Global Bifurcation for Second-Order Neumann Problem with a Set-Valued Term

Abstract

We study the global bifurcation of the differential inclusion of the form −(𝑘𝑢^′)^′+𝑔(⋅,𝑢)∈𝜇𝐹(⋅,𝑢),𝑢^′(0)=0=𝑢^′(1), where 𝐹 is a “set-valued representation” of a function with jump discontinuities along the line segment [0,1]×{0}. The proof relies on a Sturm-Liouville version of Rabinowitz's bifurcation theorem and an approximation procedure.

1. Introduction

We are concerned with the following differential inclusion which arises from a Budyko-North type energy balance climate models:

−(𝑘𝑢^′)^′𝑢(𝑥)+𝑔(𝑥,𝑢(𝑥))∈𝜇𝐹(𝑥,𝑢(𝑥)),𝑥∈(0,1)a.e._′(0)=0,𝑢^′(1)=0; (1.1) see [1–6] and the references therein. In particular, the set-valued right-hand side arises from a jump discontinuity of the albedo at the ice-edge in these models. By filling in such a gap, one arrives at the set-valued problem (1.1). As in [6], we are here interested in a considerably simplified version as compared to the situation from climate modeling; for example, a one-dimensional regular Sturm-Liouville differential operator substitutes for a two-dimensional Laplace-Beltrami operator or a singular Legendre-type operator, and the jump discontinuity is transformed to 𝑢=0 in a way, which resembles only locally the climatological problem.

Assume that

(H1)𝑘∈𝐶¹([0,1]), inf𝑘>0;(H2)𝑔∈𝐶([0,1]×ℝ), 𝑔(𝑥,⋅) strictly increasing for 𝑥∈[0,1], 𝑔₁(𝑥)∶=lim_|𝑦|→0𝑔(𝑥,𝑦)𝑦 (1.2) exists uniformly for 𝑥∈[0,1], and 𝑔₁(𝑥)>0 on [0,1],(H2^′)𝑔 satisfies that 𝑔₂(𝑥)∶=lim_|𝑦|→∞𝑔(𝑥,𝑦)𝑦 (1.3) exists uniformly for 𝑥∈[0,1];(H3)𝑓₊∈𝐶([0,1]×ℝ₊,(0,∞)), inf𝑓₊>0, 𝑓₋∈𝐶([0,1]×ℝ₋,(−∞,0)), sup𝑓₋<0.

Let 𝐹 in (1.1) be given by 𝐹(𝑥,𝑦)∶={𝑓₊(𝑥,𝑦)},𝑥∈[0,1],𝑦>0,[𝑓₋(𝑥,0),𝑓₊(𝑥,0)],𝑥∈[0,1],{𝑓₋(𝑥,𝑦)}𝑥∈[0,1],𝑦<0, (1.4) and set

𝒮∶={(𝜇,𝑤)∈ℝ×𝐶¹([0,1])∣(𝜇,𝑤)solves(1.1)}. (1.5) Throughout 𝒮 will be considered as subset of the Banach space 𝑌∶=ℝ×𝐶¹[0,1] under the norm

‖(𝜇,𝑤)‖_𝑌∶=max{|𝜇|,‖𝑤‖_∞,‖𝑤^′‖_∞}. (1.6) Let

ℤ₊∶={0,1,2,…}. (1.7)

Using a Sturm-Liouville version of Rabinowitz's bifurcation theorem and an approximation procedure, Hetzer [6] proved the following.

Theorem A (see [6, Theorem]). Let (H1)–(H3) be fulfilled. Then there exist sequences {𝐶^±_𝑛}_𝑛∈ℤ+ of unbounded, closed, connected subsets of 𝒮 with (0,𝟎)∈𝐶^±_𝑛 and the property that 𝑢 has exactly 𝑛 zeroes, which are all simple, if (𝜇,𝑢)∈𝐶^±_𝑛⧵{(0,𝟎)}. Moreover, 𝑢 is positive (negative) on an interval (0,𝑥) for some 𝑥∈(0,1], if (𝜇,𝑢)∈𝐶⁺_𝑛 ((𝜇,𝑢)∈𝐶⁻_𝑛) and 𝑢≢0.

It is easy to see from Theorem A that the effect of the discontinuity at zero is a solution branch which consists of infinitely many subbranches all meeting in (0,𝟎). Two subbranches are distinguished by the number of zeroes of the respective solutions. However, Theorem A provides no any information about the asymptotic behavior of 𝐶^±_𝑛 at infinity.

It is the purpose of this paper to study the asymptotic behavior of 𝐶^±_𝑛 at infinity, and accordingly, to determine values of 𝜇, for which there exist infinitely many nodal solutions of (1.1) (here and after, a function 𝑢∈𝐴𝐶¹[0,1] is a nodal solution of (1.1) if all of zeroes of 𝑢 are simple). To wit, we have the following.

