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ISRN Mathematical Analysis
VolumeΒ 2012Β (2012), Article IDΒ 354513, 13 pages
http://dx.doi.org/10.5402/2012/354513
Research Article

Bifurcation of Sign-Changing Solutions for π‘š-Point Boundary Value Problems

Yulian An1,2Β and Maoan Han1

1Department of Mathematics, Shanghai Normal University, Shanghai 200235, China
2Department of Mathematics, Shanghai Institute of Technology, Shanghai 201418, China

Received 13 February 2012; Accepted 21 March 2012

Academic Editors: R.Β Avery and A. I.Β Hai

Copyright Β© 2012 Yulian An and Maoan Han. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

With the help of bifurcation techniques, some multiplicity results and global structure for sign-changing solutions of some π‘š-point boundary value problems are obtained when the nonlinear term is sublinear at 0.

1. Introduction and Main Results

In this paper, we consider the π‘š-point boundary value problems:π‘’ξ…žξ…ž+πœ†π‘“(𝑒)=0,π‘‘βˆˆ(0,1),(1.1)𝑒(0)=0,𝑒(1)=π‘šβˆ’2𝑖=1π›Όπ‘–π‘’ξ€·πœ‚π‘–ξ€Έ,(1.2) where integer π‘šβ‰₯3,πœ‚π‘–βˆˆ(0,1), πœ† is a positive parameter, and π‘“βˆˆπΆ1(ℝ,ℝ). We study the multiplicity and global structure of sign-changing solutions of (1.1) and (1.2) under the assumptions:(A1)𝛼𝑖>0 for 𝑖=1,…,π‘šβˆ’2 with βˆ‘0<π‘šβˆ’2𝑖=1𝛼𝑖<1;(A2)π‘“βˆˆπΆ1(ℝ,ℝ) satisfies 𝑠𝑓(𝑠)>0 for 𝑠≠0;(A3)𝑓0∢=lim|𝑠|β†’0𝑓(𝑠)/𝑠=0;(A4)π‘“βˆžβˆΆ=lim|𝑠|β†’βˆžπ‘“(𝑠)/𝑠=0.

Multipoint boundary value problems for ordinary differential equations arise in different areas of applied mathematics and physics. The existence of solutions of second-order multipoint boundary value problems has been extensively studied in the literature, see [1–4] and the references therein. Particularly, many authors have studied the existence of sign-changing solutions for various nonlinear boundary value problems, see for example [5–10].

Recently, the global structure of solutions of nonlinear multipoint boundary value problems has also been investigated by several authors using bifurcation methods, see [7–10]. These papers dealt with the case 𝑓0∈(0,∞), and relatively little is known about the global structure of solutions when 𝑓 satisfying 𝑓0=0. The main reason is that the global bifurcation techniques cannot be used directly in this case. Very recently, [11] investigated the global structure of positive solutions for a class of boundary value problems with 𝑓0=0. However, to our knowledge there is no paper studying the global structure of sign-changing solutions for nonlinear multipoint boundary value problems under the assumption 𝑓0=0. The purpose of present paper is to fill this gap.

In this paper, we consider the global structure of nodal solutions of (1.1) and (1.2), a kind of sign-changing having a given number of zeros, when 𝑓0=π‘“βˆž=0. We find that the discussion is more complicated, when sign-changing solutions are concerned. Eigenvalue theory and Sturm’s comparison theorem play important roles in our discussion.

Now, we introduce some notations as follows.

Let π‘Œ=𝐢[0,1] with the norm β€–π‘’β€–βˆž=maxπ‘‘βˆˆ[0,1]|𝑒(𝑑)|. Let 𝑋={π‘’βˆˆπΆ1[0,1]βˆ£π‘’(0)=0,𝑒(1)=βˆ‘π‘šβˆ’2𝑖=1𝛼𝑖𝑒(πœ‚π‘–)}, and 𝐸={π‘’βˆˆπΆ2βˆ‘[0,1]βˆ£π‘’(0)=0,𝑒(1)=π‘šβˆ’2𝑖=1𝛼𝑖𝑒(πœ‚π‘–)} equipped with the norm:‖𝑒‖𝑋=maxβ€–π‘’β€–βˆž,β€–β€–π‘’ξ…žβ€–β€–βˆžξ€Ύξ€½,‖𝑒‖=maxβ€–π‘’β€–βˆž,β€–β€–π‘’ξ…žβ€–β€–βˆž,β€–β€–π‘’ξ…žξ…žβ€–β€–βˆžξ€Ύ.(1.3)

For any 𝐢1 function 𝑒, if 𝑒(π‘₯0)=0 and 𝑒′(π‘₯0)β‰ 0, then π‘₯0 is called a simple zero of 𝑒. For any integer π‘˜β‰₯1 and any 𝜈∈{+,βˆ’}, let π‘†πœˆπ‘˜,π‘‡πœˆπ‘˜βŠ‚πΆ2[0,1] be sets consisting of functions π‘’βˆˆπΆ2[0,1] satisfying the following conditions:π‘†πœˆπ‘˜: (i) 𝑒(0)=0,πœˆπ‘’β€²(0)>0; (ii) 𝑒 has only simple zeros in [0,1] and has exactly π‘˜βˆ’1 zeros in (0,1).π‘‡πœˆπ‘˜: (i) 𝑒(0)=0,πœˆπ‘’β€²(0)>0 and 𝑒′(1)β‰ 0;(ii) 𝑒′ has only simple zeros in (0,1) and has exactly π‘˜ zeros in (0,1);(iii) 𝑒 has a zero strictly between each two consecutive zeros of 𝑒′.

