A global bifurcation theorem for a multiparameter positone problem and its application to the one-dimensional perturbed Gelfand problem

A global bifurcation theorem for a multiparameter positone problem and its application to the one-dimensional perturbed Gelfand problem

Shao-Yuan Huang

, Kuo-Chih Hung

and Shin-Hwa Wang

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

2Fundamental General Education Center, National Chin-Yi University of Technology Taichung 411, Taiwan

3Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan

Received 6 May 2019, appeared 31 December 2019 Communicated by John R. Graef

Abstract. We study the global bifurcation and exact multiplicity of positive solutions

for (

u⁰⁰(x) +λfε(u) =0, −1<x <1, u(−1) =u(₁) =_0,

where λ > 0 is a bifurcation parameter, ε ∈ Θ is an evolution parameter, and Θ ≡ (σ₁,σ₂)is an open interval with 0≤σ₁<σ₂≤∞. Under some suitable hypotheses on f_ε, we prove that there existsε0∈ _Θsuch that, on the(λ,kuk_∞)-plane, the bifurcation curve is S-shaped forσ1 <ε<ε0and is monotone increasing forε0≤ε<σ2. We give an application to prove global bifurcation of bifurcation curves for the one-dimensional perturbed Gelfand problem.

Keywords: global bifurcation, multiparameter problem, S-shaped bifurcation curve, exact multiplicity, positive solution.

2010 Mathematics Subject Classification: 34B18, 74G35.

1 Introduction

We study the global bifurcation and exact multiplicity of positive solutions for the multipa- rameter positone problem

(u⁰⁰(x) +λfε(u) =0, −1<x <1,

u(−1) =u(1) =0, (1.1)

where λ > 0 is a bifurcation parameter, ε ∈ _Θ is an evolution parameter, Θ ≡ (σ₁,σ₂)is an open interval with 0 ≤ σ₁ < σ₂ ≤ ∞, and nonlinearity fε ∈ C³[0,∞). We first define some

BCorresponding author. Email:[email protected]

functions needed below:

F_ε(u) =

Z _u

0

f_ε(t)dt, whereε∈ _Θandu>_{0, (1.2)} I₁(_ε,α,u) =F_ε(α)−F_ε(u)_{, where}ε∈ _Θandα>u>_{0, (1.3)} I₂(ε,α,u) =αf_ε(α)−u f_ε(u), whereε∈ _Θandα>u>0, (1.4) I₃(ε,α,u) =α²f_ε⁰(α)−u²f_ε⁰(u), whereε∈ _Θandα>u>0,

I₄(ε,α,u) =α³f_ε⁰⁰(α)−u³f_ε⁰⁰(u), whereε∈ _Θandα>u>0.

We assume that fε satisfies hypotheses (F1)–(F6) as follows:

(F1) For any fixed ε ∈ Θ, there exists a positive number γε such that fε(0) > 0 (positone), f_ε(u)>0 on(0,∞), f_ε⁰⁰(u)>0 on[0,γ_ε), f_ε⁰⁰(u)<0 on(γ_ε,∞)and f_ε⁰⁰(γ_ε) =0. Moreover, limu→_∞(fε(u)/u) =0.

(F2) For any fixed u > 0, fε(u)is a continuously differentiable, strictly decreasing function ofε∈_Θ.

(F3) There exist two positive numbers ˜ε, ¯ε ∈(σ₁,σ₂)such that ˜ε< ε¯and the following condi- tions (i)–(iii) hold:

(i) fε(γε)−γεf_ε⁰(γε)≥0 for ¯ε≤ε< σ2. (ii) Forσ₁ <ε<ε, the function¯ Gε(_u)≡Ru

0 t³f_ε⁰⁰(_t)_dthas a positive zero κ_ε in(_0,∞)_. (iii) Forσ₁ <ε≤ε, there exists a number˜ ρ_ε ∈(_0,κ_ε]_{such that}

Hε(u)≡

Z _u

0 t fε(t)−t²f_ε⁰(t)dt

(=0 ifu=ρ_ε,

<0 ifρε <u≤κε.

(F4) Forσ₁ <ε<ε,¯

γε <ηε ≡ (

ρε ifσ₁ <ε≤ε,˜ κ_ε if ˜ε<ε< ε,¯ and

K(ε,u,v)≡ −8(I₁)²(I₂)−16(I₁)²(I₃)−4(I₁)²(I₄) +24(I₁)(I₂)²+18(I₁)(I₂)(I₃)−15(I₂)³

>0 foru∈[γε,ηε]and 0<v<u.

(F5) Forσ₁ <ε<ε, there exists a number¯ ω_ε ∈(η_ε,∞]_{such that} 3

∂

∂εI₁

(I₂)−2 ∂

∂εI₁

(I₁)−2 ∂

∂εI₂

(I₁)>0 for 0<v<u <ω_ε. Furthermore,ωε is a decreasing function on[ε, ¯˜ ε).

(F6) For ˜ε≤ε< ε,¯

2I₁(_ε,ω_ε,u)−I₂(_ε,ω_ε,u)>_{0 for 0}<u<ω_ε.

