Classification and evolution of bifurcation curves for a one-dimensional Neumann – Robin problem and its applications

We study the classification and evolution of bifurcation curves of positive solutions for the one-dimensional Neumann–Robin boundary value problem { u′′(x) + λ f (u(x)) = 0, 0 < x < 1, u′(0) = 0 and u′(1) + αu(1) = 0, where λ > 0 is a bifurcation parameter, α > 0 is an evolution parameter, and nonlinearity f satisfies f (0) ≥ 0 and f (u) > 0 for u > 0. We obtain the multiplicity of positive solutions for α > 0 and λ > 0. Applications are given.

Before going into further discussions on problems (1.1) and (1.2), we first introduce following terminologies, which also hold for S if S α is replaced by S.

Monotone increasing and strictly increasing:
We say that, on the (λ, u ∞ )-plane, the bifurcation curve S α is monotone increasing if S α is a continuous curve and for each pair of points (λ Monotone decreasing and strictly decreasing: We say that, on the (λ, u ∞ )-plane, the bifurcation curve S α is monotone decreasing if S α is a continuous curve and for each pair of points (λ 1 , u λ 1 ∞ ) and (λ 2 , u λ 2 ∞ ) of S α , u λ 1 ∞ < u λ 2 ∞ implies λ 1 ≥ λ 2 , and it is
Exactly S-shaped: We say that, on the (λ, u ∞ )-plane, the bifurcation curve S α is exactly Sshaped if S α is S-shaped and it has exactly two turning points; see Fig. 1.1 (I).
Exactly reversed S-shaped: We say that, on the (λ, u ∞ )-plane, the bifurcation curve S α is exactly reversed S-shaped if S α is reversed S-shaped and it has exactly two turning points.

⊂-shaped:
We say that, on the (λ, u ∞ )-plane, the bifurcation curve S α is ⊂-shaped if S α has at least one turning point, say (λ * , u λ * ∞ ), satisfying (i) at (λ * , u λ * ∞ ) the bifurcation curve S α turns to the right, (ii) S α initially continues to the left and eventually continues to the right.
Exactly ⊂-shaped: We say that, on the (λ, u ∞ )-plane, the bifurcation curve S α is exactly ⊂-shaped if S α is ⊂-shaped and it has exactly one turning point; see Fig. 1.1(II).
Reversed ⊂-shaped: We say that, on the (λ, u ∞ )-plane, the bifurcation curve S α is ⊂-shaped if S α has at least one turning point, say (λ * , u λ * ∞ ), satisfying (i) at (λ * , u λ * ∞ ) the bifurcation curve S α turns to the left, (ii) S α initially continues to the right and eventually continues to the left.
Exactly reversed ⊂-shaped: We say that, on the (λ, u ∞ )-plane, the bifurcation curve S α is exactly reversed ⊂-shaped if S α is reversed ⊂-shaped and it has exactly one turning point.
In Section 2 below, we study the classification and evolution of bifurcation curves S α of positive solutions for (1.1) with general nonlinearity f satisfying hypothesis (H).In addition, as applications, we study the classification and evolution of bifurcation curves S α of positive solution for (1.1) with two particular nonlinearities which satisfy hypothesis (H) with η = ∞.Huang and Wang [6] studied the evolution and qualitative behaviors of bifurcation curves S of positive solutions of one-dimensional perturbed Gelfand problem (1.2), (1.6).
(i) For p = 1, the bifurcation curve S starts at π 2 8 , 0 and goes to infinity oscillationally along the vertical line λ = π 2 8 , and it has infinitely many turning points.
(ii) For 1 < p < 2, the bifurcation curve S starts at π 2 4 , 0 and goes to infinity along the vertical line λ = 0, and it is reversed S-shaped.
(iii) For p = 2, the bifurcation curve S starts at π 2 4 , 0 and goes to infinity along the vertical line λ = 0, and it is strictly decreasing.
(iv) For p > 2, the bifurcation curve S starts at π 2 4 , 0 and goes to infinity along the vertical line λ = 0, and it is exactly reversed ⊂-shaped.
The paper is organized as follows.Section 2 contains statements of main results.Section 3 contains lemmas needed to prove the main results.Finally, Section 4 contains the proofs of the main results.

