifurcation curve ¯S of (1.5) with varyingε>0 where

S¯≡ {(λ,ku_λk_∞):λ>0 andu_λ is a positive solution of (1.5)}, (1.6) see Theorem1.2.

Figure 1.1: Graphs of bifurcation curves ¯Sof (1.4). (i)ε≥ε0 and (ii) 0<ε<ε0.

Theorem 1.2 ([6, Theorem 3.1]). Consider (1.5). Then the bifurcation curveS is continuous on the¯ (λ,ku_λk_∞)-plane, starts from(0, 0)and goes to infinity. Furthermore, there exists a critical bifurcation value ε₀ ∈ (0, 1/√

27)such that the bifurcation curve S is monotone increasing if¯ ε ≥ ε₀,and S is¯ S-shaped if0<ε<ε₀, see Figure1.1.

To the best of my knowledge, there are no manuscripts to describe the variational pro- cess for SL of (1.5) with varying ε,L > 0. Hence we start to concern this issue. In addition, references [7,8,16] provided some sufficient conditions to determine the shape of bifurcation curve or multiplicity of positive solutions of problem (1.1) with general f(u)∈ C[0,∞). How- ever, these results can not be applied in our problem (1.1) because the cubic nonlinearity f(u) defined by (1.2) is not always positive in [0,∞). So studying the problem (1.1) is worth and interesting.

By elementary analysis, we find that f(u) has unique zero βε in [0,∞). Then the main result is as follows:

Theorem 1.3(See Figure1.2). Consider(1.1). Letε₀ be defined in Theorem1.2. Then the following statements (i)–(iii) hold:

(i) For L > 0, the bifurcation curve S_L is continuous on the (λ,ku_λk_∞)-plane, starts from (0, 0) and goes to infinity along the horizontal linekuk_∞ =ρ_L,ε whereρ_L,ε ≡min{L,β_ε}.

(ii) Ifε≥ε₀, then the bifurcation curve S_Lis monotone increasing for all L>0.

(iii) If0<ε<ε₀, then there exist two positive numbers L_ε <L^˜_ε such that (a) the bifurcation curve S_Lis monotone increasing for0<L≤ Lε. (b) the bifurcation curve S_Lis S-like shaped for L_ε < L≤L^˜_ε. (c) the bifurcation curve S_Lis S-shaped for L> L^˜_ε.

Furthermore, Lε is a continuous function ofε ∈(0,ε0), lim_ε→0⁺ Lε ∈ (0,∞)andlim_ε→ε⁻

0 Lε =

∞.

Remark 1.4. By numerical simulations to bifurcation curvesS_Lof (1.1), we conjecture that the bifurcation curveS_L is also S-shaped on the(_λ,ku_λk_∞)_{-plane for} L_ε < L≤ L^˜_ε and 0<ε< ε₀. Further investigations are needed. In addition, by Theorems1.2and1.3, we make a list which shows the different properties for Minkowski-curvature problem (1.1) and semilinear problem (1.4), see Table 1.

Figure 1.2: Graphs of bifurcation curveS_Lof (1.1) forε>0.

Bifurcation curve S_L of (1.1) S¯ of (1.4)

1. Shapes (0<ε<ε0) from monotone increasing

to S-shaped with varyingε ^S-shaped

2. Shapes (ε≥ε₀) monotone increasing monotone increasing

3. Numbers of turning points

(1). from 0 to 2 varying L>0 if0<ε<ε0

(2). 0 ifε≥ ε₀

(1). 2 if0<ε<ε0

(2). 0 if ε≥ε₀

4. Continuity continuous continuous

5. Evolution

parameter(s) εand L ε

6. Starting point (0, 0) (0, 0)

7. "End point" (_∞,ρL,ε) (_∞,∞)

Table 1.1: Comparison of properties ofS_Land ¯S.

The paper is organized as follows: Section 2 contains the lemmas used for proving the main result. Section 3 contains the proof of main result (Theorem1.3). Section 4 contains the proof of assertion (2.31).

2 Lemmas

To prove Theorem1.3, we first introduce the time-map method used in Corsato [4, p. 127]. We define the time-map formula for (1.1) by

T_λ(α)≡

Z _α

0

λ[F(α)−F(u)] +1 q

{λ[F(α)−F(u)] +1}²−1

du for 0<α< β_ε andλ>0, (2.1)

where F(u) ≡ Ru

0 f(t)dt. Observe that positive solutionsu_λ ∈ C²(−L,L)∩C[−L,L] for (1.1) correspond to

ku_λk_∞= α and T_λ(α) =L.