Figure 1.1: Global bifurcation of bifurcation curves Sε of (1.1) with varying ε ∈ Θ= (σ₁,σ₂).

For any ε ∈ _{Θ, on the} (λ,kuk_∞)-plane, we study the shape and structure of bifurcation curvesS_ε of positive solutions of (1.1), defined by

S_ε ≡ {(λ,ku_λk_∞)_:λ>_{0 and}u_λis a positive solution of (1.1)}.

We say that, on the(λ,kuk_∞)-plane, the bifurcation curveSε is S-shaped ifSε is a continuous curve and there exist two positive numbersλ∗ <λ^∗ such thatS_ε hasexactly twoturning points at some points(λ^∗,ku_λ^∗k_∞)_and(λ∗,ku_λ∗k_∞)_{, and}

(i) λ∗ <λ^∗ andku_λ^∗k_∞ <ku_λ∗k_∞,

(ii) at(λ^∗,ku_λ^∗k_∞)the bifurcation curveS_ε turns to theleft, (iii) at(λ∗,ku_λ∗k_∞)the bifurcation curveSε turns to theright.

See Fig. 1.1(i).

In this paper, we mainly study the global bifurcation of bifurcation curvesSε with varying ε ∈ _Θ = (σ₁,σ₂). In Theorem2.1 for (1.1) stated below, assuming that f_ε ∈ C³[0,∞) satisfies hypotheses (F1)–(F6), we prove that there exists ε₀ ∈ _Θ such that, on the (λ,kuk_∞)-plane, the bifurcation curve Sε is S-shaped when σ₁ < ε < ε₀ and is monotone increasing when ε₀≤ε< σ₂, see Fig.1.1. In Theorem2.3 stated behind, we give an application of Theorem2.1 for (1.1) to the famous one-dimensionalperturbed Gelfand problem:

(u⁰⁰(x) +λf_ε(u) =0, −1<x <1,u(−1) =u(1) =0, fε(u) =exp ₁₊^u_εu

, (1.5)

where λ>0 is the Frank–Kamenetskii parameter or ignition parameter,ε>0 is thereciprocal activation energy parameter, u(x) is the dimensionless temperature, and the reaction term f_ε(u)in (1.5) is the temperature dependence obeying the simple Arrhenius reaction-rate law in irreversible chemical reaction kinetics, see, e.g., Gelfand [5] and Boddington et al. [2]. This is the one-dimensional case of a problem arising in the study of (steady state) solid fuel ignition models in thermal combustion theory, cf. [1,4,6].

For (1.5), it has been a long-standing conjecture on the global bifurcation of bifurcation curves S_ε with varying ε>0, see e.g. [8, Conjecture 1]. Also see [3,6,8,12,13,16,19]. Very re- cently, by developing some new time-map techniques and applying Sturm’s theorem, Huang

and Wang [8] gave a rigorous proof of this conjecture for (1.5). Their main result is stated in the next theorem.

Theorem 1.1([8, Theorem 4]). Consider(1.5) with varying ε > 0. Then the bifurcation curve S_ε starts at the origin and tends to infinity asλ→_∞,and there exists a positive critical bifurcation value ε₀(≈1/4.069≈0.245)<0.25such that the following assertions (i)–(iii) hold:

(i) (See Fig.1.1(i).) For0 < ε < ε₀, the bifurcation curve S_ε is S-shaped on the(λ,kuk_∞)-plane.

More precisely, there exist two positive numbers λ∗ < λ^∗ such that (1.5) has exactly three positive solutions forλ∗ <λ< λ^∗, exactly two positive solutions for λ= λ∗ andλ= λ^∗, and exactly one positive solution for0<λ<λ∗ andλ>λ^∗. Furthermore, all positive solutions u_λ are nondegenerate except that u_λ∗and u_λ^∗are degenerate.

(ii) (See Fig.1.1(ii).) Forε=ε₀, the bifurcation curve S_ε0 is monotone increasing on the(_λ,kuk_∞)- plane. More precisely, (1.5) has exactly one positive solution for all λ > 0. Furthermore, all positive solutions u_λ are nondegenerate except that u_λ0 is a cusp type degenerate solution for someλ=λ₀ >_0.

(iii) (See Fig.1.1(iii).) Ifε > ε₀, the bifurcation curve S_ε is monotone increasing on the(_λ,kuk_∞)- plane. More precisely, (1.5) has exactly one positive solution for all λ > 0. Furthermore, all positive solutions u_λ are nondegenerate.

Note that the definitions of degenerate and nondegenerate positive solutions and cusp type degenerate solution are defined later in Section 3.

Under somewhat different hypotheses to (F1)–(F6), the authors [9, Theorem 2.1] studied the global bifurcation and exact multiplicity of positive solutions for (1.1) and obtained the same results in Theorem 2.1. The hypotheses in [9, Theorem 2.1] can apply to a class of polynomial nonlinearities

f_ε(u) =−εu^p+bu²+cu+d, p≥3, ε,b,d>0, c≥0,

see [9, Theorem 2.1 and hypotheses (H1)–(H5)] for details. But the hypotheses in [9, Theorem 2.1] do not apply to (1.5) with f_ε(u) =exp ₁₊^u_εu

. Cf. [9, Theorem 2.1 and hypotheses (H1)–

(H5)] with Theorem2.1 under (F1)–(F6).

