Global Bifurcation in 2𝑚-Order Generic Systems of Nonlinear Boundary Value Problems

Abstract

We consider the systems of (−1)^𝑚𝑢^(2𝑚)=𝜆𝑢+𝜆𝑣+𝑢𝑓(𝑡,𝑢,𝑣),𝑡∈(0,1),𝑢^(2𝑖)(0)=𝑢^(2𝑖)(1)=0, and 0≤𝑖≤𝑚−1,(−1)^𝑚𝑣^(2𝑚)=𝜇𝑢+𝜇𝑣+𝑣𝑔(𝑡,𝑢,𝑣),𝑡∈(0,1),𝑣^(2𝑖)(0)=𝑣^(2𝑖)(1)=0,0≤𝑖≤𝑚−1, where 𝜆,𝜇∈𝑅 are real parameters. 𝑓,𝑔∶[0,1]×𝑅²→𝑅 are 𝐶^𝑘,𝑘≥3 functions and 𝑓(𝑡,0,0)=𝑔(𝑡,0,0)=0,𝑡∈[0,1]. It will be shown that if the functions, 𝑓 and 𝑔 are “generic” then the solution set of the systems consists of a countable collection of 2-dimensional, 𝐶^𝑘 manifolds.

1. Introduction

Many scientific and technological problems that are modeled mathematically by systems of ODEs, for example, mathematical models of series circuits and mechanical systems involving several springs attached in series can lead to a system of differential equations. Furthermore, such systems are often encountered in chemical, ecological, biological, and engineering applications, thereby attracting constant interest of researchers in recent years, on several aspects of the problems; we focus on one aspect here, namely, the existence of solutions, from the point of view of bifurcations.

In [1], Rynne considered the global bifurcation in generic systems of coupled nonlinear Sturm-Liouville boundary value problem: 𝐿₁𝑢∶=−𝑝₁𝑢^′′+𝑞₁()(),𝑎𝑢=𝜇𝑢+𝑢𝑓𝑥,𝑢,𝑣,𝑥∈0,1₁₀𝑢(0)+𝑏₁₀𝑢^′(0)=0,𝑎₁₁𝑢(1)+𝑏₁₁𝑢^′(1)𝐿=0,₂𝑣∶=−𝑝₂𝑣^′′+𝑞₂()(),𝑎𝑣=𝜈𝑣+𝑣𝑔𝑥,𝑢,𝑣,𝑥∈0,1₂₀𝑣(0)+𝑏₂₀𝑣^′(0)=0,𝑎₂₁𝑣(1)+𝑏₂₁𝑣^′(1)=0, (1.1) where 𝑝_𝑖∈𝐶¹[0,1], 𝑞_𝑖∈𝐶[0,1], 𝑖=1,2, and 𝑞_𝑖 are positive on [0,1]; 𝜇,𝜈∈𝑅 are parameter; 𝑓,𝑔∶[0,1]×𝑅²→𝑅 are 𝐶^𝑘, 𝑘≥3 functions. The interesting results are that if the functions 𝑓 and 𝑔 are “generic” then for all integers 𝑚,𝑛≥0, there are smooth 2-dimensional manifolds 𝒮¹_𝑚,𝒮²_𝑛, of semitrivial solutions of the system which bifurcate from the eigenvalues 𝜆⁰_𝑚,𝜇⁰_𝑛, of 𝐿₁,𝐿₂, respectively. Furthermore, there are smooth curves ℬ¹_𝑚𝑛⊂𝑆¹_𝑚, ℬ²_𝑚𝑛⊂𝑆²_𝑛, along which secondary bifurcations take place, giving rise to smooth, 2-dimensional manifolds of nontrivial solutions. It is shown that there is a single such manifold, 𝒩_𝑚𝑛, which “links” the curves ℬ¹_𝑚𝑛,ℬ²_𝑚𝑛. Nodal properties of solutions on 𝒩_𝑚𝑛 and global properties of 𝒩_𝑚𝑛 are also discussed.

Inspired by [1], in this paper, we consider the 2𝑚-order systems of coupled nonlinear boundary value problems: (−1)^𝑚𝑢^(2𝑚)()(),𝑢=𝜆𝑢+𝜆𝑣+𝑢𝑓𝑡,𝑢,𝑣,𝑡∈0,1_(2𝑖)(0)=𝑢^(2𝑖)(1)()=0,0≤𝑖≤𝑚−1,−1^𝑚𝑣^(2𝑚)()(),𝑣=𝜇𝑢+𝜇𝑣+𝑣𝑔𝑡,𝑢,𝑣,𝑡∈0,1_(2𝑖)(0)=𝑣^(2𝑖)(1)=0,0≤𝑖≤𝑚−1, (1.2) where 𝜆,𝜇∈𝑅 are real parameters. 𝑓,𝑔∶[0,1]×𝑅²→𝑅 are 𝐶^𝑘, 𝑘≥3 functions and 𝑓(𝑡,0,0)=𝑔(𝑡,0,0)=0, 𝑡∈[0,1]. It will be shown that if the functions 𝑓 and 𝑔 are “generic” then the solution set of the systems consists of a countable collection of 2-dimensional, 𝐶^𝑘 manifolds. The paper is organized as follows. In Section 2, by computing algebraic multiplicity of eigenvalue (𝜆,𝜇), we get the set of bifurcation points of problem (1.2) and obtain the existence of nontrivial solution. In Section 3, we get some genericity result.