So by definition ofS_Lin (1.3), we have that

S_L= {(λ,α): T_λ(α) =L for some 0<α<β_ε andλ>0}. (2.2) Thus, it is important to understand fundamental properties of the time-map T_λ(α)on (0,βε) in order to study the shape of the bifurcation curve S_Lof (1.1) for any fixed L> 0. Note that it can be proved that T_λ(α)is a triple differentiable function of ε ∈ (0,β_ε)for ε,λ > _{0, and} T_λ(α),T_λ⁰(α)are differentiable function ofλ>0 for 0<α< βε anda >0. The proofs are easy but tedious and hence we omit them. Similarly, we define the time-map formula for (1.5) by

T¯(α)≡ √¹ 2

Z _α

0

1

pF(α)−F(u)^{du forα}>0, (2.3) see [12, p. 779]. Then we have thatkuλk_∞ = α and ¯T(α) = √

λ. So by the definition of ¯Sin (1.6), we see that

S¯ =ⁿ(λ,α):

√

λ=_T^¯(α) _{for some}α>₀^o_{. (2.4)} For the sake of convenience, we let

A= A(_α,u)≡αf(α)−u f(u)_, B= B(_α,u)≡ F(α)−F(u)_, C=_C(_α,u)≡α²f⁰(α)−u²f⁰(_u) _{and D}=_D(_α,u)≡α³f⁰⁰(α)−u³f⁰⁰(_u)_. Obviously, we have

B(α,u) =

Z _α

u f(t)dt>0 for 0<u<α< βε (2.5) because f(_u)>_{0 for 0}<_u< β_ε.

Lemma 2.1. Consider(1.1)withε>0. Then the following statements (i)–(iii) hold:

(i) lim_α→0⁺T_λ(α) =0andlim_α→β⁻

ε T_λ(α) =_∞forλ>0.

(ii) lim_λ→0⁺

√

λT_λ⁽ⁱ⁾(α) =_T^¯(i)(α)_{andlimλ→∞T}⁰

λ(α) =_1for0<α<β_ε and i=_{1, 2, 3.}

(iii) ∂T_λ(α)/∂λ<0for0<α< β_ε andλ>0.

Proof. Since

ulim→0⁺

F(u) u² =_∞,

and by [7, Lemma 3.1], we obtain that lim_α→0⁺T_λ(α) =0. Since f(β_ε) =0, there existb,c∈_R such that f(u) = (β_ε−u)(εu²+bu+c). Since f(u) > 0 on (0,β_ε), there exists M > 0 such that 0< εu²+bu+c < M for 0 < u < βε. For 0< t < 1, by the mean-value theorem, there existsη_t∈(β_εt,β_ε)such that

B(β_ε,β_εt) =

Z _βε

βεt f(t)dt= f(η_t)β_ε(1−t) = (β_ε−η_t) εη_t²+bη_t+c

β_ε(1−t)

<(βε−βεt)Mβε(1−t) =Mβ²_ε(1−t)². (2.6) Then there existst^∗ ∈ (0, 1)such that B(βε,βεt) < 1 for t^∗ < t < 1. So by (2.5) and (2.6), we see that

lim

α→β⁻_ε

T_λ(α) = lim

α→β⁻_ε

α Z ₁

0

λB(α,αt) +1 p

λ²B²(_α,αt) +2λB(_α,αt)^dt

≥ lim

α→β⁻ε

α Z ₁

t^∗

1 p

λ²B²(α,αt) +2λB(α,αt)^dt

≥βε

Z ₁

t^∗

1

p(λ²+2λ)B(β_ε,β_εt)^dt≥ _p ¹ (λ²+2λ)M

Z ₁

t^∗

1

1−tdt= _∞,

which implies that statement (i) holds. In addition, we compute that, for 0 < α < β_ε and λ>0,

T_λ⁰(α) = ¹ α

Z _α

0

λ³B³+3λ²B²+λ(2B−A)

(λ²B²+2λB)^{3/2 du, (2.7)}

T_λ⁰⁰(α) = ¹ α²

Z _α

0

3A²B−B²C−2AB²

λ³+ 3A²−4AB−2BC λ²

(λ²B²+2λB)^{5/2 du, (2.8)}

T_λ⁰⁰⁰(α) = ¹ α³

Z _α

0

λ³ [λ²B²+2λB]^7/2

h

B² 9A²B−3B²C−B²D−12A³+9ABC λ²

+B(27A²B−12B²C−4B²D−24A³+27ABC)λ+18A²B−12B²C

−4B²D−15A³+18ABCi

du. (2.9)

So we observe that, for 0<α< βε, lim

λ→0⁺

√

λT_λ⁰(α) = ¹ α

Z _α

0

2B−A

(2B)^3/2du= T^¯0(α), lim

λ→0⁺

√

λT_λ⁰⁰(α) = ¹ α²

Z _α

0

3A²−4AB−2BC

(2B)^{5/2 du}= T^¯00(α), lim

λ→0⁺

√

λT_λ⁰⁰⁰(α) = ¹ α³

Z _α

0

18A²B−12B²C−4B²D−15A³+18ABC

(2B)^{5/2 du}=T^¯000(α).