The paper is organized as follows. Section 2 contains statements of the main results (The- orems2.1–2.4). Section 3 contains several lemmas needed to prove the main results. Section 4 contains the proofs of the main results.

2 Main results

The main results in this paper are the next Theorems2.1–2.4, in particular, Theorems 2.1and 2.3. In Theorem2.1, we prove the global bifurcation of bifurcation curvesSεand hence we are able to determine exact multiplicity of positive solutions byε∈ _Θ andλ> 0, see Fig.1.1. In Theorem2.3, we apply Theorem2.1to prove the global bifurcation of bifurcation curvesS_ε for the one-dimension perturbed Gelfand problem (1.5).

Theorem 2.1(See Fig.1.1). Consider(1.1)with varyingε∈ _Θ= (σ₁,σ₂)where0≤ σ₁< σ₂≤ _∞. Assume that f ∈ C³[0,∞)satisfies (F1)–(F6). Then the bifurcation curve Sε starts at the origin and tends to infinity asλ → _∞,and there exists a positive critical bifurcation value ε₀ ∈ (ε, ¯˜ ε)such that the following assertions (i)–(iii) hold:

(i) (See Fig. 1.1(i).) For σ₁ < ε < ε₀, the bifurcation curve S_ε is S-shaped on the (λ,kuk_∞)- plane. More precisely, there exist two positive numbersλ∗ <λ^∗ such that(1.1)has exactly three positive solutions forλ∗ < λ< λ^∗, exactly two positive solutions forλ =λ∗andλ= λ^∗, and exactly one positive solution for0<λ<λ∗andλ>λ^∗. Furthermore, all positive solutions u_λ are nondegenerate except that uλ∗ and uλ^∗ are degenerate.

(ii) (See Fig.1.1(ii).) Forε=ε₀, the bifurcation curve S_ε0 is monotone increasing on the(λ,kuk_∞)- plane. More precisely, (1.1) has exactly one positive solution u_λ for all λ > 0. Furthermore, all positive solutions u_λ are nondegenerate except that u_λ0 is a degenerate solution for some λ =λ₀ >0.In addition, u_λ0 is a cusp type degenerate solution if, for any fixed u> 0, f_ε⁰(u)is continuously differentiable atε=ε0.

(iii) (See Fig. 1.1(iii).) For ε₀ < ε < σ₂, the bifurcation curve S_ε is monotone increasing on the (λ,kuk_∞)-plane. More precisely,(1.1)has exactly one positive solution u_λ for allλ > 0. Fur- thermore, all positive solutions u_λ are nondegenerate.

Figure 2.1: The bifurcation surface Γwith the fold curveC_Γ =C₁∪C₂, and the projection ofC_ΓontoFq. B_Γ= B₁∪B₂∪ {(ε₀,λ₀)}is the bifurcation set.

We next study, in the(ε,λ,kuk_∞)-space, the shape and structure of thebifurcation surfaceΓ of (1.1), defined by

Γ≡ {(_ε,λ,ku_ε,λk_∞)_:ε,λ>_{0 and}u_ε,λ is a positive solution of (1.1)}

which has the appearance of a folded surface with thefold curve

C_Γ ≡ {(ε,λ,ku_ε,λk_∞:ε ∈_Θ, λ>0 andu_ε,λ is adegeneratepositive solution of (1.1)}. See Fig.2.1. Let Fqdenote the first quadrant of the (ε,λ)-parameter plane. We also study, on F_q, thebifurcation setof (1.1)

B_Γ ≡ {(ε,λ):ε∈_Θ, λ>0 andu_ε,λ is adegeneratepositive solution of (1.1)}. By Theorem2.1, we know that the bifurcation setB_Γ= B₁∪B₂∪ {(ε₀,λ₀)}, where

B₁≡ {(ε,λ∗(ε)):σ₁<ε< ε₀} and B₂≡ {(ε,λ^∗(ε)):σ₁<ε< ε₀}. We define the set

M≡ {(_ε,λ)_:σ₁<ε <ε₀andλ∗(ε)<λ<λ^∗(ε)}. We analyze the structure of the bifurcation setB_Γ of (1.1) in the next theorem.

Figure 2.2: The graph of the bifurcation setB_Γ = B₁∪B₂∪ {(ε₀,λ₀)}. (ε₀,λ₀)_is a cusp point ofB_Γ.

Theorem 2.2 (See Fig. 2.2). Consider (1.1) with ε ∈ _Θ = (σ₁,σ₂) where 0 ≤ σ₁ < σ₂ ≤ _∞.