Note that problem (1.2) is different from the problem (1.1). In [1], the manifolds of nontrivial solution come from the secondary bifurcations which take place along the smooth curves in the manifolds of semitrivial solutions. In this paper, by computing algebraic multiplicity of eigenvalue (𝜆,𝜇), we get the existence of nontrivial solution by first bifurcation.

2. Existence of Nontrivial Solution

Let 𝑋={𝑤∈𝐶^2𝑚[0,1]∶𝑤^(2𝑖)(0)=𝑤^(2𝑖)(1)=0,0≤𝑖≤𝑚−1}, 𝑋₀=𝑋⧵{0}, 𝑌={𝑤∈𝐶^2𝑚−1[0,1]∶𝑤^(2𝑖)(0)=𝑤^(2𝑖)(1)=0,0≤𝑖≤𝑚−1}, 𝑍=𝐶[0,1]; these spaces are endowed with their usual supnorms. We also use the space 𝐸=𝑅²×𝑋² with norm ‖(𝜆,𝜇,𝑢,𝑣)‖²=|𝜆|²+|𝜇|²+‖𝑢‖²+‖𝑣‖². For 𝑢,𝑣∈𝐶[0,1], let ⟨𝑢,𝑣⟩=∫¹₀𝑢𝑣.

Define 𝐿∶𝑋→𝑍 by 𝐿𝑢∶=(−1)^𝑚𝑢^(2𝑚),𝑢∈𝑋, (2.1) then 𝐿 is invertible and ⟨𝐿𝑢,𝑣⟩=⟨𝑢,𝐿𝑣⟩.

Let 𝐺_𝑚(𝑡,𝑠) be the Green’s function of the problem (−1)^𝑚𝑤^(2𝑚)(𝑡)(),𝑤=0,𝑡∈0,1_(2𝑖)(0)=𝑤^(2𝑖)(1)=0,0≤𝑖≤𝑚−1, (2.2) then 𝐺_𝑖(𝑡,𝑠)=¹₀𝐺(𝑡,𝜉)𝐺_𝑖−1(𝜉,𝑠)𝑑𝜉,2≤𝑖≤𝑚, (2.3) where 𝐺₁()()=𝑡()𝑠()𝑡,𝑠=𝐺𝑡,𝑠1−𝑠,0≤𝑡≤𝑠≤1,1−𝑡,0≤𝑠≤𝑡≤1. (2.4) So problem (1.2) is equivalent to 𝑢(𝑡)=𝜆¹₀𝐺_𝑚(𝑡,𝑠)𝑢(𝑠)𝑑𝑠+𝜆¹₀𝐺_𝑚(𝑡,𝑠)𝑣(𝑠)𝑑𝑠+¹₀𝐺_𝑚()𝑢(𝑠)𝑓((𝑠)(𝑠𝑣(𝑡)𝑡,𝑠𝑠,𝑢,𝑣))𝑑𝑠,=𝜇¹₀𝐺_𝑚(𝑡,𝑠)𝑢(𝑠)𝑑𝑠+𝜇¹₀𝐺_𝑚(𝑡,𝑠)𝑣(𝑠)𝑑𝑠+¹₀𝐺_𝑚(𝑡,𝑠)𝑣(𝑠)𝑔(𝑠,𝑢(𝑠),𝑣(𝑠))𝑑𝑠. (2.5) In fact, problem (2.5) can be expressed in the form 𝑢𝑣𝑢𝑣𝑢𝑣+𝑁()𝑀()=𝜆,𝐴𝐴00+𝜇00𝐴𝐴𝑢,𝑣𝑢,𝑣 (2.6) where 𝐴∶𝑌→𝑌 is given by 𝐴𝑥(𝑡)=¹₀𝐺_𝑚()𝑥(𝑠)𝑁(𝑡)𝑡,𝑠𝑑𝑠,𝑢,𝑣)(∶=¹₀𝐺_𝑚()𝑢(𝑠)𝑓((𝑠)(𝑠𝑀(𝑡)𝑡,𝑠𝑠,𝑢,𝑣))𝑑𝑠,𝑢,𝑣)(∶=¹₀𝐺_𝑚(𝑡,𝑠)𝑣(𝑠)𝑔(𝑠,𝑢(𝑠),𝑣(𝑠))𝑑𝑠, (2.7) note that 𝑁(𝑢,𝑣)/‖(𝑢,𝑣)‖→0, 𝑀(𝑢,𝑣)/‖(𝑢,𝑣)‖→0 for (𝑢,𝑣) near (0,0) in 𝑌².

