Σ-Shaped Bifurcation Curves

A. Acharya; N. Fonseka; J. Quiroa; R. Shivaji

doi:10.1515/anona-2020-0180

Article Open Access

Σ-Shaped Bifurcation Curves

A. Acharya , N. Fonseka , J. Quiroa and R. Shivaji

Published/Copyright: May 4, 2021

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From the journal Advances in Nonlinear Analysis Volume 10 Issue 1

Abstract

We study positive solutions to the steady state reaction diffusion equation of the form:

−Δu=λf(u); Ω∂u∂η+λu=0; ∂Ω

where λ > 0 is a positive parameter, Ω is a bounded domain in ℝ^N when N > 1 (with smooth boundary ∂ Ω) or Ω = (0, 1), and ∂u∂η is the outward normal derivative of u. Here f(s) = ms + g(s) where m ≥ 0 (constant) and g ∈ C²[0, r) ∩ C[0, ∞) for some r > 0. Further, we assume that g is increasing, sublinear at infinity, g(0) = 0, g′(0) = 1 and g″(0) > 0. In particular, we discuss the existence of multiple positive solutions for certain ranges of λ leading to the occurrence of Σ-shaped bifurcation diagrams. We establish our multiplicity results via the method of sub-supersolutions.

Keywords: Σ-Shaped Bifurcaion Curves; Positive Solutions; Sub-Super Solutions

MSC 2010: 35J15; 35J25; 35J60

1 Introduction

In the recent literature there has been considerable interest in reaction diffusion models where a parameter influences the equation as well as the boundary conditions. See [1, 2, 3] for recent studies in this direction. In this paper, we enhance this study to show that for certain classes of such models the bifurcation diagram (λ, ∥u∥_∞) for positive solutions is at least Σ-shaped. Namely, we study boundary value problems of the form:

−Δu=λf(u); Ω∂u∂η+λu=0; ∂Ω, (1.1)

g(0) = 0, g′(0) = 1, g″(0) > 0, and lims→∞g(s)s=0.

First, we recall some results from [3]. Namely, for k > 0, let A_k be the principal eigenvalue of the problem:

−Δϕ=Akϕ; Ω∂ϕ∂η+Aϕ=0; ∂Ω. (1.2)

Then A_k is a strictly decreasing function of k with

limk→0Ak=∞. (1.3)

Further, for a fixed λ > 0, let σ_λ,k be the principal eigenvalue and θ_λ,k > 0 on Ω be the corresponding normalized eigenfunction of:

−Δθ=(σ+λ)kθ; Ω∂θ∂η+λθ=0; ∂Ω. (1.4)

We note that σ_λ,k > 0 when λ < A_k, σ_λ,k < 0 when λ > A_k, and σ_λ,k → 0 as λ → A_k. Next, let CN=(N+1)N+12NN,R be the radius of the largest inscribed ball in Ω, v be the unique solution of

−Δv=1; Ω∂v∂η+v=0; ∂Ω, (1.5)

and let w be the unique solution of

−Δw=1; Ω∂w∂η+A12w=0; ∂Ω. (1.6)

Now, we introduce hypotheses (H₂) and (H₃).

There exist a₁ > 0, b₁ > 0 such that a₁ < b₁ and

min{Am,a1f(a1)1∥v∥∞}>max{b1f(b1)2NCNR2,Am+1,1}.
There exist a₂ > 0, b₂ > 0 such that a₂ < b₂ and

a2f(a2)1∥w∥∞≥Am+1>max{b2f(b2)2NCNR2,A12}.

We note that functions satisfying (H₁) - (H₃) are such that sf(s) has the shape as in Figure 1 (with l1l2 ≫ 1).

$Fig. 1 Shape of sf(s) $\begin{array}{} \displaystyle \frac{s}{f(s)} \end{array}$ when our hypotheses are satisfied.$

Fig. 1

Shape of sf(s) when our hypotheses are satisfied.