Furthermore, limλ→_∞T_λ⁰(α) =1. So statement (ii) holds. The statement (iii) follows immedi- ately by [7, Lemma 4.2(ii)]. The proof is complete.

Lemma 2.2. Consider(1.1)withε>0. Then the following statements (i) and (ii) hold:

(i) T_λ⁰(α)>0for0< α≤1andλ>0.

(ii) T_λ(α)has at most one critical point, a local minimum, on[_12ε⁵ _,β_ε)_.

Proof. We can see that 2B(α,u)−A(α,u)>0 for 0< u<α≤1 because 2B(α,α)−A(α,α) =0 and

∂

∂u[2B(_α,u)−A(_α,u)] =−2εu³+ _u²−1

<_{0 for 0}<_u <α<_1.

So by (2.5) and (2.7), we obtain that T_λ⁰(α)> 0 for 0 < α ≤ 1 and λ > 0. Then statement (i) holds. By (2.5), (2.7) and (2.8), we observe that, for 0< α< β_ε andλ>0,

αT_λ⁰⁰(α) +2T_λ⁰(α)

= ¹ α

Z _α

0

B⁵λ³+5B⁴λ²+λB 3A²+16B²−4AB−BC

+3A²+8B²−8AB−2BC

√

λ(λB²+2B)^{5/2 du}

> ¹ α

Z _α

0

λB 3A²+16B²−4AB−BC

+3A²+8B²−8AB−2BC

√

λ(λB²+2B)^{5/2 du}

= ¹ α

Z _α

0

λBh

3(A−B)²+5B²+B(2A−2B−C) i

+₃(A−2B)²+2B(2A−2B−C)

√

λ(λB²+2B)^{5/2 du}

> ¹ α

Z _α

0

λB²(2A−2B−C) +2B(2A−2B−C)

√

λ(λB²+2B)^{5/2 du}

= ¹ α

Z _α

0

λB²+2B

(2A−2B−C)

√

λ(λB²+2B)^{5/2 du}= ¹ α

Z _α

0

2A−2B−C

√

λ(λB²+2B)^3/2du

= ¹ 6α

Z _α

0

φ(α)−φ(u)

√

λ(λB²+2B)^{3/2du, (2.10)}

whereφ(u)≡u³(9εu−4). Clearly,φ⁰(u) =12u²(3εu−1). Since f

4 9ε

=₁+^324ε+80 729ε² >_0, we see that

1 3ε < ⁴

9ε <βε. (2.11)

So we observe that

φ(u)











<0 for 0< u< _9ε⁴,

=0 foru= _9ε⁴,

>0 for _9ε⁴ <u<β_ε,

and φ⁰(u)











<0 for 0<u< _3ε¹,

=0 foru = _3ε¹,

>0 for _3ε¹ < u<β_ε.

(2.12)

Letα∈_12ε⁵ ,βε

be given. Then we consider two cases.

Case 1. Assume that _9ε⁴ ≤α< βε. Since φ(0) =0, and by (2.12), we see thatφ(α)−φ(u)>0 for 0<u<α. So by (2.10), we obtainαT_λ⁰⁰(α) +2T_λ⁰(α)>0 forλ>0.

Case 2. Assume that _12ε⁵ ≤ α< _9ε⁴. Sinceφ(0) =0, and by (2.12), there exists ˜α∈ 0,_3ε¹ such that

φ(α)−φ(u)











<0 for 0<u <α,˜

=0 foru=α,˜

>0 for ˜α<u< α.

So by (2.10), we observe that, forλ>0, αT_λ⁰⁰(α) +2T_λ⁰(α)

> ¹

6α√ λ

"

Z _α˜

0

φ(α)−φ(u) [λB²+2B]^3/2du

+

Z _α

˜ α

φ(α)−φ(u) [λB²+2B]^3/2du

#

> ¹

6α√

λ[λB²(α, ˜α) +2B(α, ˜α)]^3/2 Z _α˜

0

[φ(α)−φ(u)]du+

Z _α

˜ α

[φ(α)−φ(u)]du

= ¹

6α√

λ[λB²(α, ˜α) +2B(α, ˜α)]^3/2

Z _α

0

[φ(α)−φ(u)]du

= ^6εα

3

5√

λ[λB²(α, ˜α) +2B(α, ˜α)]^3/2

α− ⁵ 12ε

≥0.

Thus by Cases 1–2, we have

αT_λ⁰⁰(α) +2T_λ⁰(α)>0 for 5

12ε ≤α< β_ε andλ>0. (2.13) Fixed λ > 0. If T_λ(α) has a critical point ˘α in [_12ε⁵ ,βε), by (2.13), then ˘αT_λ⁰⁰(α˘) = αT˘ _λ⁰⁰(α˘) + 2T_λ⁰(α˘)>0. It implies that T_λ(α)has at most one critical point, a local minimum, on ₅

12ε,βε

forλ>0. Then the statement (ii) holds. The proof is complete.