Theorem 1.1. Let (H1)–(H3) and (H2^′) be fulfilled. Assume that (H4)(𝑓₊)_∞(𝑥)=(𝑓₋)_∞(𝑥)=∶𝑏(𝑥)∈𝐶([0,1],(0,∞)), (1.8) where (𝑓₊)_∞(𝑥)∶=lim_𝑠→+∞𝑓₊(𝑥,𝑠)𝑠,(𝑓₋)_∞(𝑥)∶=lim_{𝑠→−∞}𝑓₋(𝑥,𝑠)𝑠. (1.9) Then for each 𝑛∈ℤ₊, 𝐶⁺_𝑛 joins (0,𝟎) with (𝜂_𝑛,∞), 𝐶⁻_𝑛 joins (0,𝟎) with (𝜂_𝑛,∞), where 𝜂_𝑛, (𝑛∈ℤ₊), is the 𝑛-th eigenvalue of the linear problem: −(𝑘𝑢^′)^′(𝑥)+𝑔₂𝑢(𝑥)𝑢(𝑥)=𝜂𝑏(𝑥)𝑢(𝑥),𝑥∈[0,1],_′(0)=0,𝑢^′(1)=0. (1.10)

Corollary 1.2. Let (H1)–(H4) and (H2^′) be fulfilled. Let 𝑘∈ℕ be fixed. Then
(1) for each 𝜇∈[𝜂_𝑘−1,𝜂_𝑘), (1.1) has infinitely many solutions: 𝑢^𝜈_𝑗,𝜈∈{+,−},𝑗∈{𝑘,𝑘+1…}, (1.11) which satisfies that 𝑢⁺_𝑗 has exactly 𝑗 simple zeroes and 𝑢⁺_𝑗 is positive on an interval (0,𝑥) for some 𝑥∈(0,1], 𝑢⁻_𝑗 has exactly 𝑗 simple zeroes and 𝑢⁻_𝑗 is negative on an interval (0,𝑥) for some 𝑥∈(0,1);
(2) for each 𝜇∈(0,𝜂₀), (1.1) has infinitely many solutions: 𝑢^𝜈_𝑗,𝜈∈{+,−},𝑗∈{0,1,2…} (1.12) which satisfies that 𝑢⁺_𝑗 has exactly 𝑗 simple zeroes, and 𝑢⁺_𝑗 is positive on an interval (0,𝑥) for some 𝑥∈(0,1], 𝑢⁻_𝑗 has exactly 𝑗 simple zeroes, and 𝑢⁻_𝑗 is negative on an interval (0,𝑥) for some 𝑥∈(0,1).

2. Notations and Preliminary Results

Recall Kuratowski's notion of lower and upper limits of sequences of sets.

Definition 2.1 (see [7]). Let 𝑋 be a metric space and let {𝑍_𝑙}_𝑙∈ℕ be a sequence of subsets of 𝑋. The set limsup_𝑙→∞𝑍_𝑙∶={𝑥∈𝑋∶liminf_𝑙→∞dist(𝑥,𝑍_𝑙)=0} (2.1) is called the upper limit of the sequence {𝑍_𝑙}, whereas liminf_𝑙→∞𝑍_𝑙∶={𝑥∈𝑋∶lim_𝑙→∞dist(𝑥,𝑍_𝑙)=0} (2.2) is called the lower limit of the sequence {𝑍_𝑙}.

Definition 2.2 (see [7]). A component of a set 𝑀 is meant a maximal connected subset of 𝑀.

Lemma 2.3 (see [7]). Suppose that 𝑌 is a compact metric space, 𝐴 and 𝐵 are nonintersecting closed subsets of 𝑌, and no component of 𝑌 intersects both 𝐴 and 𝐵. Then there exist two disjoint compact subsets 𝑌_𝐴 and 𝑌_𝐵, such that 𝑌=𝑌_𝐴∪𝑌_𝐵, 𝐴⊂𝑌_𝐴, 𝐵⊂𝑌_𝐵.

Using the above Whyburn Lemma, Ma and An [8] proved the following.

Lemma 2.4 (see [8, Lemma 2.1]). Let 𝑍 be a Banach space and let {𝐴_𝑛} be a family of closed connected subsets of 𝑍. Assume that(i)there exist 𝑧_𝑛∈𝐴_𝑛, 𝑛=1,2,…, and 𝑧^∗∈𝑍, such that 𝑧_𝑛→𝑧^∗;(ii)𝑟_𝑛=sup{‖𝑥‖∣𝑥∈𝐴_𝑛}=∞;(iii)for every 𝑅>0, (⋃^∞_𝑛=1𝐴_𝑛)∩𝐵_𝑅 is a relatively compact set of 𝑍, where 𝐵_𝑅={𝑥∈𝑍∣‖𝑥‖≤𝑅}. (2.3)
Then there exists an unbounded component 𝒞 in limsup_𝑙→∞𝐴_𝑙 and 𝑧^∗∈𝒞.

Remark 2.5. The limiting processes for sets go back at least to the work of Kuratowski [9]. Lemma 2.4 will play an important role in the proof of Theorem 1.1. It is a slight generalization of the following well-known results due to Whyburn [7].

Proposition 2.6 (Whyburn [7, page 12]). Let 𝑍 be a Banach space and let {𝐴_𝑛} be a family of closed connected subsets of 𝑍. Let liminf_𝑙→∞𝐴_𝑙≠∅ and ⋃_𝑙∈ℕ𝐴_𝑙 is relatively compact. Then limsup_𝑙→∞𝐴_𝑙 is nonempty, compact, and connected.