Remark 1.1. If π‘’βˆˆπ‘‡πœˆπ‘˜, then π‘’βˆˆπ‘†πœˆπ‘˜ or π‘’βˆˆπ‘†πœˆπ‘˜+1. The sets π‘‡πœˆπ‘˜(π‘˜=1,2,…) are open in 𝐸 and disjoint [8].

Lemma 1.2 (See [8]). Let (A1) and (A2) hold. If (πœ‡,𝑒) is a nontrivial solution of (1.1) and (1.2). Then, π‘’βˆˆπ‘‡πœˆπ‘˜ for some π‘˜,𝜈.
Let 𝕏=ℝ×𝑋 with the product topology. As in [12], we add the point {(πœ†,∞)βˆ£πœ†βˆˆβ„} to the space 𝕏. Denote πœƒβˆˆπ‘‹,πœƒ(𝑑)≑0,π‘‘βˆˆ[0,1].

The main results of this paper are as follows.

Theorem 1.3. Let (A1)–(A4) hold. Then, there exists a component π’žπœˆπ‘˜βŠ‚(0,∞)Γ—π‘‡πœˆπ‘˜ of solutions of (1.1) and (1.2), which joins (∞,πœƒ) to (∞,∞)(see Figure 1(a)) such that Projβ„π’žπœˆπ‘˜=[πœŒπœˆπ‘˜,∞) for some 𝜌𝜈k>0. Here, π’žπœˆπ‘˜ joins (∞,πœƒ) to (∞,∞) meaning that: lim(πœ†,𝑒)βˆˆπ’žπœˆπ‘˜,‖𝑒‖≀1,πœ†β†’+βˆžβ€–π‘’β€–=0,lim(πœ†,𝑒)βˆˆπ’žπœˆπ‘˜,‖𝑒‖>1,πœ†β†’+βˆžβ€–π‘’β€–=+∞.(1.4)

fig1
Figure 1

Corollary 1.4. Let (A1)–(A4) hold. Then, there exists πœ†πœˆπ‘˜β‰₯πœŒπœˆπ‘˜>0 such that (1.1) and (1.2) have at least two solutions in π‘‡πœˆπ‘˜ for πœ†βˆˆ(πœ†πœˆπ‘˜,∞).

Remark 1.5. Theorem 1.3 extends the result stated in [11]. Meanwhile, Theorem 1.3 and Corollary 1.4 do not only obtain the multiplicity of nodal solutions of (1.1) and (1.2), but also describe the global structure of these solutions.

2. Preliminary Lemmas

The following definition and lemmas about superior limit and component are important to prove Theorem 1.3.

Definition 2.1 (See [13]). Let π‘Š be a Banach space, and {πΆπ‘›βˆ£π‘›=1,2,…} be a family of subsets of π‘Š. Then, the superior limit π’Ÿ of {𝐢𝑛} is defined by: π’ŸβˆΆ=limsupπ‘›β†’βˆžπΆπ‘›=𝑛π‘₯βˆˆπ‘Šβˆ£βˆƒπ‘–ξ€ΎβŠ‚β„•,π‘₯π‘›π‘–βˆˆπΆπ‘›π‘–,suchthatπ‘₯𝑛𝑖.⟢π‘₯(2.1)

Lemma 2.2 (See [13]). Each connected subset of metric space π‘Š is contained in a component, and each component of π‘Š is closed.

Lemma 2.3 (See [11]). Let π‘Š be a Banach space and 𝐢𝑛 a family of closed connected subsets of π‘Š. Assume that:
(i) there exist π‘§π‘›βˆˆπΆπ‘›, 𝑛=1,2,…, and π‘§βˆ—βˆˆπ‘Š such that π‘§π‘›β†’π‘§βˆ—;
(ii) π‘Ÿπ‘›=∞, where π‘Ÿπ‘›=sup{β€–π‘₯β€–βˆ£π‘₯βˆˆπΆπ‘›};
(iii) for all𝑅>0,(βˆͺβˆžπ‘›=1𝐢𝑛)βˆ©π΅π‘… is a relative compact set of W, where 𝐡𝑅={π‘₯βˆˆπ‘Šβˆ£β€–π‘₯‖≀𝑅}.(2.2) Then, there exists an unbounded connected component π’ž in π’Ÿ such that π‘§βˆ—βˆˆπ’ž.