Main results
The main results in this paper are next Theorems 2.1-2.6 and 2.8.In Theorems 2.1 and 2.3 for (1.1), under (H) for nonlinearity f , for all α > 0, we present some basic properties of bifurcation curves S α on the (λ, u ∞ )-plane.In particular, in Theorem 2.1(iii), we show that, on the (λ, u ∞ )-plane, S α moves to right strictly as α increases and S α tends to the u ∞ -axis as α approaches 0 + and tends to S as α approaches infinity.In Theorem 2.1 (iv), we prove an interesting comparison result, cf.Remark 2.2 stated behind.In Theorems 2.4 and 2.5, under (H) and some suitable hypotheses on f , for α > 0, we give a classification of bifurcation curves S α on the (λ, u ∞ )-plane.In Theorems 2.6 and 2.8, as applications of Theorems 2.1-2.5, we study the classification and evolution of bifurcation curves S α for problem (1.2), (1.6) and problem (1.2), (1.7), respectively with α varying from 0 + to infinity.
We first define the number Notice that λ * is a strictly increasing function of α > 0.
then S α goes to infinity along the vertical line λ = 0 if s ∞ > 1, to infinity along the vertical line then S α goes to infinity along the horizontal line u ∞ = η if s η ≥ 1, and ends at some point λ η , η with λ η > 0 if 0 < s η < 1.
(b) As α approaches infinity, S α tends to S.
For the sake of convenience, we assume the following conditions.
(iii) (See Fig. (3a) The bifurcation curve S of (1.2) is S-shaped (resp.reversed S-shaped), are turning points of S such that S turns to the left (resp.to the right) at λ 1 , u λ 1 ∞ and S turns to the right Then S α is S-shaped (resp.reversed S-shaped) for all α > 0. Furthermore, there exist two points λ 3 , u λ 3 ∞ and λ 4 , and the portion of S α connecting λ 3 , u λ 3 ∞ and λ 4 , u λ 4 ∞ is monotone decreasing (resp.monotone increasing), where λ 3 , u λ 3 ∞ is a turning point to the left (resp.to the right) of S α and λ 4 , u λ 4 ∞ is a turning point to the right (resp. to the left) of S α .
are two positive zeros of quadratic polynomial (iii) For ε ≥ ε * , S α is strictly increasing for all α > 0. (i) For p = 1, S α has infinitely many turning points for all α > 0.
(ii) For 1 < p < 2, for any positive integer k, there exist k positive numbers such that S α has at least i turning points for 0 < α ≤ α i for i ∈ {1, 2, . . . ,k}.
Proof.(I) First, for f (u) = u b with b ≥ 0, we calculate that We observe that For b ≥ 0, by (3.15), we have that
Proof.We simply prove part (ii).The proof of part (i) is similar.We compute that where We see that φ(ρ, s) approaches zero as ρ → ∞, uniformly in s.So and hence part (ii) holds.The proof of Lemma 3.7 is complete.
The proof of Lemma 3.8 is complete.
(ii) Assume that f satisfies (D2) with positive p * < p < η and The proof of Lemma 3.10 is easy but tedious and hence we put it in Appendix D. Lemma 3.11. and be the function ∆ defined in Lemma 3.3.Then the following assertions (i) and (ii) hold.
(i) If f satisfies (D1) with positive p * < p < η, then (ii) If f satisfies (D2) with positive p * < p < η, then The proof of Lemma 3.11 is easy but tedious and hence we put it in Appendix E.
(ii) If f satisfies (C1 − ), then S α continues to the left initially.
(iii) If f satisfies (C2 + ), then S α continues to the right eventually.
(iv) If f satisfies (C2 − ), then S α continues to the left eventually.
The proof of Lemma 3.12 is complete.
The proof of Theorem 2.3 is complete.
The proof of Theorem 2.4 is complete.
The proof of Theorem 2.5 is complete.
The proof of part (ii) is complete.
The proof of Theorem 2.6 is complete.
The proof of Theorem 2.8 is complete.
The proof of Lemma 3.2 (iii) is complete.

Appendix B
Proof of Lemma 3.3.For fixed α > 0, since m ρ is defined by the equation .
Differentiating the above equation with respect to ρ, we have that .
Using notations in (3.11), we obtain that Multiplying both sides of above RHS equation by D( m ρ α ) and gathering terms containing m ρ , we obtain that Clearly, the coefficient of m ρ is always positive.Dividing both sides by the coefficient of m ρ , we obtain (3.10).In (B.1), plugging in (3.10), we obtain that The denominator is exactly equal to Φ(ρ) as defined in (3.14), and the numerator is exactly equal to (3.13) as well.The proof of Lemma 3.3 is complete. and Then, taking limit inferior to both sides of (3.2), we have that lim inf  (3.13), we obtain ∆(p) < 0 for all α > 0. The proof of part (i) is complete.Similarly, part (ii) can be proved.We omit it here.
The proof of Lemma 3.10 is complete.

Appendix E
Proof of Lemma 3.11.We compute the numerator of T (ρ) in (3.12)

Figure 1 . 1 :
Figure 1.1: (I) Exactly S-shaped bifurcation curve S α with exactly two turning points, which starts at (0, 0) and goes to infinity along the horizontal line u ∞ = η.(II) Exactly ⊂-shaped bifurcation curve S α with exactly one turning point, which starts at (λ 0 , 0) and goes to infinity along the horizontal line u ∞ = η.
η) and by(3.1),we have that H(m, ρ) − G(m, ρ) is C 3 in ρ and thus m ρ is C 3 in ρ as well.We consider m ρ as a function of α and ρ, and denote ∂ ∂ρ m ρ by m ρ .We then rewrite (3.2) as