Lemma 2.7. Let 𝑞∈𝐶([0,1],(0,∞)). Let 𝑝_𝑚∈𝐶([0,1],(0,∞)) be such that 𝑝_𝑚(𝑡)≥𝜌,𝑡∈[0,1] (2.4) for some 𝜌>0. Suppose that the sequence {(𝜇_𝑚,𝑦_𝑚)} satisfies −(𝑘𝑦^′_𝑚)^′+𝑞(𝑡)𝑦_𝑚=𝜇_𝑚𝑝_𝑚(𝑡)𝑦_𝑚,𝑦^′_𝑚(0)=𝑦^′_𝑚(1)=0 (2.5) with either (𝑦_𝑚|_𝐼)(𝑡)>0∀𝑚sufficientlylarge (2.6) or (𝑦_𝑚|_𝐼)(𝑡)<0∀𝑚sufficientlylarge, (2.7) where 𝐼∶=[𝛼,𝛽] with 𝛼<𝛽 being a given closed subinterval of (0,1). Then |𝜇_𝑚|≤𝑀₀ (2.8) for some positive constant 𝑀₀.

Proof. We only deal with the case that (𝑦_𝑚|_𝐼)(𝑡)>0 for all 𝑚 sufficiently large. The other case can be treated by the similar way. We may assume that (𝑦_𝑚|_𝐼)(𝑡)>0 for all 𝑚∈ℕ.
We divide the proof into three cases.
Case 1. Let (𝛼_𝑚,𝛽_𝑚) be a subinterval of [0,1] satisfying(i)𝐼⊂(𝛼_𝑚,𝛽_𝑚);(ii)𝑦_𝑚(𝛼_𝑚)=𝑦_𝑚(𝛽_𝑚)=0;(iii)𝑦_𝑚(𝑡)>0 for all 𝑡∈(𝛼_𝑚,𝛽_𝑚).
Let 𝜓_𝑚(𝑡) and 𝜑_𝑚(𝑡) be the unique solution of the problems:
−(𝑘𝑦^′)^′+𝑞(𝑡)𝑦=0,𝑡∈(𝛼_𝑚,𝛽_𝑚),𝑦(𝛼_𝑚)=0,𝑦^′(𝛼_𝑚)=1,−(𝑘𝑦^′)^′+𝑞(𝑡)𝑦=0,𝑡∈(𝛼_𝑚,𝛽_𝑚),𝑦(𝛽_𝑚)=0,𝑦^′(𝛽_𝑚)=−1, (2.9) respectively. Then it is easy to check 𝜓_𝑚(⋅) is nondecreasing on (𝛼_𝑚,𝛽_𝑚), 𝜑_𝑚(⋅) is nonincreasing on (𝛼_𝑚,𝛽_𝑚), and that Green's function 𝐺_𝑚(𝑡,𝑠) of −(𝑘𝑦^′)^′+𝑞(𝑡)𝑦=0,𝑡∈(𝛼_𝑚,𝛽_𝑚),𝑦(𝛼_𝑚)=𝑦(𝛽_𝑚)=0 (2.10) is explicitly given by 𝐺_𝑚1(𝑡,𝑠)=𝜑_𝑚(𝛼_𝑚)𝜓_𝑚(𝑡)𝜑_𝑚(𝑠),𝛼_𝑚≤𝑡≤𝑠≤𝛽_𝑚,𝜑_𝑚(𝑡)𝜓_𝑚(𝑠),𝛼_𝑚≤𝑠≤𝑡≤𝛽_𝑚. (2.11) Let Ψ(𝑡) and Φ(𝑡) be the unique solution of the problems: −(𝑘𝑦^′)^′+𝑞(𝑡)𝑦=0,𝑡∈(0,1),𝑦(0)=0,𝑦^′(0)=1,−(𝑘𝑦^′)^′+𝑞(𝑡)𝑦=0,𝑡∈(0,1),𝑦(1)=0,𝑦^′(1)=−1, (2.12) respectively. Then it is easy to check that Ψ(⋅) is nondecreasing on (0,1) and Φ(⋅) is nonincreasing on (0,1), and Φ(0)≥𝜑_𝑚(𝛼_𝑚),Ψ(1)≥𝜓_𝑚(𝛽_𝑚). (2.13) Let 𝜓_𝐼(𝑡) and 𝜑_𝐼(𝑡) be the unique solution of the problems −(𝑘𝑦^′)^′+𝑞(𝑡)𝑦=0,𝑡∈(𝛼,𝛽),𝑦(𝛼)=0,𝑦^′(𝛼)=1,−(𝑘𝑦^′)^′+𝑞(𝑡)𝑦=0,𝑡∈(𝛼,𝛽),𝑦(𝛽)=0,𝑦^′(𝛽)=−1, (2.14) respectively. Then, for (𝑡,𝑠)∈[𝛼+(𝛽−𝛼)/4,𝛽−(𝛽−𝛼)/4]×[𝛼+(𝛽−𝛼)/4,𝛽−(𝛽−𝛼)/4],𝐺_𝑚1(𝑡,𝑠)≥𝜓Φ(0)^𝐼(𝛼+𝛽−𝛼4)𝜑_𝐼(𝛽−𝛽−𝛼4). (2.15)
Since 𝐺_𝑚(𝑡,𝑠)𝐺_𝑚≥𝜓(𝑠,𝑠)^𝑚(𝑡)𝜓_𝑚(𝑠),𝛼_𝑚≤𝑡≤𝑠≤𝛽_𝑚,𝜑_𝑚(𝑡)𝜑_𝑚(𝑠),𝛼_𝑚≤𝑠≤𝑡≤𝛽_𝑚,≥𝜓_𝑚(𝑡)Ψ(1),𝛼_𝑚≤𝑡≤𝑠≤𝛽_𝑚,𝜑_𝑚(𝑡)Φ(0),𝛼_𝑚≤𝑠≤𝑡≤𝛽_𝑚,𝜓≥min{^𝑚(𝑡),𝜑Ψ(1)^𝑚(𝑡)Φ(0)}=∶𝛿_𝑚(𝑡), (2.16) it follows that for 𝑡∈[𝛼+(𝛽−𝛼)/4,𝛽−(𝛽−𝛼)/4], 𝑦_𝑚(𝑡)=𝜇_𝑚^𝛽𝑚_𝛼𝑚𝐺_𝑚(𝑡,𝑠)𝑝_𝑚(𝑠)𝑦_𝑚(𝑠)𝑑𝑠≥𝛿_𝑚(𝑡)𝜇_𝑚^𝛽𝑚_𝛼𝑚𝐺_𝑚(𝑠,𝑠)𝑝_𝑚(𝑠)𝑦_𝑚(𝑠)𝑑𝑠≥𝛿_𝑚(𝑡)‖(𝑦_𝑚|_{[𝛼𝑚,𝛽𝑚]})‖_∞≥𝛿_𝑚(𝑡)‖(𝑦_𝑚|_{[𝛼+(𝛽−𝛼)/4,𝛽−(𝛽−𝛼)/4]})‖_∞≥𝛿_𝐼(𝑡)‖(𝑦_𝑚|_{[𝛼+(𝛽−𝛼)/4,𝛽−(𝛽−𝛼)/4]})‖_∞, (2.17) where 𝛿_𝐼𝜓(𝑡)∶=min{^𝐼(𝑡),𝜑Ψ(1)^𝐼(𝑡)Φ(0)}. (2.18)
