**Bifurcation curves of positive solutions for the**

**Minkowski-curvature problem with cubic nonlinearity**

**Shao-Yuan Huang**

^{B}

^{1}

### and **Min-Shu Hwang**

^{2}

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_{L}of positive solutions
for the Minkowski-curvature problem

−

u^{0}(x)
q

1−(u^{0}(x))^{2}

0

=*λ*

−*εu*^{3}+u^{2}+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 exists*ε*_{0}>0 such that the bifurcation curveS_{L} is monotone increasing
for all L > 0 if*ε* ≥ *ε*0, and the bifurcation curve SL is from monotone increasing to
S-shaped for varyingL>0 if 0<*ε*<*ε*0.

**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∈C^{2}(−L,L)∩
C[−L,L]for the one-dimensional Minkowski-curvature problem

−

u^{0}(x)
q

1−(u^{0}(x))^{2}

0

=*λ*f(u), −L< x< L,
u(−L) =u(L) =0,

(1.1)

where*λ*>0 is a bifurcation parameter, L>0 is an evolution parameter and the nonlinearity
f(u)≡ −*εu*^{3}+u^{2}+u+1, *ε*>0. (1.2)

BCorresponding author. Email: syhuang@mail.ntue.edu.tw

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* _{λ}*k

_{∞}):

*λ*>0 andu

*is a positive solution of (1.1)} for L>0. (1.3) Thus this investigation is essential.*

_{λ}Before going into further discussions on problems (1.1), we give some terminologies in this
paper for the shape of bifurcation curveS_{L}on 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 *λ*_{1},ku_{λ}_{1}k_{∞}^{} and *λ*_{2},ku_{λ}_{2}k_{∞}^{} where *λ*_{1} < *λ*_{2} are two positive
numbers such that:

(a) at *λ*_{1},ku_{λ}_{1}k_{∞}^{}the bifurcation curveS_{L}turns to the right,
(b) ku_{λ}_{2}k_{∞}< ku_{λ}_{1}k_{∞},

(c) at *λ*_{2},ku_{λ}_{2}k_{∞}^{}the bifurcation curveS_{L}turns 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 *λ*_{1} < *λ*_{2} for any
two points *λ*_{i},ku_{λ}_{i}k_{∞}^{},i=1, 2, lying inS_{L}withku_{λ}_{1}k_{∞} ≤ ku_{λ}_{2}k_{∞}.

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

(−_{∆u}(x) =*λ* −*εu*^{3}+u^{2}+u+1

inΩ,

u(x) =0 on *∂*Ω, (1.4)

whereΩis a general bounded domain in**R**^{n}(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* _{λ}*k

_{∞})-plane when

*ε*>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

**R**

^{n}with 1 ≤ n ≤ 6. Hung and Wang [6]

consider the one-dimensional case

(−u^{00}(x) =*λ* −*εu*^{3}+u^{2}+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 varying*ε*>0 where

S¯≡ {(*λ,*ku* _{λ}*k

_{∞}):

*λ*>0 andu

*is a positive solution of (1.5)}, (1.6) see Theorem1.2.*

_{λ}Figure 1.1: Graphs of bifurcation curves ¯Sof (1.4). (i)*ε*≥*ε*0 and (ii) 0<*ε*<*ε*0.

**Theorem 1.2** ([6, Theorem 3.1]). Consider (1.5). Then the bifurcation curveS is continuous on the¯
(*λ,*ku* _{λ}*k

_{∞})-plane, starts from(0, 0)and goes to infinity. Furthermore, there exists a critical bifurcation value

*ε*

_{0}∈ (0, 1/√

27)such that the bifurcation curve S is monotone increasing if¯ *ε* ≥ *ε*_{0},and S is¯
S-shaped if0<*ε*<*ε*_{0}, 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 *β**ε* in [0,∞). Then the main
result is as follows:

**Theorem 1.3**(See Figure1.2). Consider(1.1). Let*ε*_{0} 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* _{λ}*k

_{∞})-plane, starts from (0, 0) and goes to infinity along the horizontal linekuk

_{∞}=

*ρ*

_{L,ε}where

*ρ*

_{L,ε}≡min{L,

*β*

*}.*

_{ε}(ii) If*ε*≥*ε*_{0}, then the bifurcation curve S_{L}is monotone increasing for all L>0.

(iii) If0<*ε*<*ε*_{0}, then there exist two positive numbers L* _{ε}* <L

^{˜}

*such that (a) the bifurcation curve S*

_{ε}_{L}is monotone increasing for0<L≤ L

*ε*. (b) the bifurcation curve S

_{L}is S-like shaped for L

*< L≤L*

_{ε}^{˜}

*. (c) the bifurcation curve S*

_{ε}_{L}is S-shaped for L> L

^{˜}

*.*

_{ε}Furthermore, L*ε* is a continuous function of*ε* ∈(0,*ε*0), lim* _{ε}*→0

^{+}L

*ε*∈ (0,∞)andlim

_{ε}_{→}

_{ε}^{−}

0 L*ε* =

∞.

**Remark 1.4.** By numerical simulations to bifurcation curvesS_{L}of (1.1), we conjecture that the
bifurcation curveS_{L} is also S-shaped on the(* _{λ,}*ku

*k*

_{λ}_{∞})

_{-plane for}L

*< L≤ L*

_{ε}^{˜}

*and 0<*

_{ε}*ε*<

*ε*

_{0}. 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_{L}of (1.1) for*ε*>0.