Define a linear operator πΏβˆΆπΈβ†’π‘Œ by:Lu∢=βˆ’π‘’ξ…žξ…ž,π‘’βˆˆπΈ.(2.3) We consider the linear eigenvalues problem:𝐿𝑒=πœ†π‘’,π‘’βˆˆπΈ.(2.4) Let πœ†π‘˜ be the π‘˜th eigenvalue of (2.4), and πœ‘π‘˜ an eigenfunction corresponding to πœ†π‘˜. The following lemma or similar result can be found in [7–9].

Lemma 2.4. Let (A1) hold. Then, 0<πœ†1<πœ†2<β‹―<πœ†π‘˜<πœ†π‘˜+1<β‹―,limπ‘˜β†’βˆžπœ†π‘˜=∞.(2.5) For each π‘˜βˆˆπ‘, algebraic multiplicity of πœ†π‘˜ is equal to 1, and the corresponding eigenfunction πœ‘π‘˜βˆˆπ‘‡+π‘˜ and is strictly positive on (0,1).

Define a map π‘‡πœ†βˆΆπ‘Œβ†’πΈ by:π‘‡πœ†ξ€œπ‘’(𝑑)=πœ†10𝐻(𝑑,𝑠)𝑓(𝑒(𝑠))𝑑𝑠,(2.6) whereβˆ‘π»(𝑑,𝑠)=𝐺(𝑑,𝑠)+π‘šβˆ’2𝑖=1π›Όπ‘–πΊξ€·πœ‚π‘–ξ€Έ,π‘ βˆ‘1βˆ’π‘šβˆ’2𝑖=1π›Όπ‘–πœ‚π‘–ξ‚»π‘‘,𝐺(𝑑,𝑠)=(1βˆ’π‘‘)𝑠,0≀𝑠≀𝑑≀1,𝑑(1βˆ’π‘ ),0≀𝑑≀𝑠≀1.(2.7)

It is clear that π‘‡πœ†βˆΆπ‘Œβ†’π‘‹ is completely continuous provided that (A1) and (A2) hold.

Lemma 2.5. Let (A1) and (A2)  hold, and {(πœ‡π‘™,𝑦𝑙)}βŠ‚(0,∞)Γ—π‘‡πœˆπ‘˜ be a sequence of solutions of (1.1) and (1.2). Assume that πœ‡π‘™β‰€πΆ0 for some constant 𝐢0>0, and limπ‘™β†’βˆžβ€–π‘¦π‘™β€–=∞. Then, limπ‘™β†’βˆžβ€–β€–π‘¦π‘™β€–β€–βˆž=∞.(2.8)

Proof. From the relation 𝑦𝑙(𝑑)=πœ‡π‘™βˆ«10𝐻(𝑑,𝑠)𝑓(𝑦𝑙(𝑠))𝑑𝑠,we conclude that π‘¦ξ…žπ‘™(𝑑)=πœ‡π‘™βˆ«10𝐻𝑑(𝑑,𝑠)𝑓(𝑦𝑙(𝑠))𝑑𝑠. Then, β€–β€–π‘¦ξ…žπ‘™β€–β€–βˆžβ‰€πΆ0ξƒ©βˆ‘1+π‘šβˆ’2𝑖=1π›Όπ‘–βˆ‘1βˆ’π‘šβˆ’2𝑖=1π›Όπ‘–πœ‚π‘–ξƒͺξ€œ10||𝑓𝑦𝑙||(𝑠)𝑑𝑠.(2.9) Equations (2.9) and (1.1) imply that {β€–π‘¦ξ…žπ‘™β€–βˆž},{β€–π‘¦π‘™ξ…žξ…žβ€–βˆž} are bounded, whenever {β€–π‘¦π‘™β€–βˆž} is bounded.

3. Proof of the Main Results

We will construct a sequence of functions {𝑓[𝑛]} which is asymptotic linear at 0 and satisfieslimπ‘›β†’βˆžsupπ‘ βˆˆR||𝑓[𝑛]||(𝑠)βˆ’π‘“(𝑠)=0,limπ‘›β†’βˆžξ€·π‘“[𝑛]ξ€Έ0∢=limπ‘›β†’βˆžξƒ©lim|𝑠|β†’0𝑓[𝑛](𝑠)𝑠ξƒͺ=0.(3.1) By means of some corresponding auxiliary equations, we can obtain a sequence of unbounded components {πΆπ‘˜πœˆ[𝑛]} via Rabinowitz’s global bifurcation theorem [14]. Based on the sequence, we can find an unbounded component πΆπœˆπ‘˜ satisfying:πΆπœˆπ‘˜βŠ‚limsupπ‘›β†’βˆžπΆπ‘˜πœˆ[𝑛],(3.2) and joining (∞,πœƒ) with (∞,∞). We do it as follows.