Set 𝛿₀∶=min{𝛿_𝐼(𝑡)∣𝑡∈[𝛼+𝛽−𝛼4,𝛽−𝛽−𝛼4]}. (2.19) Then min_{𝑡∈[𝛼+(𝛽−𝛼)/4,𝛽−(𝛽−𝛼)/4]}𝑦_𝑚(𝑡)≥𝛿₀‖(𝑦_𝑚|_{[𝛼+(𝛽−𝛼)/4,𝛽−(𝛽−𝛼)/4]})‖_∞. (2.20) By (2.5), we have that 𝑦_𝑚(𝑡)=𝜇_𝑚^𝛽𝑚_𝛼𝑚𝐺_𝑚(𝑡,𝑠)𝑝_𝑚(𝑠)𝑦_𝑚(𝑠)𝑑𝑠, (2.21) which together with (2.15) and (2.20) imply that for 𝑡∈[𝛼+(𝛽−𝛼)/4,𝛽−(𝛽−𝛼)/4], 𝑦_𝑚(𝑡)≥𝜇_𝑚𝐼𝐺_𝑚(𝑡,𝑠)𝜌𝑦_𝑚(𝑠)𝑑𝑠≥𝜇_𝑚^{𝛽−(𝛽−𝛼)/4}_{𝛼+(𝛽−𝛼)/4}𝐺_𝑚(𝑡,𝑠)𝜌𝑦_𝑚(𝑠)𝑑𝑠≥𝛿₀𝜇_𝑚^{𝛽−(𝛽−𝛼)/4}_{𝛼+(𝛽−𝛼)/4}𝐺_𝑚(𝑡,𝑠)𝜌𝑑𝑠⋅‖(𝑦_𝑚|_{[𝛼+(𝛽−𝛼)/4,𝛽−(𝛽−𝛼)/4]})‖_∞≥𝛿₀𝜇_𝑚𝜓Φ(0)^𝐼(𝛼+𝛽−𝛼4)𝜑_𝐼(𝛽−𝛽−𝛼4)𝜌^{𝛽−(𝛽−𝛼)/4}_{𝛼+(𝛽−𝛼)/4}𝑑𝑠⋅‖(𝑦_𝑚|_{[𝛼+(𝛽−𝛼)/4,𝛽−(𝛽−𝛼)/4]})‖_∞. (2.22) Therefore |𝜇_𝑚𝛿|≤(⁰𝜌𝜓Φ(0)^𝐼(𝛼+𝛽−𝛼4)𝜑_𝐼(𝛽−𝛽−𝛼4)⋅𝛽−𝛼2)⁻¹. (2.23)Case 2. Let (0,𝛽_𝑚) be a subinterval of [0,1] satisfying(i)𝐼⊂(0,𝛽_𝑚);(ii)𝑦^′_𝑚(0)=0,𝑦_𝑚(𝛽_𝑚)=0;(iii)𝑦_𝑚(𝑡)>0 for all 𝑡∈(0,𝛽_𝑚).
Let 𝜓_𝑚(𝑡) and 𝜑_𝑚(𝑡) be the unique solution of the problems:
−(𝑘𝑦^′)^′+𝑞(𝑡)𝑦=0,𝑡∈(0,𝛽_𝑚𝑦),_′(0)=0,𝑦(𝛽_𝑚)=1,−(𝑘𝑦^′)^′+𝑞(𝑡)𝑦=0,𝑡∈(0,𝛽_𝑚),𝑦(𝛽_𝑚)=0,𝑦^′(𝛽_𝑚)=−1, (2.24) respectively. Then it is easy to check that 𝜓_𝑚(⋅) is nondecreasing on (0,𝛽_𝑚), 𝜑_𝑚(⋅) is nonincreasing on (0,𝛽_𝑚), and Green's function 𝐺^∗(𝑡,𝑠) of −(𝑘𝑦^′)^′+𝑞(𝑡)𝑦=0,𝑡∈(0,𝛽_𝑚𝑦),_′(0)=𝑦(𝛽_𝑚)=0 (2.25) is explicitly given by 𝐺^∗1(𝑡,𝑠)=𝜑_𝑚(0)𝜓_𝑚(𝑡)𝜑_𝑚(𝑠),0≤𝑡≤𝑠≤𝛽_𝑚,𝜑_𝑚(𝑡)𝜓_𝑚(𝑠),0≤𝑠≤𝑡≤𝛽_𝑚. (2.26) By the similar method to prove Case 1, we may get the desired results.Case 3. Let (𝛼_𝑚,1) be a subinterval of [0,1] satisfying(i)𝐼⊂(𝛼_𝑚,1);(ii)𝑦_𝑚(𝛼_𝑚)=0,𝑦^′_𝑚(1)=0;(iii)𝑦_𝑚(𝑡)>0 for all 𝑡∈(𝛼_𝑚,1).
Using the same method to prove Case 2, with obvious changes, we may show that (2.8) is true.Case 4. Let (𝛼_𝑚,𝛽_𝑚)=(0,1). We may assume that 𝑦_𝑚(𝑡)>0 for all (0,1).
Let 𝜓(𝑡) and 𝜑(𝑡) be the unique solution of the problems −(𝑘𝑦^′)^′+𝑞(𝑡)𝑦=0,𝑡∈(0,1),𝑦(0)=0,𝑦^′(0)=1,−(𝑘𝑦^′)^′+𝑞(𝑡)𝑦=0,𝑡∈(0,1),𝑦(1)=0,𝑦^′(1)=−1, (2.27) respectively. Then, it is easy to verify that 𝜓 is strictly increasing on [0,1] and 𝜑 is strictly decreasing on [0,1]. Using the same method to deal with Case 1, we may get the desired results.