Lemma 2.3. Consider(1.1)withε>0. Then

∂

∂λ h√

λT_λ⁰(α)ⁱ>0 for0< α≤ ⁵

12ε andλ>0. (2.14)

Proof. By (2.5) and (2.7), we compute and find that

∂

∂λ h√

λT_λ⁰(α)ⁱ= ¹ 2α

Z _α

0

B² B³λ²+_5B²λ+_3A+_6B

(λB²+2B)^{5/2 du}> ¹ 2α

Z _α

0

3B²(A+2B)

(λB²+2B)^{5/2du. (2.15)} In addition, we compute that

∂

∂u [A(α,u) +2B(α,u)] = R(u),

whereR(u)≡3εu³−3(1−ε)u²−6u−4. Clearly,R⁰(u) =9εu²−6(1−ε)u−6 is a quadratic polynomial ofuwith positive leading coefficient. Furthermore,

R⁰(0) =−6<0 and R⁰ 5

12ε

≡ −^56ε+15 16ε <0.

Thus we observe thatR⁰(u)<_{0 for 0}≤u≤ _12ε⁵ . It follows that

∂

∂u[A(α,u) +2B(α,u)] =R(u)≤ R(0) =−4<0 for 0≤ u≤ ⁵ 12ε. Then we have

A(α,u) +2B(α,u)> A(α,α) +2B(α,α) =0 for 0<u< α≤ ⁵ 12ε. So by (2.15), we obtain (2.14). The proof is complete.

Lemma 2.4. Consider (1.1) with ε > 0. Let I be a closed interval in (0,β_ε). Then the following statements (i)–(iii) hold:

(i) IfT¯⁰(α)<0forα∈ I,then there existsλˇ >0such that T_λ⁰(α)<0forα∈ I and0<λ< λ.^ˇ (ii) IfαT¯⁰⁰(α) +kT¯⁰(α)<0forα∈ I and some k>0,then there existsλˆ >0such thatαT_λ⁰⁰(α) +

kT_λ⁰(α)<0forα∈ I and0<λ<λ.^ˆ

(iii) If[2αT¯⁰⁰(α) +3 ¯T⁰(α)]⁰ >0forα∈ I,then there existsλ¯ >0such that[2αT_λ⁰⁰(α) +3T_λ⁰(α)]⁰ >

0forα∈ I and0< λ<λ.^¯

Proof. (I) Assume that ¯T⁰(α)<0 forα∈ I. By Lemma2.1(ii), we have lim

λ→0⁺

√

λT_λ⁰(α) =T^¯0(α)<0 forα∈ I. (2.16) For α∈ I, by (2.16), we defineλ_α by

λ_α ≡







1 if T_λ⁰(α)<0 for allλ>0, sup{λ₁ :T_λ⁰(α)<0 for 0<λ<λ₁} if T_λ⁰(α)≥0 for someλ>0.

(2.17)

Clearly, T_λ⁰(α)<0 for α∈ I and 0< λ<λα. Let ˇλ≡inf{λα :α∈ I}. Assume that ˇλ= 0. By (2.17), there exists a sequence {α_k}_k∈N⊂ I such that

klim→∞λ_αk =_{0 and Tλ}⁰

αk(α_k)≥0 fork ∈_N. (2.18) Without loss of generality, we assume that lim_k→∞α_k = αˇ ∈ I. So by (2.16) and (2.18), we observe that

0≤ _lim

k→_∞

q λ_αkT_λ⁰

αk(α_k) = _lim

k→_∞

q λ_αkT_λ⁰

αk(αˇ) =T^¯0(αˇ)<_0, which is a contradiction. It implies that ˇλ>0. So statement (i) holds.

(II) Assume thatαT¯⁰⁰(α) +kT¯⁰(α)< 0 forα∈ I and some k >0. Let G₁(α,λ)≡ αT_λ⁰⁰(α) + kT_λ⁰(α)_{. By Lemma}2.1(ii), we see that

lim

λ→0⁺

√

λG₁(α,λ) =αT¯⁰⁰(α) +kT¯⁰(α)<0 forα∈ I. (2.19) For α∈ I, by (2.19), we defineλα by

λα ≡







1 ifG₁(α,λ)<0 for allλ>0,

sup{λ₂: G₁(α,λ)<0 for 0< λ<λ₂} ifG₁(α,λ)≥0 for someλ>0.

Clearly, G₁(_α,λ)< _{0 for} α ∈ I and 0 < λ < λ_α. Let ˆλ ≡_inf{λ_α :α ∈ I}. We use the similar argument in (I) to obtain that ˆλ>0. So statement (ii) holds.