For each π‘›βˆˆβ„•, define 𝑓[𝑛](𝑠)βˆΆβ„β†’β„ by:𝑓[𝑛]⎧βŽͺ⎨βŽͺβŽ©ξ‚€1(𝑠)=𝑓(𝑠),π‘ βˆˆπ‘›ξ‚βˆͺξ‚€1,βˆžβˆ’βˆž,βˆ’π‘›ξ‚,ξ‚€1π‘›π‘“π‘›ξ‚ξ‚ƒβˆ’1𝑠,π‘ βˆˆπ‘›,1𝑛.(3.3) Then, 𝑓[𝑛]β‹‚πΆβˆˆπΆ(ℝ,ℝ)1(ℝ⧡{Β±1/𝑛},ℝ) with𝑠𝑓[𝑛]𝑓(𝑠)>0,βˆ€π‘ β‰ 0,[𝑛]ξ€Έ0ξ‚€1=𝑛𝑓𝑛.(3.4) By (A3), it follows thatlimπ‘›β†’βˆžξ€·π‘“[𝑛]ξ€Έ0=0.(3.5) Now let us consider the auxiliary family of problems:π‘’ξ…žξ…ž+πœ†π‘“[𝑛](𝑒)=0,π‘‘βˆˆ(0,1),𝑒(0)=0,𝑒(1)=π‘šβˆ’2𝑖=1π›Όπ‘–π‘’ξ€·πœ‚π‘–ξ€Έ.(3.6) From Proposition  4.1 in [8], we obtain the following.

Lemma 3.1. Let (A1) and (A2) hold. If (πœ‡,𝑒) is a nontrivial solution of (3.6). Then, π‘’βˆˆπ‘‡πœˆπ‘˜ for some π‘˜,𝜈.

Let 𝑔[𝑛]∈𝐢(ℝ,ℝ) such that:𝑓[𝑛]𝑓(𝑒)=[𝑛]ξ€Έ0𝑒+𝑔[𝑛]ξ‚€1(𝑒)=𝑛𝑓𝑛𝑒+𝑔[𝑛](𝑒).(3.7) Note thatlim|𝑠|β†’0𝑔[𝑛](𝑠)𝑠=0.(3.8) Let us considerξ€·π‘“πΏπ‘’βˆ’πœ†[𝑛]ξ€Έ0𝑒=πœ†π‘”[𝑛](𝑒),(3.9) as a bifurcation problem from the trivial solution π‘’β‰‘πœƒ.

Equation (3.9) can be converted to the equivalent form:ξ€œπ‘’(𝑑)=10ξ€Ίπœ†ξ€·π‘“π»(𝑑,𝑠)[𝑛]ξ€Έ0𝑒(𝑠)+πœ†π‘”[𝑛]ξ€»(𝑒(𝑠))π‘‘π‘ βˆΆ=πœ†πΏβˆ’1𝑓[𝑛]ξ€Έ0𝑒(β‹…)(𝑑)+πœ†πΏβˆ’1𝑔[𝑛]ξ€»(𝑒(β‹…))(𝑑).(3.10) Note that β€–πΏβˆ’1[𝑔[𝑛](𝑒)]β€–=π‘œ(‖𝑒‖) for 𝑒 near πœƒ in 𝑋. Applying Lemma 2.4, the global bifurcation result of Rabinowitz [14] for (3.9) can be stated as follows: for each integer π‘˜β‰₯1, 𝜈∈{+,βˆ’}, there exists a continuum πΆπ‘˜πœˆ[𝑛] of solutions of (3.9) joining (πœ†π‘˜/(𝑓[𝑛])0,πœƒ) to infinity in 𝕏. Moreover, πΆπ‘˜πœˆ[𝑛]⧡(πœ†π‘˜/(𝑓[𝑛])0,πœƒ)βŠ‚(0,∞)Γ—π‘‡πœˆπ‘˜.

For properties of πΆπ‘˜πœˆ[𝑛], we give the following lemmas.

Lemma 3.2. Let (A1)–(A4) hold. Then for each fixed 𝑛, πΆπ‘˜πœˆ[𝑛] joins (πœ†π‘˜/(𝑓[𝑛])0,πœƒ) to (∞,∞) in 𝕏 (see Figure 1(b)).