We discuss the bifurcation phenomena for problem (2.6).

From [2] we know that the set {(𝜆,𝜇)∈𝑅²:(𝜆,𝜇,0,0)  is a bifurcation point for (2.6)} that is contained in the set Σ_𝐴𝜆,𝜇∈𝑅²∶𝑢𝑣=𝜆𝑢𝑣𝐴𝐴00+𝜇00𝐴𝐴hasanontrivialsolution𝑢𝑣in𝑌². (2.8)

Note that the equation defining Σ_𝐴 is equivalent to the system (),().𝐿𝑢=𝜆𝑢+𝑣𝐿𝑣=𝜇𝑢+𝑣 (2.9) If (2.9) has a nontrivial solution (^𝑢_𝑣), then 𝑢 and 𝑣 can be shown to solve 𝐿𝐿−𝜆+𝜇𝑥=0. (2.10) It is known in [3] that 𝐿 has a discrete sequence of simple eigenvalues 0<𝜆₁<𝜆₂<⋯<𝜆_𝑛→+∞. (2.11) It can thus be shown that Σ_𝐴={(𝜆,𝜇)∶𝜆+𝜇=𝜆_𝑛,𝑛=1,2,…}.

Theorem  2.1 of [2] guarantees that elements of Σ_𝐴 of odd algebraic multiplicity are bifurcation points. Suppose (𝜆,𝜇) satisfies 𝜆+𝜇=𝜆_𝑛, and 𝐿𝑥_𝑛=𝜆_𝑛𝑥_𝑛. It is easy to show that (𝜆,𝜇) has geometric multiplicity 1. In fact, since 𝑢 and 𝑣 solve (2.10), it must be the case that 𝑢=𝛼𝑥_𝑛, 𝑣=𝛽𝑥_𝑛. Then (2.9) yields 𝜆_𝑛00𝜆_𝑛𝛼𝛽𝑥_𝑛=𝛼𝛽𝑥𝜆𝜆𝜇𝜇^𝑛. (2.12) Then (2.12) has a nontrivial solution with 𝛽=(𝜇/𝜆)𝛼 if 𝜆≠0; 𝛼=0 if (𝜆,𝜇)=(0,𝜆_𝑛). So (𝜆,𝜇) has geometric multiplicity 1. We will shown that the algebraic multiplicity at (𝜆,𝜇) is also 1.

Assume for the moment that 𝜆=0, 𝜇=𝜆_𝑛. In this case, the equation 𝐿−𝜆−𝜆−𝜇𝐿−𝜇²𝑢𝑣=00 (2.13) implies 𝑢𝑣=𝐿−𝜆−𝜆0−𝜇𝐿−𝜇𝛽𝑥_𝑛, (2.14) where 𝛽∈𝑅. We have 𝑢=0 from 𝐿𝑢=0, so 𝐿𝑣=𝜇𝑣+𝛽𝑥_𝑛=𝜆_𝑛𝑣+𝛽𝑥_𝑛; then ⟨𝐿𝑣,𝑥_𝑛⟩=⟨𝜆_𝑛𝑣+𝛽𝑥_𝑛,𝑥_𝑛⟩, so 𝛽=0. It follows that the algebraic multiplicity at (0,𝜆_𝑛) is 1. By the homotopy invariance of Leray-Schauder degree, the algebraic multiplicity at (𝜆,𝜇) is odd for each (𝜆,𝜇) such that 𝜆+𝜇=𝜆_𝑛.

We have established the following result.

Theorem 2.1. If 𝜆_𝑛 is an eigenvalue of 𝐿, then the points belonging to the set {(𝜆,𝜇)∶𝜆+𝜇=𝜆_𝑛} are all bifurcation points.

From Theorem  3.3 of [2] one has the following.

Theorem 2.2. Suppose that (𝜆₀,𝜇₀)∈Σ_𝐴. Then there is a two-dimensional continuum 𝒞_𝑛 emerging in 𝑅²×𝑌² from (𝜆₀,𝜇₀,0,0). If (𝜆₀,𝜇₀)≠(0,𝜆_𝑛) or (𝜆_𝑛,0), then at least locally, all nontrivial solutions (𝜆,𝜇,𝑢,𝑣) in 𝒞_𝑛 are such that 𝑢 and 𝑣 have 𝑛−1 simple zeros in (0,1).