We now state our main results:

Theorem 1.1

Let (H₁) hold. Then (1.1) has a positive solution for λ ∈ [A_m+1, A_m). Also, a positive solution u_λ for λ < A_m and λ ≈ A_m such that ∥u_λ∥_∞ → ∞ as λ → Am− . Further, there exists λ < A_m+1 such that (1.1) has at least two positive solutions for λ ∈ [λ, A_m+1). (Here, by λ ≈ A_m, we mean λ is close to A_m.)
Let (H₁) and (H₂) hold. Then (1.1) has at least three positive solutions for

λ∈max{b1f(b1)2NCNR2,Am+1,1},min{Am,a1f(a1)1∥v∥∞}.

Theorem 1.2

Let (H₁) and (H₃) hold. Then there exists λ∗∈(max{b2f(b2)2NCNR2,A12},Am+1) such that (1.1) has at least four positive solutions for λ ∈ [λ^*, A_m+1).

Corollary 1.3

Let (H₁) - (H₃) hold. Then there exists λ^* such that (1.1) has a positive solution for λ ∈ [λ^*, A_m), a positive solution u_λ for λ < A_m and λ ≈ A_m such that ∥u_λ∥_∞ → ∞ as λ → Am− , at least four positive solutions for λ ∈ [λ^*, A_m+1) and at least three positive solutions for

λ∈max{b1f(b1)2NCNR2,Am+1,1},min{Am,a1f(a1)1∥v∥∞}.

Remark 1.1

It is easy to show that (1.1) has no positive solutions for λ ≈ 0, and when m > 0 for λ > A_m (see Appendix).

Remark 1.2

A typical f which is likely to produce such a Σ-shaped bifurcation curve is as follows: Convex on (0, α) for some α > 0 driving the bifurcation curve initially to the left, a strong concavity on (α, β) with β > α making the bifurcation curve go back to the right, a strong convexity on (β, γ) with γ > β driving the bifurcation curve back again to the left, and then a strong concavity on (γ, ∞) bringing the curve eventually to the right (see Figure 4).

Fig. 2

An expected bifurcation diagram for (1.1) when hypotheses of Theorem 1.1(b) are satisfied.

Fig. 3

An expected bifurcation diagram for (1.1) when hypotheses of Corollary 1.3 are satisfied.

Fig. 4

Shape of f producing multiplicity.

For related study of models in biology see also [4, 5].

Finally, for an example for which Theorem 1.1, Theorem 1.2, and Corollary 1.3 hold, consider

−Δu=λf(u)=λ[mu+g(u)]; Ω∂u∂η+λu=0; ∂Ω,

with

g(s)=gα,k(s)=ecsc+s−1;s≤k[eαsα+s−eαkα+k]+[eckc+k−1];s>k,

where c > 2 is a fixed number, m ≥ 0, α > 0 and k > 0 are parameters. We will discuss this example in detail in Section 4.

We present some preliminaries in Section 2. We provide proofs of Theorems 1.1 - 1.2 and Corollary 1.3 in Section 3. In Section 4, we discuss in detail the example f we introduced above and show that Theorems 1.1 - 1.2 and Corollary 1.3 hold for certain parameter values. In Section 5, when Ω = (0, 1), via the quadrature method discussed in [3], we provide approximations to the exact bifurcation diagrams via Mathematica computations for the example discussed in Section 4. Our existence and multiplicity results are established via a method of sub-supersolutions.

2 Preliminaries

In this section, we introduce definitions of a (strict) subsolution and a (strict) supersolution of (1.1), and state a sub-supersolution theorem and a three solution theorem that we will use.

By a subsolution of (1.1) we mean ψ ∈ C²(Ω) ∩ C¹(Ω) that satisfies

−Δψ≤λf(ψ); Ω∂ψ∂η+λψ≤0; ∂Ω.

By a supersolution of (1.1) we mean Z ∈ C²(Ω) ∩ C¹(Ω) that satisfies

−ΔZ≥λf(Z); Ω∂Z∂η+λZ≥0; ∂Ω.

By a strict subsolution of (1.1) we mean a subsolution which is not a solution. By a strict supersolution of (1.1) we mean a supersolution which is not a solution.