Bifurcation curve S_{L} of (1.1) S¯ of (1.4)

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

to S-shaped with varying*ε* ^{S-shaped}

2. Shapes (*ε*≥*ε*_{0}) monotone increasing monotone increasing

3. Numbers of turning points

(1). from 0 to 2 varying L>0 if0<*ε*<*ε*0

(2). 0 if*ε*≥ *ε*_{0}

(1). 2 if0<*ε*<*ε*0

(2). 0 if *ε*≥*ε*_{0}

4. Continuity continuous continuous

5. Evolution

parameter(s) *ε*and L *ε*

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

7. "End point" (_{∞,ρ}_{L,ε}) (_{∞,}_{∞})

Table 1.1: Comparison of properties ofS_{L}and ¯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}* and

*λ*>0, (2.1)

where F(u) ≡ Ru

0 f(t)dt. Observe that positive solutionsu* _{λ}* ∈ C

^{2}(−L,L)∩C[−L,L] for (1.1) correspond to

ku* _{λ}*k

_{∞}=

*α*and T

*(*

_{λ}*α*) =L.

So by definition ofS_{L}in (1.3), we have that

S_{L}= {(*λ,α*): T* _{λ}*(

*α*) =L for some 0<

*α*<

*β*

*and*

_{ε}*λ*>0}. (2.2) Thus, it is important to understand fundamental properties of the time-map T

*(*

_{λ}*α*)on (0,

*β*

*ε*) in order to study the shape of the bifurcation curve S

_{L}of (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*

*)for*

_{ε}*ε,λ*>

_{0, and}T

*(*

_{λ}*α*),T

_{λ}^{0}(

*α*)are differentiable function of

*λ*>0 for 0<

*α*<

*β*

*ε*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} ^{for}* ^{α}*>0, (2.3)
see [12, p. 779]. Then we have thatku

*λ*k

_{∞}=

*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)≡

*α*f(

*α*)−u f(u)

_{,}B= B(

*u)≡ F(*

_{α,}*α*)−F(u)

_{,}C=

_{C}(

_{α,}_{u})≡

*α*

^{2}f

^{0}(

*α*)−u

^{2}f

^{0}(

_{u})

_{and}

_{D}=

_{D}(

_{α,}_{u})≡

*α*

^{3}f

^{00}(

*α*)−u

^{3}f

^{00}(

_{u})

_{.}Obviously, we have

B(*α,*u) =

Z _{α}

u f(t)dt>0 for 0<u<*α*< *β**ε* (2.5)
because f(_{u})>_{0 for 0}<_{u}< *β** _{ε}*.

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

(i) lim* _{α}*→0

^{+}T

*(*

_{λ}*α*) =0andlim

_{α}_{→}

_{β}^{−}

*ε* T* _{λ}*(

*α*) =

_{∞}for

*λ*>0.

(ii) lim* _{λ}*→0

^{+}

√

*λT*_{λ}^{(}^{i}^{)}(*α*) =_{T}^{¯}^{(}^{i}^{)}(*α*)_{and}_{lim}_{λ}_{→}_{∞}_{T}^{0}

*λ*(*α*) =_{1}_{for}_{0}<*α*<*β** _{ε}* and i=

_{1, 2, 3.}

(iii) *∂T** _{λ}*(

*α*)/∂λ<0for0<

*α*<

*β*

*and*

_{ε}*λ*>0.

Proof. Since

ulim→0^{+}

F(u)
u^{2} =_{∞,}

and by [7, Lemma 3.1], we obtain that lim_{α}_{→}_{0}^{+}T* _{λ}*(

*α*) =0. Since f(

*β*

*) =0, there existb,c∈*

_{ε}**such that f(u) = (**

_{R}*β*

*−u)(*

_{ε}*εu*

^{2}+bu+c). Since f(u) > 0 on (0,

*β*

*), there exists M > 0 such that 0<*

_{ε}*εu*

^{2}+bu+c < M for 0 < u <

*β*

*ε*. For 0< t < 1, by the mean-value theorem, there exists

*η*

_{t}∈(

*β*

*t,*

_{ε}*β*

*)such that*

_{ε}B(*β** _{ε}*,

*β*

*t) =*

_{ε}Z _{β}_{ε}

*β**ε*t f(t)dt= f(*η*_{t})*β** _{ε}*(1−t) = (

*β*

*−*

_{ε}*η*

_{t})

*εη*

_{t}

^{2}+bη

_{t}+c

*β** _{ε}*(1−t)

<(*β**ε*−*β**ε*t)M*β**ε*(1−t) =Mβ^{2}* _{ε}*(1−t)

^{2}. (2.6) Then there existst

^{∗}∈ (0, 1)such that B(

*β*

*ε*,

*β*

*ε*t) < 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

*λ*^{2}B^{2}(* _{α,}αt*) +2λB(

*)*

_{α,}αt^{dt}

≥ lim

*α*→*β*^{−}*ε*

*α*
Z _{1}

t^{∗}

1 p

*λ*^{2}B^{2}(*α,αt*) +2λB(*α,αt*)^{dt}

≥*β**ε*

Z _{1}

t^{∗}

1

p(*λ*^{2}+2λ)B(*β** _{ε}*,

*β*

*t)*

_{ε}^{dt}≥

_{p}

^{1}(

*λ*

^{2}+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_{λ}^{0}(*α*) = ^{1}
*α*

Z _{α}

0

*λ*^{3}B^{3}+3λ^{2}B^{2}+*λ*(2B−A)