Assume that f_ε ∈ C³[0,∞)satisfies (F1)–(F6),ω_ε is a increasing function on(σ₁, ˜ε], and there exists a functionβε ∈[ρε,κε]on(σ₁, ˜ε)such thatβε is decreasing on(σ₁,ε⁰)and(ε⁰, ˜ε)for some ε⁰ ∈ (σ₁, ˜ε) respectively. Then(1.1) has exactly two positive solutions for(ε,λ) ∈ B_Γ\ {(ε₀,λ₀)}, exactly three positive solutions for(ε,λ)∈ M, and exactly one positive solution for(ε,λ)∈/(B_Γ\ {(ε₀,λ₀)})∪M.

Moreover,λ∗(ε)andλ^∗(ε)are both continuous, strictly increasing functions on(σ₁,ε₀)and satisfy 0≤ _lim

ε→σ₁⁺

λ∗(ε)≤ _lim

ε→σ₁⁺

λ^∗(ε)<λ₀ = _lim

ε→ε⁻₀

λ^∗(ε) = _lim

ε→ε⁻₀

λ∗(ε)_. In addition,lim_ε→σ⁺

1 λ∗(ε)<lim_ε→σ⁺

1 λ^∗(ε)iflim_ε→σ⁺

1 ρε <lim_ε→σ⁺

1 ωε.

Theorem 2.3. Consider (1.5) with varying ε ∈ (0,∞). Then the bifurcation curve S_ε starts at the origin and tends to infinity asλ→_∞,and there exists a positive critical bifurcation valueε₀(≈0.245) satisfying0.243≈ε˜< ε₀ <ε¯≡ 0.25,whereε˜=1/ ˜a anda˜ ≈4.107is defined in [7, (1.4)] such that all the results in Theorem1.1(i)–(iii) hold.

Theorem 2.4(See Fig.2.2). Consider(1.5)withε>0. Then(1.5)has exactly two positive solutions for (ε,λ)∈ B_Γ\ {(ε₀,λ₀)}, exactly three positive solutions for (ε,λ)∈ M, and exactly one positive solution for(ε,λ)∈/ (B_Γ\ {(ε₀,λ₀)})∪M. Moreover,λ∗(ε)andλ^∗(ε)are both continuous, strictly increasing functions on(σ₁,ε₀)and satisfy

0= lim

ε→0⁺λ∗(ε)<λ_∞= lim

ε→0⁺λ^∗(ε)<λ₀ = lim

ε→ε⁻₀

λ^∗(ε) = lim

ε→ε⁻₀

λ∗(ε) (≈2.286), where

λ_∞ ≡ max

α∈(_0,∞)

1 2e^α

ln

2e^α+2 q

e^α(e^α−1)−1 2

≈0.878.

3 Lemmas

To prove Theorem 2.1, we need the next Lemmas 3.1–3.11. We simply modify the time- map techniques used in [8,9,11,18] without applying Sturm’s theorem for Theorem 1.1 ([8, Theorem 4]). The time map formula we apply to study (1.1) takes the form as follows:

√ λ= √¹

2 Z _α

0

[Fε(α)−Fε(u)]^−1/2du≡Tε(α) forα>0 ifε ∈_Θ= (σ₁,σ₂), (3.1) where Fε(u)is defined by (1.2), see Laetsch [14]. Observe that positive solutions u_ε,λ for (1.1) correspond to

ku_ε,λk_∞ =α and T_ε(α) =√

λ. (3.2)

Thus, studying of the exact number of positive solutions of (1.1) for fixed ε∈ _Θis equivalent to studying the shape of the time mapTε(α)on(0,∞), cf. [8,9,11,18]. In this section we always assume that f_ε ∈C³[0,∞)satisfies (F1)–(F6). Notice that, since f_ε ∈C³[0,∞), it can be proved that T_ε(α)is a thrice differentiable function ofα> 0 forε ∈ _Θ. The proof is easy but tedious and consequently we omit it.

In addition, we recall that a positive solutionu_λof (1.1) isdegenerateifT_ε⁰(ku_λk_∞) =0 and isnondegenerateifT_ε⁰(ku_λk_∞)6=0. Also, adegeneratepositive solutionu_λ of (1.1) is ofcusp type if T_ε⁰⁰(ku_λk_∞) =0 andT_ε⁰⁰⁰(ku_λk_∞)6=0, see [16, p. 497] and [17, p. 214].

By (3.2), Theorem 2.1follows if lim_α→0⁺T_ε(α) =0 and lim_α→_∞T_ε(α) =∞, and there exists ε₀∈(ε, ¯˜ ε)⊂_Θ= (σ₁,σ₂)such that the following assertions (M1)–(M3) hold (See Fig.3.1):

(M1) For σ₁ <ε <ε0,Tε(α)has exactly two critical points, a local maximum at someα_M and a local minimum at someα_m (>α_M), on(0,∞).

(M2) For ε = ε₀, T_ε⁰₀(α) > 0 for α ∈ (0,∞)\ {α₀}, and T_ε⁰₀(α₀) = 0. In addition, T_ε⁰⁰₀(α₀) = 0 andT_ε⁰⁰⁰₀(α₀)6=0 if, for any fixed u>_0, f_ε⁰(u)is continuously differentiable atε=ε₀. (M3) For ε0<ε< σ2,T_ε⁰(α)>0 forα∈(0,∞).