Then the following results hold (see [6, 7]):

Lemma 2.1

Let ψ and Z be a subsolution and a supersolution of (1.1) respectively such that ψ ≤ Z. Then (1.1) has a solution u ∈ C²(Ω) ∩ C¹(Ω) such that u ∈ [ψ, Z].

Lemma 2.2

Let u₁ and u₂ be a subsolution and a supersolution of (1.1) respectively such that u₁ ≤ u₂ in Ω. Let u₂ and u₁ be a strict subsolution and a strict supersolution of (1.1) respectively such that u₂, u₁ ∈ [u₁, u₂] and u₂ ≰ u₁. Then (1.1) has at least three solutions u₁, u₂ and u₃ where u_i ∈ [u_i, u_i] for i = 1, 2 and u₃ ∈ [u₁, u₂] ∖ ([u₁, u₁] ∪ [u₂, u₂]).

3 Proofs of Theorems 1.1-1.2 and Corollary 1.3

First we construct sub-super solutions for certain λ ranges. Recall θ_λ,k and σ_λ,k (see (1.4)).

Construction of a small strict subsolution ψ₁ for λ < A_m+1 and λ ≈ A_m+1 when (H₁) is satisfied

We first note that f″(s) > 0 for s ≈ 0 since g″(0) > 0. Hence there exists A^* > 0 and s₁ > 0 such that f″(s) > A^* for s < s₁. Let ψ₁ = δ_λθ_λ,m+1 where δλ=2(m+1)σλ,m+1λA∗minΩ¯θλ,m+1. We note that σ_λ,m+1 > 0, σ_λ,m+1 → 0 as λ → Am+1− , and minΩ¯ θ_λ,m+1 ↛ 0 as λ → Am+1− . Thus δ_λ → 0⁺ as λ → Am+1− . Now by Taylor’s Theorem, we have f(ψ₁) = f(0) + f′(0)ψ₁ + f″(ζ)2ψ12=(m+1)ψ1+f″(ζ)2ψ12 for some ζ ∈ [0, ψ₁]. Then we have

−Δψ1−λf(ψ1)=δλ(σλ,m+1+λ)(m+1)θλ,m+1−λ[(m+1)δλθλ,m+1+f″(ζ)2(δλθλ,m+1)2]<δλθλ,m+1[(m+1)σλ,m+1−λA∗2δλminΩ¯θλ,m+1]=0;Ω

by our choice of δ_λ, for λ < A_m+1 and λ ≈ A_m+1 such that ψ₁ < s₁. Also, ∂ψ1∂η+λψ1=0 on ∂ Ω since θ_λ,m+1 satisfies this boundary condition. Thus, there exists λ < A_m+1 such that ψ₁ is a strict subsolution of (1.1) for λ ∈ [λ, A_m+1).

Construction of a small subsolution ψ₂ for λ ∈ [A_m+1, A_m) when (H₁) is satisfied

We note that f′(0) = m + 1, σ_λ,m+1 ≤ 0 for λ ∈ [A_m+1, A_m) and σ_λ,m+1 → 0 as λ → A_m+1. Let ψ₂ = n_λ θ_λ,m+1 with n_λ > 0. Now, consider H(s) = (σ_λ,m+1+λ)(m + 1)s-λ f(s). Then we have H(0) = 0, H′(0) = σ_λ,m+1(m + 1) ≤ 0 and H″(0) = −λ f″(0) < 0 since f″(0) > 0. This implies that −Δ ψ₂ = n_λ (σ_λ,m+1+λ)(m + 1)θ_λ,m+1 < λ f(n_λ θ_λ,m+1) = λ f(ψ₂) in Ω for n_λ ≈ 0. We also have ∂ψ2∂η+λψ2=0 on ∂ Ω since θ_λ,m+1 satisfies this boundary condition. Thus ψ₂ is a subsolution of (1.1) for n_λ ≈ 0 when λ ∈ [A_m+1, A_m).