(*λ*^{2}B^{2}+2λB)^{3/2} ^{du,} ^{(2.7)}

T_{λ}^{00}(*α*) = ^{1}
*α*^{2}

Z _{α}

0

3A^{2}B−B^{2}C−2AB^{2}

*λ*^{3}+ 3A^{2}−4AB−2BC
*λ*^{2}

(*λ*^{2}B^{2}+2λB)^{5/2} ^{du,} ^{(2.8)}

T_{λ}^{000}(*α*) = ^{1}
*α*^{3}

Z _{α}

0

*λ*^{3}
[*λ*^{2}B^{2}+2λB]^{7/2}

h

B^{2} 9A^{2}B−3B^{2}C−B^{2}D−12A^{3}+9ABC
*λ*^{2}

+B(27A^{2}B−12B^{2}C−4B^{2}D−24A^{3}+27ABC)*λ*+18A^{2}B−12B^{2}C

−4B^{2}D−15A^{3}+18ABCi

du. (2.9)

So we observe that, for 0<*α*< *β**ε*,
lim

*λ*→0^{+}

√

*λT*_{λ}^{0}(*α*) = ^{1}
*α*

Z _{α}

0

2B−A

(2B)^{3/2}^{du}= T^{¯}^{0}(*α*),
lim

*λ*→0^{+}

√

*λT*_{λ}^{00}(*α*) = ^{1}
*α*^{2}

Z _{α}

0

3A^{2}−4AB−2BC

(2B)^{5/2} ^{du}= T^{¯}^{00}(*α*),
lim

*λ*→0^{+}

√

*λT*_{λ}^{000}(*α*) = ^{1}
*α*^{3}

Z _{α}

0

18A^{2}B−12B^{2}C−4B^{2}D−15A^{3}+18ABC

(2B)^{5/2} ^{du}=T^{¯}^{000}(*α*).

Furthermore, lim*λ*→_{∞}T_{λ}^{0}(*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)with*ε*>0. Then the following statements (i) and (ii) hold:

(i) T_{λ}^{0}(*α*)>0for0< *α*≤1and*λ*>0.

(ii) T* _{λ}*(

*α*)has at most one critical point, a local minimum, on[

_{12ε}

^{5}

_{,}

*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})] =−2εu^{3}+ _{u}^{2}−1

<_{0 for 0}<_{u} <*α*<_{1.}

So by (2.5) and (2.7), we obtain that T_{λ}^{0}(*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}* and

*λ*>0,

*αT*_{λ}^{00}(*α*) +2T_{λ}^{0}(*α*)

= ^{1}
*α*

Z _{α}

0

B^{5}*λ*^{3}+5B^{4}*λ*^{2}+*λB* 3A^{2}+16B^{2}−4AB−BC

+3A^{2}+8B^{2}−8AB−2BC

√

*λ*(*λB*^{2}+2B)^{5/2} ^{du}

> ^{1}
*α*

Z _{α}

0

*λB* 3A^{2}+16B^{2}−4AB−BC

+3A^{2}+8B^{2}−8AB−2BC

√

*λ*(*λB*^{2}+2B)^{5/2} ^{du}

= ^{1}
*α*

Z _{α}

0

*λB*h

3(A−B)^{2}+5B^{2}+B(2A−2B−C)
i

+_{3}(A−2B)^{2}+2B(2A−2B−C)

√

*λ*(*λB*^{2}+2B)^{5/2} ^{du}

> ^{1}
*α*

Z _{α}

0

*λB*^{2}(2A−2B−C) +2B(2A−2B−C)

√

*λ*(*λB*^{2}+2B)^{5/2} ^{du}

= ^{1}
*α*

Z _{α}

0

*λB*^{2}+2B

(2A−2B−C)

√

*λ*(*λB*^{2}+2B)^{5/2} ^{du}= ^{1}
*α*

Z _{α}

0

2A−2B−C

√

*λ*(*λB*^{2}+2B)^{3/2}^{du}

= ^{1}
6α

Z _{α}

0

*φ*(*α*)−*φ*(u)

√

*λ*(*λB*^{2}+2B)^{3/2}^{du,} ^{(2.10)}

where*φ*(u)≡u^{3}(9εu−4). Clearly,*φ*^{0}(u) =12u^{2}(3εu−1). Since
f

4 9ε

=_{1}+^{324ε}+80
729ε^{2} >_{0,}
we see that

1
3ε < ^{4}

9ε <*β**ε*. (2.11)

So we observe that

*φ*(u)

<0 for 0< u< _{9ε}^{4},

=0 foru= _{9ε}^{4},

>0 for _{9ε}^{4} <u<*β** _{ε}*,

and *φ*^{0}(u)

<0 for 0<u< _{3ε}^{1},

=0 foru = _{3ε}^{1},

>0 for _{3ε}^{1} < u<*β** _{ε}*.

(2.12)

Let*α*∈^{}_{12ε}^{5} ,*β**ε*

be given. Then we consider two cases.

**Case 1. Assume that** _{9ε}^{4} ≤*α*< *β**ε*. Since *φ*(0) =0, and by (2.12), we see that*φ*(*α*)−*φ*(u)>0
for 0<u<*α. So by (2.10), we obtainαT*_{λ}^{00}(*α*) +2T_{λ}^{0}(*α*)>0 for*λ*>0.