3. Proof of the Results

Recall the proof of Theorem A.

By [6, Remark 1], the hypotheses (H1)–(H3) imply that

𝒮∩((−∞,0]×𝐶¹([0,1]))=(−∞,0]×{𝟎}. (3.1)

Actually, such continua can be obtained as upper limits in the sense of Kuratowski of sequences of solution continua from associated continuous problems. To this end one sets

𝑑_𝑓∶=min{inf𝑓₊,inf|𝑓₋|} (3.2) and selects an approximation sequence {𝑓_𝑙}∈𝐶([0,1]×ℝ,ℝ)^ℕ of 𝐹 satisfying

(A1)𝑓_𝑙(𝑥,𝑦)=𝑙𝑦 for 𝑥∈[0,1] and 𝑦∈[−𝑑_𝑓/2𝑙,𝑑_𝑓/2𝑙];(A2)𝑓_𝑙(𝑥,𝑦)×sgn(𝑦)≥𝑑_𝑓/2 for 𝑥∈[0,1] and |𝑦|≥𝑑_𝑓/2𝑙; 𝑓_𝑙≤𝑓₊ on [0,1]×[𝑑_𝑓/2𝑙,𝑑_𝑓/𝑙]; 𝑓_𝑙≥𝑓₋ on [0,1]×[−𝑑_𝑓/𝑙,−𝑑_𝑓/2𝑙];(A3)𝑓_𝑙(𝑥,𝑦)=𝑓₊(𝑥,𝑦) for 𝑥∈[0,1] and 𝑦≥𝑑_𝑓/𝑙; 𝑓_𝑙(𝑥,𝑦)=𝑓₋(𝑥,𝑦) for 𝑥∈[0,1] and 𝑦≤−𝑑_𝑓/𝑙;(A4){𝑓_𝑙(𝑥,𝑦)}_𝑙∈ℕ is nondecreasing in 𝑙 for (𝑥,𝑦)∈[0,1]×(0,∞); {𝑓_𝑙(𝑥,𝑦)}_𝑙∈ℕ is nonincreasing in 𝑙 for (𝑥,𝑦)∈[0,1]×(−∞,0).

Remark 3.1. Let 𝜉(𝑥,𝑢)∶=𝑔(𝑥,𝑢)−𝑔₁(𝑥)𝑢. (3.3) We may show that there exists a positive constant 𝛾, independent of 𝑙, such that for each 𝑙∈ℕ, 𝑓_𝑙(𝑥,𝑢)𝑢−𝜉(𝑥,𝑢)𝛾𝑢≥𝜌₀,∀𝛾≥𝛾 (3.4) for some constant 𝜌₀>0.
In fact, it is easy to see from the definition of 𝑓_𝑙 that 𝑓_𝑙(𝑥,𝑢)𝑢≥𝜌₁,𝑢≠0 (3.5) for some positive constant 𝜌₁, independent of 𝑙.
Applying (H2) and (H2^′), it concludes that 0≤|𝜉(𝑥,𝑢)𝑢|≤𝜌₂ (3.6) for some positive constant 𝜌₂. Therefore, if we take 𝛾∶=2𝜌₂𝜌₁,𝜌₀=𝜌₁2, (3.7) then (3.4) holds.