(III) Assume that[2αT¯⁰⁰(α) +3 ¯T⁰(α)]⁰ >0 forα∈ I. LetG₂(α,λ)≡[2αT⁰⁰(α) +3T⁰(α)]⁰. By Lemma2.1(ii), we see that

lim

λ→0⁺

√

λG2(α,λ) = lim

λ→0⁺

h 2α√

λT_λ⁰⁰⁰(α) +5√

λT_λ⁰⁰(α)ⁱ=2αT¯⁰⁰⁰(α) +5 ¯T⁰⁰(α)

= [2αT¯⁰⁰(α) +3 ¯T⁰(α)]⁰ >0 forα∈ I. (2.20)

Forα∈ I, by (2.20), we defineλ_α by

λα≡







1 if G₂(_α,λ)<_{0 for all}λ>_0,

sup{λ₃ :G₂(α,λ)<0 for 0<λ<λ₃} if G₂(α,λ)≥0 for someλ>0.

Clearly,G₂(α,λ) < 0 for α∈ I and 0 < λ < λ_α. Let ¯λ ≡ inf{λ_α : α∈ I}. We use the similar argument in (I) to obtain that ¯λ>0. So statement (iii) holds. The proof is complete.

Lemma 2.5. Consider (1.5) with ε > 0. Let ε₀ be defined in Theorem 1.2. Then the following statements (i)–(iii) hold:

(i) T¯⁰(α)≥0for0<α< β_ε andε ≥ε₀.

(ii) [2αT¯⁰⁰(α) +3 ¯T⁰(α)]⁰ >0for _3ε¹ ≤α≤ _12ε⁵ andε≤ε₀.

(iii) There existsεˆ ∈ (0,ε0)such that T¯⁰(α) ≥ 0 for 0 < α ≤ _3ε¹ and εˆ ≤ ε < ε0. Furthermore, ˆ

ε<√

31/1000.

Proof. The statement (i) follows immediately by Theorem 1.2 and (2.4). The statement (ii) follows immediately by [6, Lemma 3.5]. By [11, Theorem 2.1], there exists ˆε>0 satisfying

ˆ ε<

r 31 1000 < ε₀ such that

T¯⁰ 1

3ε











<0 for 0<ε< ε,ˆ

=0 forε =ε,ˆ

>0 for ˆε <ε<ε₀.

(2.21) By Theorem1.2, (2.4) and [6, Lemma 3.3], we see that, for 0< ε< ε₀, there exist two positive numbersα∗ <α^∗ <β_ε such that

T¯⁰(α)











>0 on (0,α∗)∪(α^∗,βε),

=0 whenα=α∗ or α=α^∗,

<0 for (α∗,α^∗).

(2.22)

Since f is a convex function on 0,_3ε¹

, and by [15, Lemma 3.2], we see that ¯T(α) is either strictly increasing on 0,_3ε¹

, or strictly increasing and then strictly decreasing on 0,_3ε¹ . So by (2.21) and (2.22), we observe that _3ε¹ ≤ α∗ for ˆε ≤ ε < ε0. It follows that ¯T⁰(α) ≥ 0 for 0<α≤ _3ε¹ and ˆε≤ε <ε₀. So the statement (iii) holds. The proof is complete.

Lemma 2.6. Consider(1.5)with0<ε≤εˆwhereεˆis defined in Lemma2.5. ThenαT¯⁰⁰(α) +T^¯0(α)<

0for1≤α≤1.7.

Proof. Let ¯A≡ε α⁴−u⁴

, ¯B≡α³−u³, ¯C≡α²−u²and ¯D≡α−u. We compute that αT¯⁰⁰(α) +T^¯0(α) = ¹

4√ 2α

Z _α

0

N₁(α,u)

[F(α)−F(u)]^{5/2du, (2.23)} where

N₁(α,u)≡ ¹ 72

9 ¯A²+4 ¯B²+36 ¯D²−6 ¯AB¯+198 ¯AD¯ −120 ¯BD¯ +36 ¯AC¯−12 ¯BC¯ −36 ¯CD¯ .

Letα∈[1, 1.7],u∈(0,α)andε∈(0, ˜ε]be given. By Lemma [11, Lemma 3.6], we have A¯ < ^4εα

3 B¯ and D¯ > ¹

3α²B¯ > ¹ 3α²

3 4εαA¯

= ^A¯ 4α³ε. Then

1<α²< ^α

2+αu+u²D¯

D¯ = ^B¯

D¯ <3α²≤3(1.7)²=8.67, (2.24) A¯ < ^4εα

3 B¯ < ^4ˆε

3 (1.7)B^¯ = ^34ˆε

15 B¯ and D¯ > ^A¯

4α³ε > ^A¯ 4(1.7)³εˆ

= ²⁵⁰ 4913ˆε

A.¯ (2.25) In addition, by Lemma 2.5(iii), we compute and find that

34 15εˆ− ²

3 < ³⁴ 15

r 31 1000−²

3(≈ −0.26)<0, (2.26) 198

34 15εˆ−²⁰

33

<198 34

15 r 31

1000− ²⁰ 33

!