**Case 2.** Assume that _{12ε}^{5} ≤ *α*< _{9ε}^{4}. Since*φ*(0) =0, and by (2.12), there exists ˜*α*∈ 0,_{3ε}^{1}
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, for*λ*>0,
*αT*_{λ}^{00}(*α*) +2T_{λ}^{0}(*α*)

> ^{1}

6α√
*λ*

"

Z _{α}_{˜}

0

*φ*(*α*)−*φ*(u)
[*λB*^{2}+2B]^{3/2}^{du}

+

Z _{α}

˜
*α*

*φ*(*α*)−*φ*(u)
[*λB*^{2}+2B]^{3/2}^{du}

#

> ^{1}

6α√

*λ*[*λB*^{2}(*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*[*λB*^{2}(*α, ˜α*) +2B(*α, ˜α*)]^{3/2}

Z _{α}

0

[*φ*(*α*)−*φ*(u)]du

= ^{6εα}

3

5√

*λ*[*λB*^{2}(*α, ˜α*) +2B(*α, ˜α*)]^{3/2}

*α*− ^{5}
12ε

≥0.

Thus by Cases 1–2, we have

*αT*_{λ}^{00}(*α*) +2T_{λ}^{0}(*α*)>0 for 5

12ε ≤*α*< *β** _{ε}* and

*λ*>0. (2.13) Fixed

*λ*> 0. If T

*(*

_{λ}*α*) has a critical point ˘

*α*in [

_{12ε}

^{5},

*β*

*ε*), by (2.13), then ˘

*αT*

_{λ}^{00}(

*α*˘) =

*αT*˘

_{λ}^{00}(

*α*˘) + 2T

_{λ}^{0}(

*α*˘)>0. It implies that T

*(*

_{λ}*α*)has at most one critical point, a local minimum, on

_{5}

12ε,*β**ε*

for*λ*>0. Then the statement (ii) holds. The proof is complete.

**Lemma 2.3.** Consider(1.1)with*ε*>0. Then

*∂*

*∂λ*
h√

*λT*_{λ}^{0}(*α*)^{i}>0 for0< *α*≤ ^{5}

12ε and*λ*>0. (2.14)

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

*∂*

*∂λ*
h√

*λT*_{λ}^{0}(*α*)^{i}= ^{1}
2α

Z _{α}

0

B^{2} B^{3}*λ*^{2}+_{5B}^{2}*λ*+_{3A}+_{6B}^{}

(*λB*^{2}+2B)^{5/2} ^{du}> ^{1}
2α

Z _{α}

0

3B^{2}(A+2B)

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

*∂*

*∂u* [A(*α,*u) +2B(*α,*u)] = R(u),

whereR(u)≡3εu^{3}−3(1−*ε*)u^{2}−6u−4. Clearly,R^{0}(u) =9εu^{2}−6(1−*ε*)u−6 is a quadratic
polynomial ofuwith positive leading coefficient. Furthermore,

R^{0}(0) =−6<0 and R^{0}
5

12ε

≡ −^{56ε}+15
16ε <0.

Thus we observe thatR^{0}(u)<_{0 for 0}≤u≤ _{12ε}^{5} . 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*)<0for*α*∈ I,then there exists*λ*ˇ >0such that T_{λ}^{0}(*α*)<0for*α*∈ I and0<*λ*< *λ.*^{ˇ}
(ii) If*α*T¯^{00}(*α*) +kT¯^{0}(*α*)<0for*α*∈ I and some k>0,then there exists*λ*ˆ >0such that*αT*_{λ}^{00}(*α*) +

kT_{λ}^{0}(*α*)<0for*α*∈ I and0<*λ*<*λ.*^{ˆ}

(iii) If[2αT¯^{00}(*α*) +3 ¯T^{0}(*α*)]^{0} >0for*α*∈ I,then there exists*λ*¯ >0such that[2αT_{λ}^{00}(*α*) +3T_{λ}^{0}(*α*)]^{0} >

0for*α*∈ I and0< *λ*<*λ.*^{¯}

Proof. (I) Assume that ¯T^{0}(*α*)<0 for*α*∈ I. By Lemma2.1(ii), we have
lim

*λ*→0^{+}

√

*λT*_{λ}^{0}(*α*) =T^{¯}^{0}(*α*)<0 for*α*∈ I. (2.16)
For *α*∈ I, by (2.16), we define*λ** _{α}* by

*λ** _{α}* ≡

1 if T_{λ}^{0}(*α*)<0 for all*λ*>0,
sup{*λ*_{1} :T_{λ}^{0}(*α*)<0 for 0<*λ*<*λ*_{1}} if T_{λ}^{0}(*α*)≥0 for some*λ*>0.

(2.17)

Clearly, T_{λ}^{0}(*α*)<0 for *α*∈ I and 0< *λ*<*λ**α*. Let ˇ*λ*≡inf{*λ**α* :*α*∈ I}. Assume that ˇ*λ*= 0. By
(2.17), there exists a sequence {*α*_{k}}_{k}_{∈}** _{N}**⊂ I such that

klim→∞*λ*_{α}_{k} =_{0} _{and} _{T}_{λ}^{0}

*α*k(*α*_{k})≥0 fork ∈** _{N.}** (2.18)
Without loss of generality, we assume that lim

_{k}

_{→}

_{∞}

*α*

_{k}=

*α*ˇ ∈ I. So by (2.16) and (2.18), we observe that

0≤ _{lim}

k→_{∞}

q
*λ*_{α}_{k}T_{λ}^{0}

*α*k(*α*_{k}) = _{lim}

k→_{∞}

q
*λ*_{α}_{k}T_{λ}^{0}

*α*k(*α*ˇ) =T^{¯}^{0}(*α*ˇ)<_{0,}
which is a contradiction. It implies that ˇ*λ*>0. So statement (i) holds.