Construction of a subsolution ψ₃ for λ < A_m and λ ≈ A_m such that ∥ψ₃∥_∞ → ∞ as λ → Am− when (H₁) is satisfied

Let m > 0 and ψ₃ = ϵ_λ θ_λ,m where ϵλ=λg(minΩ¯θλ,m)mσλ,m∥θλ,m∥∞. We note that ϵ_λ > 0 since σ_λ,m > 0 for λ < A_m. Further, ϵ_λ → ∞ as λ → Am− since σ_λ,m → 0⁺ as λ → Am− and minΩ¯ θ_λ,m ↛ 0. This implies that ∥ψ₃∥_∞ → ∞ as λ → Am− . Now we have

−Δψ3−λf(ψ3)=ϵλ[(λ+σλ,m)mθλ,m]−λ[mϵλθλ,m+g(ϵλθλ,m)]=ϵλmσλ,mθλ,m−λg(ϵλθλ,m)≤ϵλmσλ,m∥θλ,m∥∞−λg(ϵλθλ,m)=λ[g(minΩ¯θλ,m)−g(ϵλθλ,m)]≤0;Ω

for λ ≈ A_m, since ϵ_λ > 1 for λ ≈ A_m and g is increasing. Hence, we have −Δ ψ₃ ≤ λ f(ψ₃) in Ω. Also, on the boundary we have ∂ψ3∂η+λψ3=0 since θ_λ,m satisfies this boundary condition. Consequently ψ₃ is a subsolution of (1.1) such that ∥ψ₃∥_∞ → ∞ as λ → Am− .

Next, let m = 0. Here we can show (1.1) has a subsolution ψ₃ such that ∥ψ₃∥_∞ → ∞ as λ → ∞ by using a well known result in [8] for semipositone problems. Namely, define h ∈ C²([0, ∞)) such that h(0) < 0, h(s) ≤ f(s) for s ∈ (0, ∞) and lims→∞ h(s) > 0. Then the boundary value problem

−Δw=λh(w); Ω, w=0; ∂Ω,

has a solution w_λ > 0 for λ ≫ 1 such that ∥w_λ∥_∞ → ∞ as λ → ∞. Since by the Hopf maximum principle ∂w¯λ∂η < 0 on ∂ Ω, it is easy to show that ψ₃ = w_λ is a subsolution of (1.1) for λ ≫ 1 such that ∥ψ₃∥_∞ → ∞ as λ → ∞.

Construction of a strict subsolution ψ₄ for λ>bf(b)2NCNR2 where b = b₁ when (H₂) is satisfied and b = b₂ when (H₃) is satisfied

Here we construct a strict subsolution ψ₄ for λ>bf(b)2NCNR2 using the iteration of a subsolution ψ̃ created originally in [9] and later also used in [10]. Namely, the authors in [10] take ψ to be the solution of:

−ψ″(r)−N−1rψ′(r)=λf(w(r)); r∈(0,R)ψ′(0)=0=ψ(R), (3.1)

where R is the radius of the largest inscribed ball, B_R, in Ω (see Figure 5) and w(r) = b ρ(r) with

Fig. 5

Largest inscribed ball in Ω.

ρ(r)=1;r∈[0,ϵ]1−1−R−rR−ϵβα; r∈(ϵ,R],α,β>1.

When λ>bf(b)2NCNR2 for certain choices of α > 1, β > 1, and ϵ ∈ (0, 1) it was proven that (see [9] for details) ψ ≥ w on [0, R] and hence a subsolution of (3.1) since f is increasing. Now since f(0) = 0 it follows that

ψ~=ψ; BR0; Ω∖BR,

is a strict subsolution of:

−Δu=λf(u); Ωu=0; ∂Ω,

for λ>bf(b)2NCNR2 such that ∥ ψ̃∥_∞ ≥ b.

Now let ψ₄ be the first iteration of ψ̃, namely, ψ₄ be the solution to the problem:

−Δψ4=λf(ψ~); Ω∂ψ4∂η+λψ4=0; ∂Ω.