It is easy to see thanks to (H2) and (A1) that

−(𝑘𝑣^′)^′(𝑥)+𝑔(𝑥,𝑣(𝑥))=𝜇𝑓_𝑙𝑣(𝑥,𝑣(𝑥)),𝑥∈[0,1],_′(0)=0,𝑣^′(1)=0( (3.8_𝑙)) falls into the scope of the Sturm-Liouville version of the celebrated Rabinowitz bifurcation theorem (cf. [10] for a more general, but somewhat different setting).

Indeed, denote the strictly increasing sequence of simple eigenvalues of

−(𝑘𝜓^′)^′(𝑥)+𝑔₁𝜓(𝑥)𝜓(𝑥)=𝜆𝜓(𝑥),𝑥∈[0,1],_′(0)=0,𝜓^′(1)=0, (3.9) by {𝜆_𝑛}_𝑛∈ℤ+ and set

𝜇_𝑛,𝑙𝜆∶=^𝑛𝑙. (3.10) Then (𝜇_𝑛,𝑙,𝟎) is a bifurcation point of the solution set of (3.8_𝑙) for every 𝑛∈ℤ₊, and for each (𝑛,𝑙)∈ℤ₊×ℕ, there exist two unbounded closed connected subsets 𝐶^±_𝑛,𝑙 of the solution set of (3.8_𝑙) with the following.

(a)𝐶⁺_𝑛,𝑙∩𝐶⁻_𝑛,𝑙={(𝜇_𝑛,𝑙,𝟎)}. Moreover, (𝜇_𝑛,𝑙,𝟎) is the only bifurcation point contained in 𝐶^±_𝑛,𝑙.(b)If (𝜇,𝜗)∈𝐶⁺_𝑛,𝑙 and 𝜗≢0, then 𝜗 possesses exactly 𝑛 simple zeroes (and no multiple zeroes) in (0,1) and is positive on (0,𝛿) for some 𝛿>0.(c)If (𝜇,𝜗)∈𝐶⁻_𝑛,𝑙 and 𝜗≢0, then 𝜗 possesses exactly 𝑛 simple zeroes (and no multiple zeroes) in (0,1) and is negative on (0,𝛿) for some 𝛿>0.

Combining the above with the fact

lim_𝑙→∞(𝜇_𝑛,𝑙,𝟎)=(0,𝟎) (3.11) and utilizing Lemma 2.4, it concludes that there exists an unbounded component 𝐶^𝜈_𝑛 with

(0,𝟎)∈𝐶^𝜈_𝑛,𝐶^𝜈_𝑛⊆limsup_𝑙→∞𝐶^𝜈_𝑛,𝑙,𝜈∈{+,−}. (3.12)

As an immediate consequence of [6, Lemma 4-6], we have the following

Lemma 3.2. If (𝜇,𝑢)∈𝐶^±_𝑛, then (𝜇,𝑢) is a solution of (1.1) and 𝑢∈𝑊^2,∞([0,1]). Moreover, if (𝜇,𝑢)∈𝐶⁺_𝑛 with 𝑢≢0, 𝑢 has exactly 𝑛 simple zeroes in [0,1], and 𝑢 is positive on an interval (0,𝑥) for some 𝑥∈(0,1]; if (𝜇,𝑢)∈𝐶⁻_𝑛 with 𝑢≢0, 𝑢 has exactly 𝑛 simple zeroes in [0,1], and 𝑢 is negative on an interval (0,𝑥) for some 𝑥∈(0,1].

Lemma 3.3. Let (H1)–(H4), (H2^′) and (A1)–(A4) be fulfilled. Then for each (𝑛,𝑙)∈ℤ₊×ℕ, the connected component 𝐶^±_𝑛,𝑙 joins (𝜇_𝑛,𝑙,𝟎) with (𝜂_𝑛,∞).