Proof. We divide the proof into two steps.
Step 1. We show that sup{πœ†βˆ£(πœ†,𝑒)βˆˆπΆπ‘˜πœˆ[𝑛]}=∞. Assume on the contrary that sup{πœ†βˆ£(πœ†,𝑒)βˆˆπΆπ‘˜πœˆ[𝑛]}=∢𝐢0<∞. Let {(πœ‡π‘™,𝑦𝑙)}βŠ‚πΆπ‘˜πœˆ[𝑛] be such that: ||πœ‡π‘™||+β€–β€–π‘¦π‘™β€–β€–β†’βˆž.(3.11) Similar to the argument of Lemma 2.5, we conclude that β€–π‘¦π‘™β€–βˆžβ†’βˆž.
Since (πœ‡π‘™,𝑦𝑙)βˆˆπ’žπ‘˜πœˆ[𝑛], we have π‘¦π‘™ξ…žξ…ž(𝑑)+πœ‡π‘™π‘“[𝑛]𝑦𝑙𝑦(𝑑)=0,π‘‘βˆˆ(0,1),𝑙(0)=0,𝑦𝑙(1)=π‘šβˆ’2𝑖=1π›Όπ‘–π‘¦π‘™ξ€·πœ‚π‘–ξ€Έ.(3.12) Set 𝑣𝑙(𝑑)=𝑦𝑙(𝑑)/β€–π‘¦π‘™β€–βˆž. Then, β€–π‘£π‘™β€–βˆž=1, and π‘£π‘™ξ…žξ…ž(𝑑)+πœ‡π‘™π‘“[𝑛]𝑦𝑙(𝑑)β€–β€–π‘¦π‘™β€–β€–βˆž=0,π‘‘βˆˆ(0,1).(3.13) Using lim|𝑒|β†’0𝑓(𝑒)/𝑒=0, we can show that limπ‘™β†’βˆž||𝑓[𝑛]𝑦𝑙||(𝑑)β€–β€–π‘¦π‘™β€–β€–βˆž=0.(3.14) The proof is similar to that of Theorem  1 in [12], and therefore we omit it. Equations (3.13) and (3.14) imply that β€–π‘£π‘™ξ…žξ…žβ€–βˆžβ‰€π‘€ for some constant 𝑀>0, independent of 𝑙. Hence, {𝑣𝑙} has a convergent subsequence in 𝑋. Without loss of generality, we assume that there exists (πœ‡βˆ—,π‘£βˆ—)∈[0,𝐢0]×𝑋 with: β€–β€–π‘£βˆ—β€–β€–βˆž=1,(3.15) such that limπ‘™β†’βˆžξ€·πœ‡π‘™,𝑣𝑙=ξ€·πœ‡βˆ—,π‘£βˆ—ξ€Έ,inβ„Γ—π‘Œ.(3.16) Note that (3.12) is equivalent to 𝑣𝑙(𝑑)=πœ‡π‘™ξ€œ10𝑓𝐻(𝑑,𝑠)[𝑛]𝑦𝑙(𝑠)β€–β€–π‘¦π‘™β€–β€–βˆžπ‘‘π‘ ,π‘‘βˆˆ(0,1).(3.17) Combining this with (3.16) and using (3.14) and the Lebesgue dominated convergence theorem, we have π‘£βˆ—(𝑑)=πœ‡βˆ—ξ€œ10𝐻(𝑑,𝑠)0𝑑𝑠=0,π‘‘βˆˆ(0,1).(3.18) This contradicts (3.15). Therefore, ξ€½supπœ†βˆ£(πœ†,𝑦)βˆˆπ’žπ‘˜πœˆ[𝑛]ξ€Ύ=∞.(3.19)
Step 2. We show that sup{β€–π‘’β€–βˆžβˆ£(πœ†,𝑒)βˆˆπΆπ‘˜πœˆ[𝑛]}=∞. On the contrary, assume that sup{β€–π‘’β€–βˆžβˆ£(πœ†,𝑒)βˆˆπΆπ‘˜πœˆ[𝑛]}=𝑀0<∞. Then, there exists a sequence {(πœ‡π‘™,𝑦𝑙)}βŠ‚πΆπ‘˜πœˆ[𝑛] such that πœ‡π‘™β€–β€–π‘¦βŸΆβˆž,π‘™β€–β€–βˆžβ‰€π‘€0.(3.20) From Remark 1.1, we can take a subsequences of {(πœ‡π‘™,𝑦𝑙)}, still denoted by {(πœ‡π‘™,𝑦𝑙)}, such that {𝑦𝑙}βŠ‚π‘‡πœˆπ‘˜β‹‚π‘†πœˆπ‘˜ or {𝑦𝑙}βŠ‚π‘‡πœˆπ‘˜β‹‚π‘†πœˆπ‘˜+1. Without loss of generality, we suppose that {𝑦𝑙}βŠ‚π‘‡πœˆπ‘˜β‹‚π‘†πœˆπ‘˜. When {𝑦𝑙}βŠ‚π‘‡πœˆπ‘˜β‹‚π‘†πœˆπ‘˜+1 is considered, the proof is similar. We omit it.