Then we have − Δ (ψ₄ − ψ̃) ≥ 0 and ∂(ψ4−ψ~)∂η+λ(ψ4−ψ~)=−∂ψ~∂η>0 by the Hopf maximum principle. This implies that ψ₄ > ψ̃ in Ω. Hence, ψ₄ is a strict subsolution of (1.1) for λ>bf(b)2NCNR2.

Construction of a large supersolution Z₁ for λ < A_m when (H₁) is satisfied

Let m > 0. Choose Z₁ = M θ_λ,m for M > 0. Then −Δ Z₁ − λ f(Z₁) = M(σ_λ,m + λ) m θ_λ,m − λ [mMθ_λ,m + g(M θ_λ,m)] = mMθ_λ,m σλ,m−λg(Mθλ,m)mMθλ,m > 0 in Ω for M ≫ 1 since σ_λ,m > 0 for λ < A_m and g(s)s → 0 as s → ∞. Further, ∂Z1∂η+λZ1=0 on ∂ Ω since θ_λ,m satisfies this boundary condition. Hence, Z₁ is a supersolution of (1.1) for M ≫ 1.

Next, let m = 0. Here we choose Z₁ = M e_λ, where e_λ is the unique solution of −Δ e = 1 in Ω and ∂e∂η+λe=0 on ∂ Ω. Note e_λ > 0 on Ω. Then − Δ Z₁ − λ f(Z₁) = M − λ g(Me_λ) ≥ M1−λg(M∥eλ∥∞)M∥eλ∥∞∥eλ∥∞ > 0 for M ≫ 1 since g is increasing and g(s)s → 0 as s → ∞. Also, ∂Z1∂η+λZ1=0 on ∂ Ω since e_λ satisfies this boundary condition. Hence, Z₁ is a supersolution of (1.1) for M ≫ 1.

Construction of a strict supersolution Z₂ for λ < A_m+1 when (H₁) is satisfied

Let Z₂ = m_λθ_λ,m+1 and l(s) = (σ_λ,m+1 + λ)(m + 1)s − λ f(s). We note that σ_λ,m+1 > 0 for λ < A_m+1. Then we have l(0) = 0 and l′(0) = (σ_λ,m+1 + λ)(m + 1) − λ f′(0) = σ_λ,m+1(m + 1) > 0 since f′(0) = m + 1. This implies that −Δ Z₂ = m_λ(σ_λ,m+1 + λ)(m + 1)θ_λ,m+1 > λ f(m_λ θ_λ,m+1) = λ f(Z₂) in Ω for m_λ ≈ 0. On the boundary, we have ∂Z2∂η+λZ2=0 since θ_λ,m+1 satisfies this boundary condition. Thus Z₂ with m_λ ≈ 0 is a strict supersolution of (1.1) for λ < A_m+1.

Construction of a strict supersolution Z₃ for λ∈1,a1f(a1)1∥v∥∞ when (H₂) is satisfied

Let Z3=a1v∥v∥∞ where v is as in (1.5). Then −ΔZ3=a1∥v∥∞>λf(a1)≥λf(Z3) since λ<a1f(a1)1∥v∥∞ and f is increasing. Further, Z₃ satisfies ∂Z3∂η+λZ3=a1∥v∥∞∂v∂η+λa1v∥v∥∞>a1∥v∥∞[∂v∂η+v]=0 on ∂Ω since λ > 1. Thus Z₃ is a strict supersolution of (1.1) for λ∈1,a1f(a1)1∥v∥∞.

Construction of a strict supersolution Z₄ for λ∈A12,a2f(a2)1∥w∥∞ when (H₃) is satisfied

Let Z4=a2w∥w∥∞ where w is as in (1.6). Then −ΔZ4=a2∥w∥∞>λf(a2)≥λf(Z4) since λ<a2f(a2)1∥w∥∞ and f is increasing. Further, Z₄ satisfies ∂Z4∂η+λZ4=a2∥w∥∞∂w∂η+λa2w∥w∥∞>a2∥w∥∞[∂w∂η+A12w]=0 on ∂Ω since λ > A12 . Thus Z₄ is a strict supersolution of (1.1) for λ∈A12,a2f(a2)1∥w∥∞.