The main difficulty to obtain the above assertions (M1)–(M3) is to prove theexactnumber of critical points of the time map T_ε(α) on (0,∞) for all ε ∈ _Θ = (σ₁,σ₂). Notice that by [15, Proposition 1.1.2], we see that if fε ∈ C³[0,∞), then Tε(α) ∈ C³(0,∞). By (3.1), we compute that

T_ε⁰(α) = ¹ 2√

2α Z _α

0

θ(α)−θ(u)

[F_ε(α)−F_ε(u)]^{3/2du forα}>0, (3.3) whereθ(u)≡2F_ε(u)−u f_ε(u)_.

Figure 3.1: Graphs ofT_ε(α)on (0,∞)with varyingε∈_Θ= (σ₁,σ₂).

Lemma 3.1. Consider(1.1). For any fixedε ∈ _Θ = (σ₁,σ₂)with0 ≤ σ₁ < σ₂ ≤ ∞, the following assertions (i)–(ii) hold:

(i) lim_α→0⁺Tε(α) =0andlimα→_∞Tε(α) =_∞.

(ii) Forε ∈ _Θ, either T_ε(α)is strictly increasing on(_0,γ_ε], or T_ε(α)is strictly increasing and then strictly decreasing on(0,γε].

Proof. By (F1), we obtain that f_ε(0) > 0 on [0,∞) and lim_u→_∞(f_ε(u)/u) = 0. Thus assertion (i) follows by [14, Theorems 2.6 and 2.9]. By (F1) again, f_ε⁰⁰(u)> 0 on[0,γε)and f_ε⁰⁰(γε) =0, then assertion (ii) follows by [14, Theorem 3.2].

The proof of Lemma3.1is complete.

Lemma 3.2. Consider(1.1) with ε ∈ _Θ = (σ₁,σ₂)where0 ≤ σ₁ < σ₂ ≤ ∞. For any fixed α> 0, T_ε(α)is a continuous, strictly increasing function ofε∈_Θ.

Proof. By (F2), for any fixed u > 0, f_ε(u) is a continuous function of ε ∈ _Θ. Thus T_ε(α) is a continuous function of ε ∈ _Θ by [14, Theorem 2.4]. By (F2) again, for any fixed u > 0, f_ε1(u) > f_ε2(u) if σ₁ < ε₁ < ε₂ < σ₂. By (3.1), we directly obtain that T_ε1(α) < T_ε2(α) if σ₁< ε₁<ε₂<σ₂.

The proof of Lemma3.2is complete.

Lemma 3.3. Consider(1.1) with σ₁ < ε < ε.¯ Thenκ_ε > γ_ε andκ_ε is a continuous function of ε on (σ₁, ¯ε). Furthermore,

G_ε(u)











>₀ if0<u<κ_ε,

=0 if u=κ_ε,

<0 if u>κε.

(3.4) Proof. By (F1), we compute and observe that

G_ε(0) =0 and G⁰_ε(u)

= ^∂Gε(u)

∂u

= u³f_ε⁰⁰(u)











>0 if 0<u<γε,

=_{0 if}u=γ_ε,

<0 ifu>γ_ε.

(3.5)

So for σ₁ < ε < ε, by (F3) (ii), we observe that¯ G_ε(u)has a unique positive zero κ_ε(>γ_ε)on (0,∞)such that (3.4) holds. Since G⁰_ε(κε)< 0 by (3.5) and by the Implicit Function Theorem, κ_ε is a continuous function ofεon (σ₁, ¯ε).

The proof of Lemma3.3is complete.

Lemma 3.4. Consider(1.1)withε∈_Θ= (σ₁,σ₂)where0≤σ₁<σ₂ ≤_∞.Then one of the following assertions (i)–(ii) holds:

(i) θ⁰(u)>0for u>0and u6= γε.

(ii) There exist two positive numbers p₁(ε)< p₂(ε), dependent onε, such that p₁(ε)<γ_ε < p₂(ε) and

θ⁰(u) = fε(u)−u f_ε⁰(u)











>0 for u∈ (0,p₁(ε))∪(p₂(ε),∞),

=0 for u∈ {p₁(ε),p₂(ε)},

<₀ for u∈ (p₁(ε)_,p₂(ε))_.

(3.6)

Furthermore, if α ∈ (p₁(ε)_,p₂(ε)] satisfyingθ(α) ≥ 0, then there existsα¯ ∈ [_0,p₁(ε))such thatθ(α¯) =θ(α). See Fig.3.2.

Figure 3.2: Graphs ofθ(u)on [0,∞). (i) θ(u)≥ 0 for allu> 0. (ii)θ(u)< 0 for someu >0.

Proof. By (F1), we observe that

θ⁰⁰(u) =−u²f_ε⁰⁰(u)







<0 if 0<u<γ_ε,

=0 ifu=γε,

>0 ifu>γε.