Note that (πœ‡π‘™,𝑦𝑙) satisfies the autonomous equation: π‘¦π‘™ξ…žξ…ž+πœ‡π‘™π‘“[𝑛]𝑦𝑙=0,π‘‘βˆˆ(0,1).(3.21) Therefore, the graph of 𝑦𝑙 consists of a sequence of positive and negative bumps, together with a truncated bump at the right end of the interval [0,1], with the following properties (ignoring the truncated bump) (see [8]): all the positive (respectively, negative) bumps (i) have the same shape (the shapes of the positive and negative bumps may be different); (ii) attain the same maximum (minimum) value.
Let 0=𝜏0𝑙<𝜏1𝑙<β‹―<πœπ‘™π‘˜βˆ’1(3.22) denote the zeros of 𝑦𝑙 in [0,1]. Then, after taking a subsequence if necessary, limπ‘™β†’βˆžπœπ‘—π‘™βˆΆ=πœπ‘—βˆž,π‘—βˆˆ{0,1,…,π‘˜βˆ’1}. Clearly, 𝜏0∞=0. Set πœπ‘˜βˆž=1. We can choose at least one subinterval (πœπ‘—βˆž,πœβˆžπ‘—+1)β‰œπΌπ‘—βˆž which is of length at least 1/π‘˜ for some π‘—βˆˆ{0,1,…,π‘˜βˆ’1}. Then, for this 𝑗, πœπ‘™π‘—+1βˆ’πœπ‘—π‘™>3/4π‘˜ if 𝑙 is large enough. Put (πœπ‘—π‘™,πœπ‘™π‘—+1)β‰œπΌπ‘—π‘™.
Obviously, for the above given π‘˜,𝜈, and 𝑗, 𝑦𝑙(𝑑) have the same sign on 𝐼𝑗𝑙 for all 𝑙. Without loss of generality, we assume 𝑦𝑙(𝑑)>0,π‘‘βˆˆπΌπ‘—π‘™.(3.23) Armed with the information on the shape of 𝑦𝑙, it is easy to show that for the above given 𝐼𝑗𝑙, ‖𝑦𝑙‖𝐼𝑗𝑙,∞∢=max𝐼𝑗𝑙𝑦𝑙(𝑑)≀𝑀0,𝑙=1,2,….
Let 𝜎 be a constant with 0<𝜎<3/8π‘˜. Since 𝑦𝑙 is concave on 𝐼𝑗𝑙, we have 𝑦𝑙‖‖𝑦(𝑑)β‰₯πœŽπ‘™β€–β€–πΌπ‘—π‘™,βˆžξ‚ƒπœ,βˆ€π‘‘βˆˆπ‘—π‘™+𝜎,πœπ‘™π‘—+1ξ‚„.βˆ’πœŽ(3.24) Then, there must exist constants 𝛼,𝛽 with [𝛼,𝛽]βŠ‚πΌπ‘—βˆž and 𝑙0 such that 𝑦𝑙‖‖𝑦(𝑑)β‰₯πœŽπ‘™β€–β€–πΌπ‘—π‘™,∞>0,uniformlyfor[]π‘‘βˆˆπ›Ό,𝛽and𝑙>𝑙0.(3.25) On the other hand, note that 𝑓[𝑛]𝑦𝑙(𝑑)𝑦𝑙𝑓(𝑑)β‰₯inf[𝑛](𝑠)π‘ βˆ£0<𝑠≀𝑀0ξ‚Όξ‚€πœ>0,π‘‘βˆˆπ‘—π‘™,πœπ‘™π‘—+1.(3.26) Using the relation: π‘¦π‘™ξ…žξ…ž(𝑑)+πœ‡π‘™π‘“[𝑛]𝑦𝑙(𝑑)𝑦𝑙𝑦(𝑑)π‘™ξ‚€πœ(𝑑)=0,π‘‘βˆˆπ‘—π‘™,πœπ‘™π‘—+1,(3.27) and Sturm’s comparison theorem, we deduce that 𝑦𝑙 must change its sign on (𝛼,𝛽) if 𝑙 is sufficiently large, contradicting (3.25). Therefore, limπ‘™β†’βˆžβ€–β€–π‘¦π‘™β€–β€–βˆž=∞.(3.28) Hence, πΆπ‘˜πœˆ[𝑛] joins (πœ†π‘˜/(𝑓[𝑛])0,πœƒ) to (∞,∞) in 𝕏.