Now we prove Theorems 1.1-1.2 and Corollary 1.3

Proof of Theorem 1.1

Let M be as in the construction of the supersolution Z₁ and n_λ be as in the construction of the subsolution ψ₂. We choose M ≫ 1 and n_λ ≈ 0 such that Z₁ ≥ ψ₂. By Lemma 2.1, (1.1) has a positive solution u_λ ∈ [ψ₂, Z₁] for λ ∈ [A_m+1, A_m).

Recall the subsolution ψ₃ of (1.1). Now we choose M ≫ 1 such that ψ₃ ≤ Z₁. Hence, recalling that ∥ψ₃∥_∞ → ∞ as λ → Am− , by Lemma 2.1, (1.1) has a positive solution u_λ ∈ [ψ₃, Z₁] such that ∥u_λ∥_∞ → ∞ as λ → Am− .

Next, let λ ∈ [λ, A_m+1) where λ be as in the construction of the strict subsolution ψ₁. We note that ψ₀ = 0 is a solution and hence a subsolution of (1.1). Recall the strict supersolution Z₂ of (1.1). Now we choose m_λ small enough such that ∥Z₂∥_∞ < ∥ψ₁∥_∞. Next, we choose M ≫ 1 such that ψ₁ ≤ Z₁ and Z₂ ≤ Z₁ (see Figure 6). By Lemma 2.2, (1.1) has at least two positive solutions u₁ ∈ [ψ₁, Z₁] and u₂ ∈ [ψ₀, Z₁] ∖ ([ψ₀, Z₂] ∪ [ψ₁, Z₁]) for λ ∈ [λ, A_m+1).

Fig. 6
Subsolutions ψ₀, ψ₁ and supersolutions Z₁, Z₂.
Recall the strict subsolution ψ₄ when b = b₁ and the strict supersolution Z₃ of (1.1). Now we choose n_λ small enough such that ψ₂ ≤ ψ₄ and ψ₂ ≤ Z₃. Next we choose M ≫ 1 such that ψ₄ ≤ Z₁ and Z₃ ≤ Z₁ (see Figure 7). We note that ∥ψ₄∥_∞ ≥ b₁ > a₁ = ∥Z₃∥_∞. By Lemma 2.2, (1.1) has at least three positive solutions for λ∈max{b1f(b1)2NCNR2,Am+1,1},min{Am,a1f(a1)1∥v∥∞}. We note that in the construction of ψ₂, ψ₄, Z₁, and Z₃, the intersection of intervals of λ is max{b1f(b1)2NCNR2,Am+1,1},min{Am,a1f(a1)1∥v∥∞}. This completes the proof.

Fig. 7
Subsolutions ψ₂, ψ₄ and supersolutions Z₁, Z₃.

Proof of Theorem 1.2

Let λ^* = λ and ψ₀ be as in the proof of Theorem 1.1. Recall the strict supersolution Z₄ and the strict subsolution ψ₄ when b = b₂. First we choose λ∗>max{b2f(b2)2NCNR2,A12},λ∗<Am+1, and λ^* ≈ A_m+1 (making δ_λ ≈ 0) such that ψ₁ < ψ₄ and ψ₁ < Z₄ for λ ∈ [λ^*, A_m+1). Next, we choose m_λ small enough such that ∥Z₂∥_∞ < ∥ψ₁∥_∞. Further, we can choose M ≫ 1 such that ψ₁ ≤ Z₁ and Z₂ ≤ Z₁ (see Figure 8). By Lemma 2.2, (1.1) has a positive solution u₁ ∈ [ψ₀, Z₁] ∖ ([ψ₀, Z₂] ∪ [ψ₁, Z₁]) for λ ∈ [λ^*, A_m+1). We also have ψ₄ ≤ Z₁, Z₄ ≤ Z₁ for M ≫ 1 and ∥ψ₄∥_∞ ≥ b₂ > a₂ = ∥Z₄∥_∞ (see Figure 8). Again, by Lemma 2.2, (1.1) has at least three positive solutions u₂ ∈ [ψ₁, Z₄], u₃ ∈ [ψ₄, Z₁], and u₄ ∈ [ψ₁, Z₁] ∖ ([ψ₁, Z₄] ∪ [ψ₄, Z₁]) for λ ∈ [λ^*, A_m+1). Hence (1.1) has at least four positive solutions for λ ∈ [λ^*, A_m+1). This completes the proof.