Proof. Assume that {(𝑟_𝑘,𝑦_𝑘)}⊂𝐶⁺_𝑛,𝑙 for some fixed (𝑛,𝑙)∈ℤ₊×ℕ with |𝑟_𝑘|+‖𝑦_𝑘‖_𝐶1→∞. (3.13) The case {(𝑟_𝑘,𝑦_𝑘)}⊂𝐶⁻_𝑛,𝑙 can be treated by the same way.
We divide the proof into two steps.
Step 1. We show that if there exists a constant number 𝑀>0 such that 𝑟_𝑘∈(0,𝑀], (3.14) then 𝐶⁺_𝑛,𝑙 joins (𝜇_𝑛,𝑙,𝟎) with (𝜂_𝑛,∞). In this case it follows that ‖𝑦_𝑘‖_𝐶1→∞. (3.15) Define 𝜁_𝑙(𝑟,𝑥,𝑢)∶=𝑟[𝑓_𝑙(𝑥,𝑢)−𝑏(𝑥)𝑢]−[𝑔(𝑥,𝑢)−𝑔₂(𝑥)𝑢]. (3.16) Then {(𝑟_𝑘,𝑦_𝑘)} satisfies the problem: −(𝑘𝑦^′_𝑘)^′(𝑥)+𝑔₂(𝑥)𝑦_𝑘(𝑥)=𝑟_𝑘𝑏(𝑥)𝑦_𝑘(𝑥)+𝜁_𝑙(𝑟_𝑘,𝑥,𝑦_𝑘𝑦(𝑥)),𝑥∈[0,1],_′𝑘(0)=0,𝑦^′_𝑘(1)=0.( (3.17_𝑙)) Set 𝜁̃_𝑙(𝑢)=max_{0≤|𝑠|≤𝑢,𝑥∈[0,1],𝑟∈[0,𝑀]}|𝜁_𝑙(𝑟,𝑥,𝑠)| (3.18) then 𝜁̃ is nondecreasing, and (H4) and (H2^′) yields lim_𝑢→∞𝜁̃(𝑢)𝑢=0. (3.19)
Now, we divide (3.17_𝑙) by ||𝑦_𝑘||_𝐶¹ and set 𝑦_𝑘=(𝑦_𝑘/||𝑦_𝑘||_𝐶¹). Since 𝑦_𝑘 is bounded in 𝐶²[0,1], after taking a subsequence if necessary, we have that 𝑦_𝑘→𝑦 for some 𝑦∈𝐶¹[0,1] with ||𝑦||_𝐶¹=1. Moreover, from the definition of 𝑓_𝑙 and (3.19) and the fact that 𝜁̃ is nondecreasing, we have that lim_𝑘→∞|𝜁_𝑙(𝑟_𝑘,𝑥,𝑦_𝑘(𝑥))|‖𝑦_𝑘‖_𝐶1=0 (3.20) since |𝜁_𝑙(𝑟_𝑘,𝑥,𝑦_𝑘(𝑥))|‖𝑦_𝑘‖_𝐶1≤𝜁̃(|𝑦_𝑘(𝑥)|)‖𝑦_𝑘‖_𝐶1≤𝜁̃(‖𝑦_𝑘‖_∞)‖𝑦_𝑘‖_𝐶1≤𝜁̃(‖𝑦_𝑘‖_𝐶1)‖𝑦_𝑘‖_𝐶1. (3.21) By standard limit procedure, we get −(𝑘𝑦^′)^′(𝑥)+𝑔₂(𝑥)𝑦(𝑥)=𝑟𝑏(𝑥)𝑦(𝑥),𝑥∈[0,1],𝑦^′(0)=0,𝑦^′(1)=0, (3.22) where 𝑟∶=lim_𝑘→∞𝑟_𝑘, again choosing a subsequence and relabeling if necessary. Moreover, the fact that 𝑦_𝑘, 𝑘∈ℤ₊, has exactly 𝑛 simple zeroes in [0,1] implies that 𝑦 has exactly 𝑛 simple zeroes in [0,1], too. Therefore 𝑟=𝜂_𝑛.Step 2. We show that there exists a constant 𝑀 such that 𝑟_𝑘∈(0,𝑀], for all 𝑛. Suppose there is no such 𝑀, choosing a subsequence and relabeling if necessary, it follows that lim_𝑘→∞𝑟_𝑘=∞. (3.23)
Let 𝜏(1,𝑘)<⋯<𝜏(𝑛,𝑘) (3.24) denote the zeroes of 𝑦_𝑘, and set 0=𝜏(0,𝑘),𝜏(𝑛+1,𝑘)=1. (3.25) Then, after taking a subsequence if necessary, lim_𝑘→∞𝜏(𝑙,𝑘)∶=𝜏(𝑙,∞),𝑙∈{0,1,…,𝑛+1}. (3.26) We claim that for all 𝑙∈{0,1,…,𝑛}𝜏(𝑙+1,∞)−𝜏(𝑙,∞)=0. (3.27)
Suppose on the contrary that there exists 𝑙₀∈{0,1,…,𝑛} such that 𝜏(𝑙₀,∞)<𝜏(𝑙₀+1,∞). (3.28) Define a function 𝑝∶[0,1]→ℝ by 𝑝_𝑙𝑓(𝑥)∶=^𝑙(𝑥,𝑦_𝑘(𝑥))𝑦_𝑘−(𝑥)𝜉(𝑥,𝑦_𝑘(𝑥))𝑟_𝑘𝑦_𝑘(𝑥),𝑥∈[0,1],𝑦_𝑘(𝑥)≠0,𝑙,𝑦_𝑘(𝑥)=0. (3.29) Then by Remark 3.1, there exists 𝜌₀, such that 𝑝_𝑙(𝑥)≥𝜌₀,𝑥∈[0,1]. (3.30) Now we choose a closed interval 𝐼⊂(𝜏(𝑙₀,∞),𝜏(𝑙₀+1,∞)) with positive length, then we know from Lemma 2.7 that 𝑦_𝑘 (after taking a subsequence if necessary) must change sign on 𝐼. However, this contradicts the fact that for all 𝑘 sufficiently large, we have 𝐼⊂(𝜏(𝑙₀,𝑘),𝜏(𝑙₀+1,𝑘)) and (−1)^𝑙0𝜈𝑦_𝑘(𝑥)>0,𝑥∈(𝜏(𝑙₀,𝑘),𝜏(𝑙₀+1,𝑘)). (3.31) Therefore, (3.27) holds.
On the other hand, it follows 1=𝜏(𝑛+1,𝑘)−𝜏(0,𝑘)=^𝑛_𝑙=0(𝜏(𝑙+1,𝑘)−𝜏(𝑙,𝑘)) (3.32) that 1=^𝑛_𝑙=0(𝜏(𝑙+1,∞)−𝜏(𝑙,∞)) (3.33) which contradicts (3.27).
Therefore |𝑟_𝑘|≤𝑀 (3.34) for some constant number 𝑀>0, independent of 𝑘∈ℕ.