Lemma 3.3. Let (A1)–(A4)  hold. Then, there exists πœŒπœˆπ‘˜>0 such that ξƒ©βˆžξšπ‘›=1πΆπ‘˜πœˆ[𝑛]ξƒͺ0,πœŒπœˆπ‘˜ξ€Έξ€ΈΓ—π‘‹=βˆ….(3.29)

Proof. The proof is similar to that of Lemma  4.3 in [11]. We omit it.

Lemma 3.4. Let (A1)–(A4) hold, and let πœŒπœˆπ‘˜ be as in Lemma 3.3. Then, there exist 𝑛0βˆˆβ„• and ξπœ†πœˆπ‘˜β‰₯πœŒπœˆπ‘˜>0 such that for any ξπœ†πœ†>πœˆπ‘˜ and π‘’βˆˆπΆπ‘˜πœˆ[𝑛]: πΆπ‘˜πœˆ[𝑛]βˆ©ξ‚†ξπœ†(πœ†,𝑒)βˆ£πœ†β‰₯πœˆπ‘˜;β€–π‘’β€–βˆžξ‚‡=1=βˆ…,βˆ€π‘›>𝑛0.(3.30)

Proof. Suppose on the contrary that there exists {(πœ‡π‘™,𝑦𝑙⋃)}βŠ‚(βˆžπ‘›=1πΆπ‘˜πœˆ[𝑛])β‹‚((0,∞)×𝑋) such that limπ‘™β†’βˆžπœ‡π‘™β€–β€–π‘¦=∞,π‘™β€–β€–βˆž=1.(3.31)
Now, the method used in the proof of Lemma 3.2, Step 2, is still valid. Let 𝜎 be a constant with 0<𝜎<3/8π‘˜. Taking subsequences again if necessary, still denoted by {(πœ‡π‘™,𝑦𝑙)}, such that {𝑦𝑙}βŠ‚π‘‡πœˆπ‘˜β‹‚π‘†πœˆπ‘˜. Without loss of generality, we can also derive an interval [𝛼,𝛽]βŠ‚πΌπ‘—βˆž and 𝑙0 such that 1β‰₯𝑦𝑙‖‖𝑦(𝑑)β‰₯πœŽπ‘™β€–β€–πΌπ‘—π‘™,∞=𝜎,uniformlyfor[]π‘‘βˆˆπ›Ό,𝛽and𝑙>𝑙0.(3.32) It is easy to find an integer 𝑛0βˆˆβ„• such that 1/𝑛0<𝜎. This implies that 𝑓inf[𝑛](𝑠)π‘ ξ‚Όξ‚»βˆ£πœŽ<𝑠≀1=inf𝑓(𝑠)π‘ ξ‚Όβˆ£πœŽ<𝑠≀1,βˆ€π‘›>𝑛0.(3.33) Note that for 𝑛>𝑛0, 𝑓[𝑛]𝑦𝑙(𝑑)𝑦𝑙(𝑑)β‰₯inf𝑓(𝑠)π‘ ξ‚Όβˆ£πœŽ<𝑠≀1>0,uniformlyforπ‘‘βˆˆ(𝛼,𝛽)and𝑙>𝑙0.(3.34) Combining these facts and the relation: π‘¦π‘™ξ…žξ…ž(𝑑)+πœ‡π‘™π‘“[𝑛]𝑦𝑙(𝑑)𝑦𝑙𝑦(𝑑)𝑙(𝑑)=0,π‘‘βˆˆ(𝛼,𝛽),(3.35) and Sturm’s comparison theorem, we conclude that 𝑦𝑙 must change its sign on (𝛼,𝛽) if 𝑙 is large enough. This contradicts (3.32), and the proof is done.

Lemma 3.5. Let (A1)–(A4) hold, and let 𝑛0 be as in Lemma 3.4. Then, there exist πœ†πœˆπ‘˜β‰₯πœŒπœˆπ‘˜>0 and πœ–βˆˆ(0,1/2) such that for any πœ†>πœ†πœˆπ‘˜ and π‘’βˆˆπΆπ‘˜πœˆ[𝑛]: πΆπ‘˜πœˆ[𝑛]βˆ©ξ€½(πœ†,𝑒)βˆ£πœ†β‰₯πœ†πœˆπ‘˜;1βˆ’2πœ–β‰€β€–π‘’β€–βˆžξ€Ύβ‰€1+2πœ–=βˆ…,βˆ€π‘›>𝑛0.(3.36)

Proof. Similar to the proof of Lemma 3.4, we can find a constant ξ‚πœ†πœˆπ‘˜>0 such that β€–π‘’β€–βˆžβ‰ 1 provided that ξ‚πœ†(πœ†,𝑒)∈(πœˆπ‘˜,∞)Γ—π‘‡πœˆπ‘˜ being a solution of (1.1) and (1.2).
Let πœ†πœˆπ‘˜ξπœ†=max{πœˆπ‘˜,ξ‚πœ†πœˆπ‘˜}+1. We claim that there exists πœ–βˆˆ(0,1/2) such that (3.36) holds. Suppose on the contrary that there exists πœ‡ξ€½ξ€·π‘™,π‘¦π‘™βŠ‚ξƒ©ξ€Έξ€Ύβˆžξšπ‘›=1πΆπ‘˜πœˆ[𝑛]ξƒͺξ™πœ†ξ€·ξ€·πœˆπ‘˜ξ€Έξ€Έ,,βˆžΓ—π‘‹(3.37) satisfying limπ‘™β†’βˆžπœ‡π‘™=πœ‡βˆ—β‰₯πœ†πœˆπ‘˜limπ‘™β†’βˆžβ€–β€–π‘¦π‘™β€–β€–βˆž=1.(3.38) We can discuss two cases.
Case 1. If πœ‡βˆ—<∞. {𝑦𝑙} is compact in 𝑋 implies that there exists a subsequence, still denoted by {𝑦𝑙}, such that limπ‘™β†’βˆžπ‘¦π‘™=π‘¦βˆ—βˆˆπ‘‡πœˆπ‘˜,β€–π‘¦βˆ—β€–βˆž=1.(3.39) Obviously, (πœ‡βˆ—,π‘¦βˆ—) is a solution of (1.1) and (1.2). It is impossible.Case 2. If πœ‡βˆ—=∞. Taking subsequences again if necessary, still denoted by {(πœ‡π‘™,𝑦𝑙)}, such that 1/2β‰€β€–π‘¦π‘™β€–βˆžβ‰€3/2. Using the same argument as Lemma 3.4, we can find a contradiction.