(II) Assume that*α*T¯^{00}(*α*) +kT¯^{0}(*α*)< 0 for*α*∈ I and some k >0. Let G_{1}(*α,λ*)≡ *αT*_{λ}^{00}(*α*) +
kT_{λ}^{0}(*α*)_{. By Lemma}2.1(ii), we see that

lim

*λ*→0^{+}

√

*λG*_{1}(*α,λ*) =*α*T¯^{00}(*α*) +kT¯^{0}(*α*)<0 for*α*∈ I. (2.19)
For *α*∈ I, by (2.19), we define*λ**α* by

*λ**α* ≡

1 ifG_{1}(*α,λ*)<0 for all*λ*>0,

sup{*λ*_{2}: G_{1}(*α,λ*)<0 for 0< *λ*<*λ*_{2}} ifG_{1}(*α,λ*)≥0 for some*λ*>0.

Clearly, G_{1}(* _{α,}λ*)<

_{0 for}

*α*∈ I and 0 <

*λ*<

*λ*

*. Let ˆ*

_{α}*λ*≡

_{inf}{

*λ*

*:*

_{α}*α*∈ I}. We use the similar argument in (I) to obtain that ˆ

*λ*>0. So statement (ii) holds.

(III) Assume that[2αT¯^{00}(*α*) +3 ¯T^{0}(*α*)]^{0} >0 for*α*∈ I. LetG_{2}(*α,λ*)≡[2αT^{00}(*α*) +3T^{0}(*α*)]^{0}. By
Lemma2.1(ii), we see that

lim

*λ*→0^{+}

√

*λG*2(*α,λ*) = lim

*λ*→0^{+}

h 2α√

*λT*_{λ}^{000}(*α*) +5√

*λT*_{λ}^{00}(*α*)^{i}=2αT¯^{000}(*α*) +5 ¯T^{00}(*α*)

= [2αT¯^{00}(*α*) +3 ¯T^{0}(*α*)]^{0} >0 for*α*∈ I. (2.20)

For*α*∈ I, by (2.20), we define*λ** _{α}* by

*λ**α*≡

1 if G_{2}(* _{α,}λ*)<

_{0 for all}

*λ*>

_{0,}

sup{*λ*_{3} :G_{2}(*α,λ*)<0 for 0<*λ*<*λ*_{3}} if G_{2}(*α,λ*)≥0 for some*λ*>0.

Clearly,G_{2}(*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 *ε*_{0} be defined in Theorem 1.2. Then the following
statements (i)–(iii) hold:

(i) T¯^{0}(*α*)≥0for0<*α*< *β** _{ε}* and

*ε*≥

*ε*

_{0}.

(ii) [2αT¯^{00}(*α*) +3 ¯T^{0}(*α*)]^{0} >0for _{3ε}^{1} ≤*α*≤ _{12ε}^{5} and*ε*≤*ε*_{0}.

(iii) There exists*ε*ˆ ∈ (0,*ε*0)such that T¯^{0}(*α*) ≥ 0 for 0 < *α* ≤ _{3ε}^{1} and *ε*ˆ ≤ *ε* < *ε*0. 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 ˆ*ε*>0 satisfying

ˆ
*ε*<

r 31
1000 < *ε*_{0}
such that

T¯^{0}
1

3ε

<0 for 0<*ε*< *ε,*ˆ

=0 for*ε* =*ε,*ˆ

>0 for ˆ*ε* <*ε*<*ε*_{0}.

(2.21)
By Theorem1.2, (2.4) and [6, Lemma 3.3], we see that, for 0< *ε*< *ε*_{0}, there exist two positive
numbers*α*∗ <*α*^{∗} <*β** _{ε}* such that

T¯^{0}(*α*)

>0 on (0,*α*∗)∪(*α*^{∗},*β**ε*),

=0 when*α*=*α*∗ or *α*=*α*^{∗},

<0 for (*α*∗,*α*^{∗}).

(2.22)

Since f is a convex function on
0,_{3ε}^{1}

, and by [15, Lemma 3.2], we see that ¯T(*α*) is either
strictly increasing on 0,_{3ε}^{1}

, or strictly increasing and then strictly decreasing on 0,_{3ε}^{1}
. So
by (2.21) and (2.22), we observe that _{3ε}^{1} ≤ *α*∗ for ˆ*ε* ≤ *ε* < *ε*0. It follows that ¯T^{0}(*α*) ≥ 0 for
0<*α*≤ _{3ε}^{1} and ˆ*ε*≤*ε* <*ε*_{0}. So the statement (iii) holds. The proof is complete.

**Lemma 2.6.** Consider(1.5)with0<*ε*≤*ε*ˆwhere*ε*ˆis defined in Lemma2.5. Then*α*T¯^{00}(*α*) +T^{¯}^{0}(*α*)<

0for1≤*α*≤1.7.

Proof. Let ¯A≡*ε α*^{4}−u^{4}

, ¯B≡*α*^{3}−u^{3}, ¯C≡*α*^{2}−u^{2}and ¯D≡*α*−u. We compute that
*α*T¯^{00}(*α*) +T^{¯}^{0}(*α*) = ^{1}

4√ 2α

Z _{α}

0

N_{1}(*α,*u)

[F(*α*)−F(u)]^{5/2}^{du,} ^{(2.23)}
where

N_{1}(*α,*u)≡ ^{1}
72

9 ¯A^{2}+4 ¯B^{2}+36 ¯D^{2}−6 ¯AB¯+198 ¯AD¯ −120 ¯BD¯ +36 ¯AC¯−12 ¯BC¯ −36 ¯CD¯
.