(3.7) Assume that θ⁰(γ_ε) ≥ 0. It is easy to see that assertion (i) holds by (3.7). Assume that θ⁰(γ_ε)<0. Clearly, θ⁰(0) = f_ε(0)>0 by (F1). We assert that

ulim→_∞θ⁰(u)>0. (3.8)

So by (3.7) and (3.8), there exist two positive numbers p₁(ε) < p₂(ε) such that p₁(ε) < γ_ε <

p2(ε)and (3.6) holds. If α∈ (p₁(ε),p2(ε)]satisfying θ(α) ≥ 0, then there exists ¯α ∈ [0,p₁(ε)) such that θ(α¯) = θ(α). See Fig.3.2(i)–(ii). Next, we prove assertion (3.8). Letv ∈ [γε,∞) be given. Sinceθ⁰(u)is strictly increasing foru>γ_ε by (3.7), we observe that, foru≥v,

fε(v)

v − ^fε(u)

u =

Z _u

v

d dt

−fε(t) t

dt=

Z _u

v

θ⁰(t)

t² dt<θ⁰(u)

Z _u

v

1

t²dt= ^u−v uv θ⁰(u). So by (F1) and (F2), we see that

ulim→_∞θ⁰(u)≥ lim

u→_∞

f_ε(v)

v − ^fε(u) u

uv u−v

= f_ε(v)>0.

Thus (3.8) holds. Then assertion (ii) holds.

The proof of Lemma3.4is complete.

Lemma 3.5. Consider(1.1)withσ₁ <ε≤ε. Then˜ ρ_ε is a continuous function ofεon(σ₁, ˜ε].

Proof. Since Hε(0) = 0 and H_ε⁰(u) = uθ⁰(u) for u > 0, and by (F3) (iii) and Lemma 3.4, we observe that p₁(ε)and p₂(ε)exist forσ₁ <ε≤ε. It follows that˜

θ⁰(p₁(ε)) =θ⁰(p₂(ε)) =_{0 for}σ₁<ε≤ ε.˜ (3.9) By integration by parts, (F3) (iii) and (3.4), we obtain that

0=2H_ε(ρ_ε) =ρ²_εθ⁰(ρ_ε) +G_ε(ρ_ε)≥ ρ²_εθ⁰(ρ_ε). (3.10) So by Lemma3.4, we see that p₁(ε)<ρ_ε ≤ p2(ε)forσ₁<ε ≤ε, and˜

H_ε⁰(u) =uθ⁰(u)











>_{0 for}u∈(0,p₁(ε))∪(p₂(ε)_,∞),

=0 foru∈ {p₁(ε),p₂(ε)},

<0 foru∈(p₁(ε),p₂(ε)).

(3.11)

So by (3.11), we observe that ρε is the unique zero of Hε(u) on (0,p2(ε)]. By Lemma 3.4, we see that p₁(ε) < γε < p₂(ε) for σ₁ < ε ≤ ε. By (3.7), we further see that˜ θ⁰⁰(p₁(ε)) > 0 and θ⁰⁰(p₂(ε)) > 0 forσ₁ < ε ≤ ε. So by the Implicit Function Theorem and (3.9), we obtain that˜ p₁(ε)and p2(ε) are continuous functions of ε on (σ₁, ˜ε]. Let ˇε ∈ (σ₁, ˜ε] be given. We choose a sequence {ε_n}_n∈N ⊂ (σ₁, ˜ε]/{εˇ} such that lim_n→_∞ε_n = ε. Sinceˇ p₁(ε_n) < ρε_n < p₂(ε_n) for n∈_Nby (3.11), we see that

0< p₁(εˇ)≤lim inf

n→_∞ ρεn ≤lim sup

n→_∞

ρεn ≤ p2(εˇ). (3.12) In addition, there exist two subsequences{ε_1,n}_n∈Nand{ε_2,n}_n∈Nof{ε_n}_n∈N such that

nlim→_∞ρ_ε1,n =lim inf

n→_∞ ρ_εn and lim

n→_∞ρ_ε2,n =lim sup

n→_∞

ρ_εn. So by continuity ofHε(u)foruandε, we observe that

H_εˇ(lim inf

n→_∞ ρ_εn) = lim

n→_∞H_ε1,n(ρ_ε1,n) =0, (3.13) H_εˇ(lim sup

n→_∞

ρ_εn) = lim

n→_∞H_ε2,n(ρ_ε2,n) =0. (3.14) So by (3.12)–(3.14), we further observe that lim sup_n→∞ρε_n and lim inf_n→_∞ρε_n are two zeros of H_εˇ(u)on(0,p₂(ε)]. Moreover,

lim sup

n→_∞

ρεn =lim inf

n→_∞ ρεn = lim

n→_∞ρεn =ρˇε. Thus the functionρ_ε is a continuous atε=ε.ˇ

The proof of Lemma3.5is complete.

Lemma 3.6. Consider (1.1) with ε ∈ _Θ = (σ₁,σ₂) where0 ≤ σ₁ < σ₂ ≤ ∞. Then the following assertions (i)–(iii) hold:

(i) Forε¯≤ε<σ₂, T_ε⁰(α)>0forα>0.