(≈ −40.98)<−0.40, (2.27) 1− ⁵

34ˆε − ²⁵⁰

4913ˆε <1− ⁵ 34

q 31 1000

− ²⁵⁰ 4913

q 31 1000

(≈ −0.88)<0. (2.28) By (2.24)–(2.28), we observe that

N₁(α,u) = ¹

72 9 ¯A²+4 ¯B²+36 ¯D²−6 ¯AB¯+198 ¯AD¯ −120 ¯BD¯ +36 ¯AC¯−12 ¯BC¯−36 ¯CD¯

= ¹ 72

9 ¯A

A¯− ²

3B¯

+198 ¯D

A¯ −²⁰ 33B¯

+36 ¯C

A¯−¹

3B¯−D^¯

+4 ¯B²+36 ¯D²

< ¹ 72

9 ¯AB¯

34 15εˆ−²

3

+_{198 ¯}BD¯ 34

15εˆ−²⁰ 33

+_{36 ¯}AC¯

1− ⁵ 34ˆε

− ²⁵⁰ 4913ˆε

+_{4 ¯}B²+_{36 ¯}D²

< ¹ 72

−_{40 ¯}BD¯ +_{4 ¯}B²+_{36 ¯}D²

= ^D¯

2

18

"B¯ D¯ −₅

2

−₁₆

#

< ^D¯

2

18 h

(1−5)²−16i

=0.

So by (2.23), we obtain thatαT¯⁰⁰(α) +T^¯0(α) < 0 for 1≤ α ≤ 1.7 and 0< ε ≤ ε. The proof isˆ complete.

Lemma 2.7. Consider(1.5)with0.07≤ε≤ ε. Thenˆ αT¯⁰⁰(α) + ⁵₂T^¯0(α)<0for1.7≤α≤ _3ε¹. Proof. We compute that

αT¯⁰⁰(α) + ⁵

2T¯⁰(α) = ¹ 4√

2α Z _α

0

N₂(α,u)

[F(α)−F(u)]^{5/2du, (2.29)} where

N₂(α,u)≡ ¹ 144

−_{9 ¯}A²+_{42 ¯}AB¯ +_{450 ¯}AD¯ +_{126 ¯}AC¯−_{16 ¯}B²−_{240 ¯}BD¯

−60 ¯BC¯+288 ¯D²+36 ¯CD¯

. (2.30)

Then we assert that

N₂(α,u)<0 for 0<u<α, 1.7≤α≤ ¹

3ε and 0.07≤ε≤ε.ˆ (2.31) The proof of assertion (2.31) is easy but tedious. Thus, we put it in Appendix. So by (2.29)–

(2.31), we see thatαT¯⁰⁰(α) + ⁵₂T^¯0(α)<0 for 1.7≤α≤ _3ε¹ and 0.07≤ ε≤ε.ˆ Lemma 2.8. Consider(1.5)with0<ε<0.07. ThenT¯⁰(α)<0for1.7≤ α≤ _3ε¹. Proof. We compute that

T¯⁰(α) = ¹ 2√

2α Z _α

0

2B(α,u)−A(α,u)

B^3/2(α,u) ^du= ¹ 2√

2α Z _α

0

θ(α)−θ(u)

B^3/2(α,u) ^{du, (2.32)} whereθ(u)≡2F(u)−u f(u)for 0≤u< β_ε. Since 0 <ε<0.07, and by [11, Lemma 3.1], there exists p ∈ 0,_3ε¹

such that θ⁰(u) > 0 for(0,p)and θ⁰(u) < 0 for p,_3ε¹

. Let α ∈ 1.7,_3ε¹ be given. Assume that θ(α) ≤ 0, see Figure2.1(i). Since θ(0) = 0, we see that θ(α)−θ(u) < 0 for 0<u<α. So by (2.32), we obtain that ¯T⁰(α)<0. Assume thatθ(α)>0, see Figure2.1(ii).

We compute and find that

θ⁰(1.7) = 2εu³−u²+1

u=1.7= ⁴⁹¹³

500 ε−¹⁸⁹

100 <0 for 0< ε<0.07.

Since 1.7≤α≤ _3ε¹, there exists ¯α∈(0,p)such that

θ(α)−θ(u)











>0 for 0<u<α,¯

=0 foru=α,¯

<0 for ¯α<u<α.

Figure 2.1: Graphs ofθ(u)on[0,α]where 1.7≤ α≤ _3ε¹ and 0<ε<0.07.