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

3 B¯ and D¯ > ^{1}

3α^{2}B¯ > ^{1}
3α^{2}

3 4εαA¯

= ^{A}^{¯}
4α^{3}*ε*.
Then

1<*α*^{2}< ^{α}

2+*αu*+u^{2}D¯

D¯ = ^{B}^{¯}

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

3 B¯ < ^{4ˆ}^{ε}

3 (1.7)B^{¯} = ^{34ˆ}^{ε}

15 B¯ and D¯ > ^{A}^{¯}

4α^{3}*ε* > ^{A}^{¯}
4(1.7)^{3}*ε*ˆ

= ^{250}
4913ˆ*ε*

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

34
15*ε*ˆ− ^{2}

3 < ^{34}
15

r 31
1000−^{2}

3(≈ −0.26)<0, (2.26) 198

34
15*ε*ˆ−^{20}

33

<198 34

15 r 31

1000− ^{20}
33

!

(≈ −40.98)<−0.40, (2.27)
1− ^{5}

34ˆ*ε* − ^{250}

4913ˆ*ε* <1− ^{5}
34

q 31 1000

− ^{250}
4913

q 31 1000

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

N_{1}(*α,*u) = ^{1}

72 9 ¯A^{2}+4 ¯B^{2}+36 ¯D^{2}−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 ¯B^{2}+36 ¯D^{2}

< ^{1}
72

9 ¯AB¯

34
15*ε*ˆ−^{2}

3

+_{198 ¯}BD¯
34

15*ε*ˆ−^{20}
33

+_{36 ¯}AC¯

1− ^{5}
34ˆ*ε*

− ^{250}
4913ˆ*ε*

+_{4 ¯}B^{2}+_{36 ¯}D^{2}

< ^{1}
72

−_{40 ¯}BD¯ +_{4 ¯}B^{2}+_{36 ¯}D^{2}

= ^{D}^{¯}

2

18

"B¯
D¯ −_{5}

2

−_{16}

#

< ^{D}^{¯}

2

18 h

(1−5)^{2}−16i

=0.

So by (2.23), we obtain that*α*T¯^{00}(*α*) +T^{¯}^{0}(*α*) < 0 for 1≤ *α* ≤ 1.7 and 0< *ε* ≤ *ε. The proof is*ˆ
complete.

**Lemma 2.7.** Consider(1.5)with0.07≤*ε*≤ *ε. Then*ˆ *α*T¯^{00}(*α*) + ^{5}_{2}T^{¯}^{0}(*α*)<0for1.7≤*α*≤ _{3ε}^{1}.
Proof. We compute that

*α*T¯^{00}(*α*) + ^{5}

2T¯^{0}(*α*) = ^{1}
4√

2α
Z _{α}

0

N_{2}(*α,*u)

[F(*α*)−F(u)]^{5/2}^{du,} ^{(2.29)}
where

N_{2}(*α,*u)≡ ^{1}
144

−_{9 ¯}A^{2}+_{42 ¯}AB¯ +_{450 ¯}AD¯ +_{126 ¯}AC¯−_{16 ¯}B^{2}−_{240 ¯}BD¯

−60 ¯BC¯+288 ¯D^{2}+36 ¯CD¯

. (2.30)

Then we assert that

N_{2}(*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 that*α*T¯^{00}(*α*) + ^{5}_{2}T^{¯}^{0}(*α*)<0 for 1.7≤*α*≤ _{3ε}^{1} and 0.07≤ *ε*≤*ε.*ˆ
**Lemma 2.8.** Consider(1.5)with0<*ε*<0.07. ThenT¯^{0}(*α*)<0for1.7≤ *α*≤ _{3ε}^{1}.
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)}
where*θ*(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,

_{3ε}

^{1}

such that *θ*^{0}(u) > 0 for(0,p)and *θ*^{0}(u) < 0 for p,_{3ε}^{1}

. Let *α* ∈ ^{}1.7,_{3ε}^{1}
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 ¯*T^{0}(*α*)<0. Assume that*θ*(*α*)>0, see Figure2.1(ii).

We compute and find that

*θ*^{0}(1.7) = 2εu^{3}−u^{2}+1

u=1.7= ^{4913}

500 *ε*−^{189}

100 <0 for 0< *ε*<0.07.

Since 1.7≤*α*≤ _{3ε}^{1}, 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 of*θ*(u)on[0,*α*]where 1.7≤ *α*≤ _{3ε}^{1} 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

uθ^{0}(u)du= ^{α}^{8εα}

3−5α^{2}+10
40√

2B^{3/2}(*α, ¯α*) ^{.} ^{(2.33)}
Since

*∂*

*∂u* 8εu^{3}−5u^{2}+10

=2u(12εu−5)<0 for 1.7≤ u≤ ^{1}
3ε,

we see that, for 1.7≤u ≤ _{3ε}^{1} and 0<*ε*<0.07,
8εu^{3}−5u^{2}+_{10}< 8εu^{3}−5u^{2}+_{10}^{}

u=1.7= ^{4913}
125 *ε*− ^{89}

20 <_{0.}

So by (2.33), we obtain that ¯T^{0}(*α*)<0. The proof is complete.