Fig. 8

Subsolutions ψ₀, ψ₁, ψ₄ and supersolutions Z₁, Z₂, Z₄.

Proof of Corollary 1.3

We note that the proof of Corollary 1.3 is an immediate consequence of the proof of Theorem 1.1 and Theorem 1.2.

4 Example

In this section, we provide an example for which Theorems 1.1 - 1.2 and Corollary 1.3 hold. Consider

−Δu=λf(u)=λ[mu+g(u)]; Ω∂u∂η+λu=0; ∂Ω, (4.1)

where

g(s)=gα,k(s)=ecsc+s−1;s≤k[eαsα+s−eαkα+k]+[eckc+k−1];s>k.

Here c > 2 is a fixed number, m ≥ 0, α > 0 and k > 0 are parameters. It is easy to verify that (H₁) is satisfied.

We first consider the case when m = 0. Since kf(k)=keckc+k−1⟶∞ as k ⟶ ∞, there exists k₀ > 0 (independent of α) such that for k > k₀

kf(k)>max{A1,1}.max{∥v∥∞,∥w∥∞}. (4.2)

Let k > k₀. Next, for α > k, since αf(α)=α[eα2−eαkα+k]+[eckc+k−1]⟶0 as α ⟶ ∞, there exists α₀(k) ( > k) such that for α > α₀(k)

A1>αf(α).2NCNR2. (4.3)

Thus, choosing a₁ = a₂ = k, b₁ = b₂ = α, by (4.2), (4.3), it is easy to see that both (H₂) and (H₃) are also satisfied when k > k₀ and α > α₀(k). Hence Theorems 1.1 - 1.2 and Corollary 1.3 hold for this example when k > k₀ and α > α₀(k).

By continuity, it follows that Theorems 1.1-1.2 and Corollary 1.3 also hold for this example when k > k₀, α > α₀(k) and m ≈ 0.

5 Approximation to the exact bifurcation diagrams for (4.1) when Ω = (0, 1)

In this case, we note that the solutions of (4.1) can be completely analyzed by the quadrature method discussed in [3]. Here, (4.1) reduces to

−u″=λf(u); (0,1)−u′(0)+λu(0)=0u′(1)+λu(1)=0, (5.1)

and the positive solutions to (5.1) are symmetric about x = 12 . Namely, the solutions take the shape as in Figure 9.

Fig. 9

The shape of the solutions of (5.1).

Further, the exact bifurcation diagrams for positive solutions to (5.1) are described by the equations:

λ=2(∫qρdsF(ρ)−F(s))2 (5.2)

and

2[F(ρ)−F(q)]=q2 (5.3)

where, ρ = u( 12 ), q = u(0) = u(1), and F(s)=∫0sf(t)dt.

Below we provide some bifurcation diagrams for the example discussed in the previous section via Mathematica computation of (5.2)-(5.3). In fact, we obtain exact Σ-shaped bifurcation curves for certain parameter values.

Fig. 10

The local view of the bifurcation diagrams near the bifurcation point (A₁, 0) when m = 0 and c = 2.5.

Fig. 11

The local view of the bifurcation diagrams near the bifurcation point (A_1.01, 0) when m = 0.01 and c = 2.5.

Appendix

Proof of Remark 1.1

First, we show the non-existence of positive solutions for λ ≈ 0. Let u be a positive solution of (1.1). Then by the Green’s second identity we obtain:

0=∫Ω[θλ,m+1Δu−uΔθλ,m+1]dx=∫Ω[−λf(u)+u(σλ,m+1+λ)(m+1)]θλ,m+1dx≥∫Ω[−λMu+u(σλ,m+1+λ)(m+1)]θλ,m+1dx=∫Ωλ(m+1)σλ,m+1λ−[M−(m+1)]uθλ,m+1dx (.4)

where M > (m + 1) is such that f(s) ≤ Ms for all s ∈ [0, ∞). Now for λ < A_m+1, σ_λ,m+1 > 0, and limλ→0σλ,m+1λ=∞ (see [11]). This contradicts (.4) for λ ≈ 0 and hence (1.1) has no positive solution for λ ≈ 0.