So by (2.32) and similar argument of [14, (3.11)], we observe that T¯⁰(α)< ¹

2√

2αB^3/2(α, ¯α)

Z _α

0

uθ⁰(u)du= ^{α 8εα}

3−5α²+10 40√

2B^3/2(α, ¯α) ^{. (2.33)} Since

∂

∂u 8εu³−5u²+10

=2u(12εu−5)<0 for 1.7≤ u≤ ¹ 3ε,

we see that, for 1.7≤u ≤ _3ε¹ and 0<ε<0.07, 8εu³−5u²+₁₀< 8εu³−5u²+₁₀

u=1.7= ⁴⁹¹³ 125 ε− ⁸⁹

20 <_0.

So by (2.33), we obtain that ¯T⁰(α)<0. The proof is complete.

Lemma 2.9. Consider(1.1)with0<ε<ε₀. Then there existsξε >0such that Γε ≡ {λ>0 :T_λ⁰(α)<0for someα∈(0,βε)}= (0,ξε).

Proof. Letε ∈ (0,ε₀)be given. By (2.22), there exist two positive numbers α∗ < α^∗ < β_ε such that

lim

λ→0⁺

√

λT_λ⁰(α) =T^¯0(α)











>0 on (0,α∗)∪(α^∗,β_ε),

=0 whenα=α∗ orα^∗,

<_{0 on} (α∗,α^∗).

(2.34)

Then we divide this proof into the next four steps.

Step 1. We prove thatα∗ < _12ε⁵ . Assume thatα∗≥ _12ε⁵ . By (2.34) and Lemma2.3, we see that 0≤T^¯0(α) = _lim

λ→0⁺

√

λT_λ⁰(α)<√

λT_λ⁰(α) _{for 0}<α≤ ⁵

12ε andλ>_{0. (2.35)} By Lemma 2.2(ii) and (2.35), we further see that T_λ⁰(α) > 0 for 0 < α < βε for λ > 0. So by (2.34), we obtain that

0≤ lim

λ→0⁺

√ λT_λ⁰

α∗+α^∗ 2

=_T^¯0

α∗+α^∗ 2

<_0, which is a contradiction. It implies that α∗< _12ε⁵ .

Step 2. We prove that, for α ∈ (α∗,α^∗)∩ 0,_12ε⁵

, there exists a continuously differential function ˜λα >0 ofαsuch that

√

λT_λ⁰(α)











<0 if 0<λ<λ^˜_α,

=0 ifλ=λ^˜α,

>_{0 if}λ>λ^˜_α.

(2.36)

By Lemma2.1(ii), we see that lim

λ→_∞

√

λT_λ⁰(α) =_∞·1=_∞ forα∈(0,βε). (2.37) By (2.34), (2.37), Lemma2.3and implicit function theorem, we observe that, forα∈(α∗,α^∗)∩

0,_12ε⁵

, there exists a continuously differential function ˜λα >0 of αsuch that (2.36) holds.

Step 3. We prove that

ξε ≡sup

λ˜α :α∈(α∗,α^∗)∩

0, 5 12ε

∈ (0,∞). Clearly,ξ_ε >0. By (2.34) and Lemma2.3, we see that

0= lim

λ→0⁺

√

λT_λ⁰(α∗)< T_λ⁰₌₁(α∗).

So by Lemma2.3 and continuity ofT_λ⁰₌₁(α)with respect toα, there existsδ>0 such that 0< T_λ⁰₌₁(α)≤√

λT_λ⁰(α) _forα∗< α<α∗+δ< ⁵

12ε andλ≥_1,

from which it follows that ˜λα <1 for α∗ < α<α∗+δ. Thus lim_α→α⁺_∗ λ˜α ≤ 1< ∞. By similar argument, we obtain that

lim

α→(α^∗)⁻

λ˜α <_∞ if α^∗< ⁵ 12ε. So by Step 2, we observe thatξ_ε ∈(0,∞).

Step 4. We prove that Γε = (_0,ξ_ε)_{. Let} λ₁ ∈ (_0,ξ_ε). There exists α₁ ∈ (α∗,α^∗)∩ _0,12ε⁵ _such that λ₁ < λ^˜α₁. Then by (2.36), we see that T_λ⁰

1(α₁) < 0, which implies that λ₁ ∈ _Γε. Thus (0,ξ_ε) ⊆ _Γε. Letλ₂ ∈ _Γε. There existsα₂ ∈ (0,β_ε) such thatT_λ⁰

2(α₂) < 0. Next, we consider two cases.