**Lemma 2.9.** Consider(1.1)with0<*ε*<*ε*_{0}. Then there exists*ξ**ε* >0such that
Γ*ε* ≡ {*λ*>0 :T_{λ}^{0}(*α*)<0for some*α*∈(0,*β**ε*)}= (0,*ξ**ε*).

Proof. Let*ε* ∈ (0,*ε*_{0})be given. By (2.22), there exist two positive numbers *α*∗ < *α*^{∗} < *β** _{ε}* such
that

lim

*λ*→0^{+}

√

*λT*_{λ}^{0}(*α*) =T^{¯}^{0}(*α*)

>0 on (0,*α*∗)∪(*α*^{∗},*β** _{ε}*),

=0 when*α*=*α*∗ or*α*^{∗},

<_{0} _{on} (*α*∗,*α*^{∗}).

(2.34)

Then we divide this proof into the next four steps.

**Step 1. We prove that***α*∗ < _{12ε}^{5} . Assume that*α*∗≥ _{12ε}^{5} . By (2.34) and Lemma2.3, we see that
0≤T^{¯}^{0}(*α*) = _{lim}

*λ*→0^{+}

√

*λT*_{λ}^{0}(*α*)<√

*λT*_{λ}^{0}(*α*) _{for 0}<*α*≤ ^{5}

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

0≤ lim

*λ*→0^{+}

√
*λT*_{λ}^{0}

*α*∗+*α*^{∗}
2

=_{T}^{¯}^{0}

*α*∗+*α*^{∗}
2

<_{0,}
which is a contradiction. It implies that *α*∗< _{12ε}^{5} .

**Step 2. We prove that, for** *α* ∈ (*α*∗,*α*^{∗})∩ 0,_{12ε}^{5}

, there exists a continuously differential
function ˜*λ**α* >0 of*α*such that

√

*λT*_{λ}^{0}(*α*)

<0 if 0<*λ*<*λ*^{˜}* _{α}*,

=0 if*λ*=*λ*^{˜}*α*,

>_{0} _{if}*λ*>*λ*^{˜}* _{α}*.

(2.36)

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

*λ*→_{∞}

√

*λT*_{λ}^{0}(*α*) =_{∞}·1=_{∞} for*α*∈(0,*β**ε*). (2.37)
By (2.34), (2.37), Lemma2.3and implicit function theorem, we observe that, for*α*∈(*α*∗,*α*^{∗})∩

0,_{12ε}^{5}

, there exists a continuously differential function ˜*λ**α* >0 of *α*such that (2.36) holds.

**Step 3. We prove that**

*ξ**ε* ≡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*_{λ}^{0}(*α*∗)< T_{λ}^{0}_{=}_{1}(*α*∗).

So by Lemma2.3 and continuity ofT_{λ}^{0}_{=}_{1}(*α*)with respect to*α, there existsδ*>0 such that
0< T_{λ}^{0}_{=}_{1}(*α*)≤√

*λT*_{λ}^{0}(*α*) _{for}*α*∗< *α*<*α*∗+*δ*< ^{5}

12ε and*λ*≥_{1,}

from which it follows that ˜*λ**α* <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 that*ξ** _{ε}* ∈(0,∞).

**Step 4. We prove that** Γ*ε* = (_{0,}*ξ** _{ε}*)

_{. Let}

*λ*

_{1}∈ (

_{0,}

*ξ*

*). There exists*

_{ε}*α*

_{1}∈ (

*α*∗,

*α*

^{∗})∩

_{0,}

_{12ε}

^{5}

^{}

_{such}that

*λ*

_{1}<

*λ*

^{˜}

*α*

_{1}. Then by (2.36), we see that T

_{λ}^{0}

1(*α*_{1}) < 0, which implies that *λ*_{1} ∈ _{Γ}* _{ε}*. Thus
(0,

*ξ*

*) ⊆*

_{ε}_{Γ}

*. Let*

_{ε}*λ*

_{2}∈

_{Γ}

*. There exists*

_{ε}*α*

_{2}∈ (0,

*β*

*) such thatT*

_{ε}

_{λ}^{0}

2(*α*_{2}) < 0. Next, we consider
two cases.

**Case 1.** Assume that _{12ε}^{5} <*α*^{∗}. By (2.34) and Lemma2.3, we see that
0≤ lim

*λ*→0^{+}

√

*λT*_{λ}^{0}(*α*)<

√

*λT*_{λ}^{0}(*α*) for*α*∈(0,*α*∗]and*λ*>0. (2.38)
By Steps 2 and 3, we see that

√

*λT*_{λ}^{0}(*α*)≥0 for*α*∈

*α*∗, 5
12ε

if*λ*≥*ξ**ε*. (2.39)

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

T_{λ}^{0}(*α*)>0 for 5

12ε ≤ *α*< *β** _{ε}* and

*λ*≥

*ξ*

*. (2.40) So by (2.38)–(2.40), we obtain that T*

_{ε}

_{λ}^{0}(

*α*)≥0 for

*α*∈ (0,

*β*

*)if*

_{ε}*λ*≥

*ξ*

*. It implies that*

_{ε}*λ*

_{2}<

*ξ*

*. ThusΓ*

_{ε}*ε*⊆(

_{0,}

*ξ*

*)*

_{ε}_{.}

**Case 2.** Assume that*α*^{∗} < _{12ε}^{5} . By (2.34) and Lemma2.3, we see that
0≤ lim

*λ*→0^{+}

√

*λT*_{λ}^{0}(*α*)<√

*λT*_{λ}^{0}(*α*) for*α*∈ (0,*α*∗]∪

*α*^{∗}, 5
12ε

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

√

*λT*_{λ}^{0}(*α*)≥_{0} _{for}*α*∈(*α*∗,*α*^{∗}) if*λ*≥*ξ** _{ε}*. (2.42)
By (2.41) and Lemma2.2(ii), we see that