(ii) Forσ₁< ε<ε,¯

T_ε⁰⁰(α) + ²

αT_ε⁰(α)>0 forα≥κε. (3.15) Moreover, T_ε(α)has at most one critical point, a local minimum, on[κ_ε,∞).

(iii) For σ₁< ε≤ε, T˜ _ε⁰(α)<0forρε ≤α≤κε.

Proof. (I) We prove assertion (i). By (F3) (i) and (3.7), we observe that, for ¯ε≤ ε<σ₂, θ⁰(u)> θ⁰(γ_ε) = f_ε(γ_ε)−γ_εf_ε⁰(γ_ε)≥0 foru>0 and u6=γ_ε.

It follows thatθ(α)−θ(u)>0 forα>u>0. So by (3.3), we see thatT_ε⁰(α)>0 forα>0ε. So assertion (i) holds.

Figure 3.3: Graphs ofφ(u)on [0,∞). (i) φ(u)>0 for some u> 0. (ii)φ(u)≤0 for all u≥0.

(II) We prove assertion (ii). We compute and observe that T_ε⁰⁰(α) + ²

αT_ε⁰(α) = √¹ 2α²

Z _α

0 3

2[θ(α)−θ(u)]²+ [F_ε(α)−F_ε(u)][φ(α)−φ(u)]

[F(α)−F(u)]^{5/2 du}

≥ √¹ 2α²

Z _α

0

φ(α)−φ(_u)

[Fε(α)−Fε(u)]^{3/2du, (3.16)} whereφ(_u)≡uθ⁰(_u)−θ(_u), see [10, (3.12)]. We obtain that

φ(0) =0 and φ⁰(u) =uθ⁰⁰(u) =−u²f_ε⁰⁰(u)











<0 for 0≤u< γ_ε,

=0 foru =γε,

>0 foru >γ_ε.

(3.17)

Let α∈ [κ_ε,∞) be given. By Lemma3.3, we see thatα≥ κ_ε > γ_ε for σ₁ < ε < ε. If¯ φ(α) ≥_0, by (3.17), we see thatφ(α)−φ(u) >0 for 0< u < α, and hence (3.15) holds by (3.16). While if φ(α) < 0, there exists ξ_α ∈ (0,γ_ε) such thatφ(ξ_α) = φ(α). See Fig. 3.3. So by [10, (3.15)], (F3) (ii) and (3.4),

T_ε⁰⁰(α) + ²

αT_ε⁰(α)> −1

√2α²[F_ε(α)−F_ε(ξ_α)]^3/2Gε(α)≥0,

and hence (3.15) holds. Assume thatT_ε(α)has a critical pointα₁∈[κ_ε,∞). By (3.15),T_ε⁰⁰(α₁)>

0. SoTε(α)has at most one critical point, a local minimum, on[κ_ε,∞). Therefore, assertion (ii) holds.

(III) We prove assertion (iii). By (F3) (iii), we see that ρ_ε ≤ κ_ε for σ₁ < ε ≤ ε. We fix˜ ε ∈ (σ₁, ˜ε]and α∈ [ρε,κε]. Assume thatθ(α) ≤0. By assertion (ii) of Lemma 3.4, we see that θ(α)−θ(u)<0 for 0< u< α, see Fig.3.2(ii). It follows that T_ε⁰(α)<0 by (3.3). Assume that θ(α)>0. By integration by parts and (F3) (ii)–(iii), we observe that

0≥2Hε(κ_ε) =κ²_εθ⁰(κ_ε) +Gε(κ_ε) =κ_ε²θ⁰(κ_ε).

So by (3.10), we have that p₁(ε) < ρε ≤ α ≤ κε ≤ p₂(ε). Assume thatθ(α)> 0. By assertion (ii) of Lemma3.4, there exists ¯α∈ (0,p₁(ε))such thatθ(α¯) =θ(α). It follows that

θ(α)−θ(u)











>0 foru∈(0, ¯α),

=0 foru=α,¯

<0 foru∈(α,¯ α).

So by (3.3) and (F3) (iii), we obtain that T_ε⁰(α) = ¹

2√ 2α

Z _α

0

θ(α)−θ(u) [Fε(α)−Fε(u)]^3/2du

= ¹

2√ 2α

( Z _α¯

0

θ(α)−θ(u)

[Fε(α)−Fε(u)]^3/2du+

Z _α

¯ α

θ(α)−θ(u) [Fε(α)−Fε(u)]^3/2du

)

< ¹

2√

2α[Fε(α)−Fε(α¯)]^3/2 _{Z α¯}

0

[θ(α)−θ(u)]du+

Z _α

¯ α

[θ(α)−θ(u)]du

= ¹

2√

2α[F_ε(α)−F_ε(α¯)]^3/2

αθ(α)−

Z _α

0 θ(u)du

= ¹

2√

2α[F_ε(α)−F_ε(α¯)]^3/2

Z _α

0 uθ⁰(u)du= ¹

2√

2α[F_ε(α)−F_ε(α¯)]^3/2Hε(α)≤0.