Next, when m > 0, if u is a positive solution of (1.1), then again by the Green’s second identity we obtain:

0=∫Ω[θλ,mΔu−uΔθλ,m]dx=∫Ω[−λf(u)+u(σλ,m+λ)m]θλ,mdx≤∫Ω[−λmu+u(σλ,m+λ)m]θλ,mdx=∫Ωmσλ,muθλ,mdx (.5)

since f(s) ≥ ms on [0, ∞). Now if λ > A_m then σ_λ,m < 0 which contradicts (.5). Hence (1.1) has no positive solution for λ > A_m.

Conflict of interest: The authors state no conflict of interest.

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Received: 2020-12-02

Accepted: 2021-03-15

Published Online: 2021-05-04

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/anona-2020-0180

Keywords for this article

Σ-Shaped Bifurcaion Curves; Positive Solutions; Sub-Super Solutions

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Σ-Shaped Bifurcation Curves

Article

Abstract

1 Introduction

Theorem 1.1

Theorem 1.2

Corollary 1.3

Remark 1.1

Remark 1.2

2 Preliminaries

Lemma 2.1

Lemma 2.2

3 Proofs of Theorems 1.1-1.2 and Corollary 1.3

Construction of a small strict subsolution ψ1 for λ < Am+1 and λ ≈ Am+1 when (H1) is satisfied

Construction of a small subsolution ψ2 for λ ∈ [Am+1, Am) when (H1) is satisfied

Construction of a subsolution ψ3 for λ < Am and λ ≈ Am such that ∥ψ3∥∞ → ∞ as λ → Am− when (H1) is satisfied

Construction of a strict subsolution ψ4 for λ>bf(b)2NCNR2 where b = b1 when (H2) is satisfied and b = b2 when (H3) is satisfied

Construction of a large supersolution Z1 for λ < Am when (H1) is satisfied

Construction of a strict supersolution Z2 for λ < Am+1 when (H1) is satisfied

Construction of a strict supersolution Z3 for λ∈1,a1f(a1)1∥v∥∞ when (H2) is satisfied

Construction of a strict supersolution Z4 for λ∈A12,a2f(a2)1∥w∥∞ when (H3) is satisfied

Now we prove Theorems 1.1-1.2 and Corollary 1.3

Proof of Theorem 1.1

Proof of Theorem 1.2

Proof of Corollary 1.3

4 Example

5 Approximation to the exact bifurcation diagrams for (4.1) when Ω = (0, 1)

Appendix

Proof of Remark 1.1

References

Articles in the same Issue

Articles in the same Issue

Articles in the same Issue

Construction of a small strict subsolution ψ₁ for λ < A_m+1 and λ ≈ A_m+1 when (H₁) is satisfied

Construction of a small subsolution ψ₂ for λ ∈ [A_m+1, A_m) when (H₁) is satisfied

Construction of a subsolution ψ₃ for λ < A_m and λ ≈ A_m such that ∥ψ₃∥_∞ → ∞ as λ → Am− when (H₁) is satisfied

Construction of a strict subsolution ψ₄ for λ>bf(b)2NCNR2 where b = b₁ when (H₂) is satisfied and b = b₂ when (H₃) is satisfied

Construction of a large supersolution Z₁ for λ < A_m when (H₁) is satisfied

Construction of a strict supersolution Z₂ for λ < A_m+1 when (H₁) is satisfied

Construction of a strict supersolution Z₃ for λ∈1,a1f(a1)1∥v∥∞ when (H₂) is satisfied

Construction of a strict supersolution Z₄ for λ∈A12,a2f(a2)1∥w∥∞ when (H₃) is satisfied