T_{λ}^{0}(*α*)>_{0} _{for} ^{5}

12ε ≤*α*< *β** _{ε}* and

*λ*>

_{0.}

_{(2.43)}So by (2.41)–(2.43), we obtain that T

_{λ}^{0}(

*α*)≥0 for

*α*∈ (0,

*β*

*)if*

_{ε}*λ*≥

*ξ*

*. It implies that*

_{ε}*λ*

_{2}<

*ξ*

*. ThusΓ*

_{ε}*ε*⊆(0,

*ξ*

*ε*).

By the above discussions, we obtain thatΓ*ε* = (0,*ξ** _{ε}*). The proof is complete.

**Lemma 2.10.** Consider (1.1) with 0 < *ε* < *ε*_{0}. Then there exists *κ**ε* ∈ (0,*ξ**ε*)such that T* _{λ}*(

*α*)has exactly two critical points, a local maximum at

*α*

_{M}(

*λ*)and a local minimum at

*α*

_{m}(

*λ*) (>

*α*

_{M}(

*λ*)), on (0,

*β*

*)if0<*

_{ε}*λ*<

*κ*

*.*

_{ε}Proof. Let*ε*∈(0,*ε*_{0})be given. By (2.34) and Lemma2.1(ii), there exists*λ*_{1} >0 such that
T_{λ}^{0}

*α*∗+*α*^{∗}
2

<0 for 0<*λ*<*λ*_{1}. (2.44)
We divide this proof into the next four steps.

**Step 1.** We prove that there exists*λ*_{2} ∈ (0,*λ*_{1})such that, for 0 <*λ* <*λ*_{2}, either T_{λ}^{0}(*α*)> 0 on
0,_{3ε}^{1}

, orT* _{λ}*(

*α*)has exactly one critical point, a local maximum, on 0,

_{3ε}

^{1}

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

T_{λ}^{0}(*α*)>0 for 0< *α*≤1 and*λ*>0. (2.45)

### 0 α ^{0} α

*T* _{λ} ^{(} α ) *T* _{λ} ^{(} ^{α} ^{)}

### (i) (ii)

### 1

^{1}

_{3ε}

### 1

^{1}

_{3ε}

Figure 2.2: Graphs ofT* _{λ}*(

*α*)

_{on}(

_{0,}

_{3ε}

^{1}]

_{for 0}<

*λ*<

*λ*

_{2}. Then we consider the following three cases.

**Case 1.** Assume that ˆ*ε*≤*ε*<*ε*_{0}. By Lemmas2.1(ii),2.3and2.5(iii), we see that
0≤T^{¯}^{0}(*α*) = lim

*λ*→0^{+}

√

*λT*_{λ}^{0}(*α*)<√

*λT*_{λ}^{0}(*α*) for 1< *α*≤ ^{1}

3ε and*λ*>0.

So by (2.45),T_{λ}^{0}(*α*)>0 on 0,_{3ε}^{1}

for*λ*>0, see Figure 2.2(i).

**Case 2.** Assume that 0.07 ≤ *ε* < *ε. By (2.21), Lemmas*ˆ 2.1(ii),2.4(ii),2.6 and 2.7, there exists
*λ*_{2}∈(0,*λ*_{1})such that

T_{λ}^{0}
1

3ε

<0 and *αT*_{λ}^{00}(*α*) +K(*α*)T_{λ}^{0}(*α*)<0 for 1≤*α*≤ ^{1}

3ε and 0<*λ*<*λ*_{2}, (2.46)
where K(*α*) ≡ 1 if 1 ≤ *α* ≤ 1.7, and K(*α*) ≡ 5/2 if 1.7 < *α* ≤ _{3ε}^{1}. By (2.45) and (2.46), there
exists *α** _{λ}*∈ 1,

_{3ε}

^{1}

such thatT_{λ}^{0}(*α** _{λ}*) =0 for 0<

*λ*<

*λ*

_{2}. Furthermore,

*α** _{λ}*T

_{λ}^{00}(

*α*

*) =*

_{λ}*α*

*T*

_{λ}

_{λ}^{00}(

*α*

*) +K(*

_{λ}*α*

*)T*

_{λ}

_{λ}^{0}(

*α*

*)<0 for 0<*

_{λ}*λ*<

*λ*

_{2}. Thus T

*(*

_{λ}*α*)has exactly one local maximum at

*α*

*on 0,*

_{λ}_{3ε}

^{1}

for 0< *λ*<*λ*_{2}, see Figure2.2(ii).

**Case 3.** Assume that 0< *ε* <0.07. By Lemmas2.4,2.6 and2.8, there exists*λ*_{2} ∈ (0,*λ*_{1})such
that

*αT*_{λ}^{00}(*α*) +T_{λ}^{0}(*α*)<0 for 1≤*α*≤1.7 and 0<*λ*<*λ*_{2}, (2.47)
T_{λ}^{0}(*α*)<0 for 1.7≤*α*≤ ^{1}

3ε and 0<*λ*<*λ*_{2}. (2.48)