Now we are in the position to prove Theorem 1.1.

Proof of Theorem 1.1. We only prove that 𝐶⁺_𝑛 has the desired property, the case of 𝐶⁻_𝑛 can be treated by the same way.
Assume that {(𝜇_𝑘,𝑧_𝑘)}⊂𝐶⁺_𝑛 is a sequence with |𝜇_𝑘|+‖𝑧_𝑘‖_𝐶¹→∞. (3.35) We claim that lim_𝑘→∞(𝜇_𝑘,𝑧_𝑘)=(𝜂_𝑛,∞). (3.36)
Assume on the contrary that (3.36) is not true. We divide the proof into two cases.
Case 1. lim_𝑘→∞𝜇_𝑘≠𝜂_𝑛.In this case, we may take a subsequence of {𝜇_𝑘}, denote it by {𝜇_𝑘} again, with the property that there exists 𝜀₀>0, such that for each 𝑘∈ℕ, |𝜇_𝑘−𝜂_𝑛|≥𝜖₀. (3.37)
Since {(𝜇_𝑘,𝑧_𝑘)}⊂𝐶⁺_𝑛, it follows that for each 𝑘∈ℤ₊, there exists a sequence {(𝛾_𝑘𝑗,𝑧_𝑘𝑗)}⊂𝐶⁺_{𝑛,𝑘𝑗}, such that lim_𝑗→∞𝛾_𝑘𝑗=𝜇_𝑘,lim_𝑗→∞𝑧_𝑘𝑗=𝑧_𝑘. (3.38)
Now let us consider the sequence {(𝛾_𝑘𝑘,𝑧_𝑘𝑘)}. Obviously, we have that (𝛾_𝑘𝑘,𝑧_𝑘𝑘)∈𝐶⁺_{𝑛,𝑘𝑘},|𝛾_𝑘𝑘|+‖𝑧_𝑘𝑘‖_𝐶1→∞. (3.39) ?Equation (3.39) implies that −(𝑘𝑧^′_𝑘𝑘)^′(𝑥)+𝑔₂(𝑥)𝑧_𝑘𝑘(𝑥)=𝛾_𝑘𝑘𝑏(𝑥)𝑧_𝑘𝑘(𝑥)+𝜁_𝑘𝑘(𝛾_𝑘𝑘,𝑥,𝑧_𝑘𝑘𝑧(𝑥)),𝑥∈[0,1],_′𝑘𝑘(0)=0,𝑧^′_𝑘𝑘(1)=0, (3.40) Noticing that 𝜌₀ in (3.30) is independent of 𝑙 and using Remark 3.1 and the method to prove Lemma 3.3 and with obvious changes, we may show that {𝛾_𝑘𝑘} is bounded, and subsequently lim_𝑘→∞𝛾_𝑘𝑘=𝜂_𝑛. (3.41) However, this contradicts (3.37).
Case 2. lim_𝑘→∞||𝑧_𝑘||_𝐶¹≠∞. In this case, after taking a subsequence of {𝑧_𝑘} and relabeling if necessary, we may assume that ‖𝑧_𝑘‖_𝐶¹≤𝑀₀ (3.42) for some constant 𝑀₀>0. Equation (3.35) together with (3.42) implies lim_𝑘→∞𝜇_𝑘=+∞. (3.43) Using the same notations as those in Case 1, we have from (3.43) that lim_𝑘→∞𝛾_𝑘𝑘=+∞. (3.44) Combining this with (3.40) and using Remark 3.1 and the similar method to prove Step 2 of Lemma 3.3 and noticing that 𝜌₀ in (3.30) is independent of 𝑙, it concludes that {𝛾_𝑘𝑘} is bounded. This is a contradiction.

Remark 3.4. It is easy to see from Theorem 1.1 and its proof that the “jumping” of 𝐹 at 𝑢=0: 𝑓₊(𝑥,0)−𝑓₋(𝑥,0)(=∶Δ(𝑥)) does not affect the asymptotic behavior of 𝐶^±_𝑛 at infinity. In other words, for any nonnegative function Δ(𝑥), the asymptotic behavior of 𝐶^±_𝑛 at infinity is the same.

Acknowledgement

This work supported by the NSFC (no. 10671158), the NSF of Gansu Province (no. ZS051-A25-016), NWNU-KJCXGC-03-17, the Spring-sun program (no. Z2004-1-62033), SRFDP (no. 20060736001), and the SRF for ROCS, SEM (2006[311]).