Case 1. Assume that _12ε⁵ <α^∗. By (2.34) and Lemma2.3, we see that 0≤ lim

λ→0⁺

√

λT_λ⁰(α)<

√

λT_λ⁰(α) forα∈(0,α∗]andλ>0. (2.38) By Steps 2 and 3, we see that

√

λT_λ⁰(α)≥0 forα∈

α∗, 5 12ε

ifλ≥ξε. (2.39)

By (2.39) and Lemma2.2, we see that

T_λ⁰(α)>0 for 5

12ε ≤ α< β_ε andλ≥ξ_ε. (2.40) So by (2.38)–(2.40), we obtain that T_λ⁰(α)≥0 for α∈ (0,β_ε)if λ≥ ξ_ε. It implies thatλ₂ < ξ_ε. ThusΓε ⊆(_0,ξ_ε)_.

Case 2. Assume thatα^∗ < _12ε⁵ . By (2.34) and Lemma2.3, we see that 0≤ lim

λ→0⁺

√

λT_λ⁰(α)<√

λT_λ⁰(α) forα∈ (0,α∗]∪

α^∗, 5 12ε

andλ>0. (2.41) By Steps 2 and 3, we see that

√

λT_λ⁰(α)≥_{0 for}α∈(α∗,α^∗) ifλ≥ξ_ε. (2.42) By (2.41) and Lemma2.2(ii), we see that

T_λ⁰(α)>_{0 for} ⁵

12ε ≤α< β_ε andλ>_{0. (2.43)} So by (2.41)–(2.43), we obtain that T_λ⁰(α)≥0 for α∈ (0,β_ε)if λ≥ ξ_ε. It implies thatλ₂ < ξ_ε. ThusΓε ⊆(0,ξε).

By the above discussions, we obtain thatΓε = (0,ξ_ε). The proof is complete.

Lemma 2.10. Consider (1.1) with 0 < ε < ε₀. Then there exists κε ∈ (0,ξε)such that T_λ(α)has exactly two critical points, a local maximum atα_M(λ)and a local minimum atα_m(λ) (>α_M(λ)), on (0,β_ε)if0< λ<κ_ε.

Proof. Letε∈(0,ε₀)be given. By (2.34) and Lemma2.1(ii), there existsλ₁ >0 such that T_λ⁰

α∗+α^∗ 2

<0 for 0<λ<λ₁. (2.44) We divide this proof into the next four steps.

Step 1. We prove that there existsλ₂ ∈ (0,λ₁)such that, for 0 <λ <λ₂, either T_λ⁰(α)> 0 on 0,_3ε¹

, orT_λ(α)has exactly one critical point, a local maximum, on 0,_3ε¹

, see Figure2.2. By Lemma2.2(i), we have

T_λ⁰(α)>0 for 0< α≤1 andλ>0. (2.45)

0 α ⁰ α

T _λ ⁽ α ) T _λ ^{( α )}

(i) (ii)

1

1

Figure 2.2: Graphs ofT_λ(α)_on(_0,3ε¹]_{for 0}<λ<λ₂. Then we consider the following three cases.

Case 1. Assume that ˆε≤ε<ε₀. By Lemmas2.1(ii),2.3and2.5(iii), we see that 0≤T^¯0(α) = lim

λ→0⁺

√

λT_λ⁰(α)<√

λT_λ⁰(α) for 1< α≤ ¹

3ε andλ>0.

So by (2.45),T_λ⁰(α)>0 on 0,_3ε¹

forλ>0, see Figure 2.2(i).

Case 2. Assume that 0.07 ≤ ε < ε. By (2.21), Lemmasˆ 2.1(ii),2.4(ii),2.6 and 2.7, there exists λ₂∈(0,λ₁)such that

T_λ⁰ 1

3ε

<0 and αT_λ⁰⁰(α) +K(α)T_λ⁰(α)<0 for 1≤α≤ ¹

3ε and 0<λ<λ₂, (2.46) where K(α) ≡ 1 if 1 ≤ α ≤ 1.7, and K(α) ≡ 5/2 if 1.7 < α ≤ _3ε¹. By (2.45) and (2.46), there exists α_λ∈ 1,_3ε¹

such thatT_λ⁰(α_λ) =0 for 0<λ<λ₂. Furthermore,

α_λT_λ⁰⁰(α_λ) =α_λT_λ⁰⁰(α_λ) +K(α_λ)T_λ⁰(α_λ)<0 for 0<λ<λ₂. Thus T_λ(α)has exactly one local maximum atα_λon 0,_3ε¹

for 0< λ<λ₂, see Figure2.2(ii).

Case 3. Assume that 0< ε <0.07. By Lemmas2.4,2.6 and2.8, there existsλ₂ ∈ (0,λ₁)such that

αT_λ⁰⁰(α) +T_λ⁰(α)<0 for 1≤α≤1.7 and 0<λ<λ₂, (2.47) T_λ⁰(α)<0 for 1.7≤α≤ ¹

3ε and 0<λ<λ₂. (2.48)