So assertion (iii) holds.

The proof of Lemma3.6is complete.

Lemma 3.7. Consider(1.1) with ε ∈ _Θ = (σ₁,σ₂)where0 ≤ σ₁ < σ₂ ≤ ∞. For any fixed α> 0, T_ε⁰(α)is a continuously differentiable function of ε ∈ Iα. Furthermore, _∂ε^∂T_ε⁰(α) > _{0for 0} < α < ω_ε andσ₁ <ε<ε.¯

Proof. First, for any fixed α > 0, it can be proved that T_ε⁰(α) is a continuously differentiable function ofε ∈ Iα. The proof is easy but tedious and consequently we omit it. Secondly, by (1.3), (1.4), (3.3) and (F5), we compute and obtain that, for 0<α<ωε,

∂

∂εT_ε⁰(α) = ¹ 4√

2α Z _α

0

3 _∂ε^∂I₁

(I₂)−2 _∂ε^∂I₁

(I₁)−2 _∂ε^∂I₂ (I₁) [Fε(α)−Fε(u)]^{5/2 du}

>_0.

The proof of Lemma3.7is complete.

Lemma 3.8. Consider(1.1)withε˜< ε<ε.¯ Assume thatγ_ε <η_ε. Then [αT_ε⁰⁰(α)]⁰ >0forγ_ε ≤α≤ η_εand one of the following assertions (i)–(iii) holds:

(i) T_ε⁰(α)is a strictly increasing function ofαon[γ_ε,η_ε]. (ii) T_ε⁰(α)is a strictly decreasing function ofαon[γ_ε,η_ε].

(iii) T_ε⁰(α)is a strictly decreasing and then strictly increasing function ofαon[γε,ηε]. Proof. By (F4), we compute and observe that

αT_ε⁰⁰(α)⁰ = ¹ 8√

2α² Z _α

0

K(ε,α,u)

[Fε(α)−Fε(u)]^7/2du>0 forγε ≤α≤ηε.

It follows that αT_ε⁰⁰(α)is a strictly increasing function ofα∈[γ_ε,η_ε]. So we observe that there are three cases:

Case 1. T_ε⁰⁰(α)>0 forα∈[γ_ε,η_ε]. Case 2. T_ε⁰⁰(α)<_{0 for}α∈[γ_ε,η_ε)_.

Case 3. T_ε⁰⁰(α) < 0 for α ∈ [γε, ˇα), T_ε⁰⁰(α) > 0 for α ∈ (α,ˇ ηε], and T_ε⁰⁰(αˇ) = 0 for some ˇ

α∈(γε,ηε).

So by Cases 1–3, assertions (i)–(iii) hold.

The proof of Lemma3.8is complete.

Lemma 3.9. Consider(1.1)withσ₁ <ε<ε.¯ Either one of the following assertions (i)–(ii) holds:

(i) T_ε(α)is a strictly increasing function on(0,∞).

(ii) Tε(α)has exactly one local maximum and exactly one local minimum on(0,∞).

Proof. We fixε ∈ (σ₁, ¯ε). Assume that assertion (i) does not hold. By Lemma3.1(i),T_ε(α)has a local maximum and a local minimum on(_0,∞). Assume thatT_ε(α)has two local maximum at some positive numbers α_M1 < α_M2. Then there exists αm ∈ (α_M1,α_M2) such thatTε(αm) is the local minimum value. We consider four cases:

Case 1. ˜ε< ε<ε¯andγε <ηε. Case 2. ˜ε< ε<ε¯andγε ≥ηε. Case 3. σ₁<ε≤ ε˜andγ_ε <η_ε. Case 4. σ₁<ε≤ ε˜andγ_ε ≥η_ε.

If Case 1 holds, by Lemmas3.1(ii) and3.6(ii), we observe thatγε ≤αm < α_M2 <κε =ηε. It is a contradiction by Lemma3.8. If Case 2 holds, by Lemma3.6(ii), we observe that 0< α_M1 <

α_M2 <κ_ε =η_ε ≤γ_ε. It is a contradiction by Lemma3.1(ii). If Case 3 holds, by Lemmas3.1(ii) and3.6(ii)–(iii), we observe thatγε ≤αm <α_M2 <ρε = ηε. It is a contradiction by Lemma3.8.

If Case 4 holds, by Lemma 3.6(ii)–(iii), we observe that 0<α_M1 < α_M2 < ρ_ε =η_ε ≤ γ_ε. It is a contradiction by Lemma3.1(ii). SoT_ε(α)has exactly one local maximum.

Assume that Tε(α) has two local minimum at some positive numbers αm₁ < αm₂. By Lemma 3.1(i), then there exist α_M1 ∈ (0,α_m1) and α_M2 ∈ (α_m1,α_m2) such that T_ε(α_M1) and T_ε(α_M2)are the local maximum values. By previous discussion, we obtain a contradiction. So Tε(α)has exactly one local minimum.

By above,T_ε(α)has exactly one local maximum and exactly one local minimum on(0,∞). The proof of Lemma3.9is complete.