Proof of Theorem 1.3. We will prove that the superior limit of πΆπ‘˜πœˆ[𝑛] contains an unbounded component π’žπœˆπ‘˜βŠ‚(0,∞)Γ—π‘‡πœˆπ‘˜ of solutions of (1.1) and (1.2), which joins (∞,πœƒ) to (∞,∞). For π‘Ÿ>0, introduce Ξ©π‘Ÿ=ξ€½π‘’βˆˆπ‘Œβˆ£β€–π‘’β€–βˆžξ€Ύ.<π‘Ÿ(3.40) Set Ξ“πœˆπ‘˜ξ€·[∢=0,∞)Γ—π‘‡πœˆπ‘˜ξ€Έβ§΅ξ€½(πœ‚,𝑒)βˆ£πœ‚β‰₯πœ†πœˆπ‘˜;π‘’βˆˆπ‘‡πœˆπ‘˜,β€–π‘’β€–βˆžξ€Ύ,Σ≀1+πœ–πœˆπ‘˜ξ€½(∢=πœ‚,𝑒)βˆ£πœ‚β‰₯πœ†πœˆπ‘˜;π‘’βˆˆπ‘‡πœˆπ‘˜,β€–π‘’β€–βˆžξ€Ύ.≀1βˆ’πœ–(3.41) Let 𝑛0 and πœ– be as in Lemma 3.5. Firstly, for each given nonnegative integer 𝑝=0,1,2,…, and 𝑛β‰₯𝑛0 with (πœ†π‘˜/(𝑓[𝑛])0)β‰₯πœ†πœˆπ‘˜+𝑝, we define the connected subset, (πœπ‘˜πœˆ[𝑛])𝑝, in πΆπ‘˜πœˆ[𝑛] satisfying (see Figure 2(a)):
(i)(πœπ‘˜πœˆ[𝑛])π‘βŠ‚(πΆπ‘˜πœˆ[𝑛]⧡(πœ†πœˆπ‘˜+𝑝,∞)Γ—Ξ©1βˆ’πœ–);
(ii)(πœπ‘˜πœˆ[𝑛])𝑝 joins {πœ†πœˆπ‘˜+𝑝}Γ—Ξ©1βˆ’πœ– with infinity in Ξ“πœˆπ‘˜.By Lemmas 2.2 and 2.3, limsupπ‘›β†’βˆž(πœπ‘˜πœˆ[𝑛])𝑝 contains a component (πœπœˆπ‘˜)𝑝 joining {πœ†πœˆπ‘˜+𝑝}Γ—Ξ©1βˆ’πœ– with infinity in Ξ“πœˆπ‘˜ (see Figure 2(b)).
It is easy to verify that if (πœ†,𝑒)∈(πœπœˆπ‘˜)𝑝(𝑝=0,1,2,…), then (πœ†,𝑒) is a solution of (1.1) and (1.2), and π‘’βˆˆπ‘‡πœˆπ‘˜.
Next, by using Lemma 2.3 and the method in [11] (see (4.22)–(4.30) in [11]), we can find a component π’žπœˆπ‘˜ in limsupπ‘β†’βˆž(πœπœˆπ‘˜)𝑝, which is unbounded both in Ξ“πœˆπ‘˜ and Ξ£πœˆπ‘˜.
Finally, we show that π’žπœˆπ‘˜ joins (∞,πœƒ) with (∞,∞). This will be done by the following three steps.
Step 1. We show that limπœ†β†’+βˆžβ€–π‘’β€–βˆž=0 for (πœ†,𝑒)∈(π’žπœˆπ‘˜βˆ©Ξ£πœˆπ‘˜).
Suppose on the contrary that there exists {(πœ‡π‘™,𝑦𝑙)}βŠ‚π’žπœˆπ‘˜ with β€–π‘¦π‘™β€–βˆžβ‰€1βˆ’πœ–, and πœ‡π‘™β€–β€–π‘¦β†’+∞,π‘™β€–β€–βˆžβ‰₯π‘Ž,(3.42) for some constant π‘Ž>0. Applying the method of proving Lemma 3.2, we can deduce a contradiction.
Step 2. We show that sup{πœ†βˆ£(πœ†,𝑒)∈(π’žπœˆπ‘˜βˆ©Ξ“πœˆπ‘˜)}=∞. By a similar argument as Lemma 3.2, we can get the conclusion.Step 3. We show that limπœ†β†’+βˆžβ€–π‘’β€–βˆž=+∞ for (πœ†,𝑒)∈(π’žπœˆπ‘˜βˆ©Ξ“πœˆπ‘˜).
On the contrary, suppose that there exists {(πœ‡π‘™,𝑦𝑙)}βŠ‚(π’žπœˆπ‘˜βˆ©Ξ“πœˆπ‘˜) with πœ‡π‘™β€–β€–π‘¦β†’+∞,1<π‘™β€–β€–βˆžβ‰€π‘€,(3.43) for some constant 𝑀>0. The proof can be done by the same argument as Lemma 3.2.

This completes the proof of Theorem 1.3.

fig2
Figure 2

Proof of Corollary 1.4. The result can be directly obtained by Theorem 1.3.

Acknowledgments

This paper is supported by NSFC (no. 10971139); China Postdoctoral Fund (no. 2011M500615); Scientific Innovation Projection of Shanghai Education Department (no. 11YZ225); SIT-YJ2009-16.

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