3. Genericity Result

We begin by stating the basic transversality theorems as given in [4] (see Theorems 1.1, 1.2 and Remark A1.1 in the appendix of [4]). Let 𝑋,𝑌,𝑍 be real, separable Banach spaces. Let 𝑉⊂𝑋,𝑊⊂𝑌 be open sets, and let 𝐹∶𝑉×𝑊→𝑍 be a 𝐶^𝑘, 𝑘≥1 mapping such that for every 𝑤₀∈𝑊, 𝐹(⋅,𝑤₀) is a Fredholm mapping of index 𝑙<𝑘, that is, for all (𝑣⁰,𝑤⁰)∈𝑉×𝑊; the linear operator 𝐷_𝑣𝐹(𝑣⁰,𝑤⁰) is Fredholm with index 𝑙<𝑘 (where 𝐷𝐹∶𝑋×𝑌→𝑍,𝐷_𝑣𝐹∶𝑋→𝑍, 𝐷_𝑤𝐹∶𝑌→𝑍, will denote, resp., the Fréchet derivative of 𝐹). We say that 0 is a regular value of 𝐹 if the operator 𝐷𝐹(𝑣⁰,𝑤⁰) is onto every point (𝑣⁰,𝑤⁰) such that 𝐹(𝑣⁰,𝑤⁰)=0. Also, a subset of a topological space is said to be residual if it contains the intersection of a countable collection of open dense sets. Note that the intersection of a countable collection of residual sets is also residual.

Theorem 3.1 (see [4]). If 0 is a regular value of 𝐹, then the set Θ=𝑤⁰∈𝑊∶0isaregularvalueof𝐹⋅,𝑤⁰ (3.1) is a residual subset of 𝑊. For every 𝑤⁰∈Θ, the set 𝑣∈𝑉∶𝐹𝑣,𝑤⁰=0 (3.2) is the disjoint union of a finite, or countable, collection of connected 𝐶^𝑘 submanifolds of 𝑋 of dimension 𝑙.

A property of the elements of a topological space is said to be generic if it holds for all elements in a residual subset of the space. Since we wish to discuss properties of the systems (1.2) which hold for “generic” functions 𝑓 and 𝑔, we need an appropriate space of functions and a topology on this space. Let ℱ be the set of all real valued 𝐶^𝑘 functions defined on [0,1]×𝑅². We define a topology on ℱ as follows. For any 𝑓∈ℱ and any continuous, positive function 𝜖∶𝑅²→𝑅₊, we define an 𝜖-neighbourhood of 𝑓 by 𝑈_𝜖𝑓=ℎ∈ℱ∶_{|𝛼|≤𝑘}𝐷^𝛼𝑓𝑡,𝜂,𝜁−𝐷^𝛼ℎ𝑡,𝜂,𝜁<𝜀𝜂,𝜁,𝑡,𝜂,𝜁∈[0,1]×𝑅² (3.3) (here 𝐷^𝛼 is the usual multi-index notation for partial derivatives); a subset ℰ⊂ℱ is defined to be open if and only if for every 𝑓∈ℰ there is a function 𝜖 such that the 𝑈_𝜖(𝑓) of 𝑓 lies in ℰ.

Let ℱ₁⊂ℱ denote the set of functions such that 𝑓(⋅,0,0)=0. Let 𝒩(𝑓,𝑔) denote the set of nontrivial solutions of (1.2), we get the following result.

Theorem 3.2. There is a residual set 𝒢⊂ℱ²₁ such that if (𝑓,𝑔)∈𝒢, then the set 𝒩 consists of a countable collection of 2-dimensional, 𝐶^𝑘 manifolds.

Proof . Let 𝐸₀=𝑅²×𝑋²₀ and, for any (𝑓,𝑔)∈ℱ²₁, define a 𝐶^𝑘 function 𝐺(𝑓,𝑔)∶𝐸₀→𝑍² by 𝐺=()()𝑓,𝑔𝜆,𝜇,𝑢,𝑣,𝐿𝑢−𝜆𝑢−𝜆𝑣−𝑓⋅,𝑢,𝑣𝑢,𝐿𝑣−𝜇𝑢−𝜇𝑣−𝑔⋅,𝑢,𝑣𝑣,𝜆,𝜇,𝑢,𝑣∈𝐸₀. (3.4) Clearly, 𝒩=𝐺(𝑓,𝑔)⁻¹(0). For any 𝜌>1, let 𝐴_𝜌=𝜆,𝜇,𝑢,𝑣∈𝐸₀∶𝜆,𝜇,𝑢,𝑣<𝜌,‖𝑢‖>𝜌⁻¹,‖𝑣‖>𝜌⁻¹, (3.5) and let 𝐴_𝜌 be the closure of 𝐴_𝜌. For any integer 𝑟≥1, let 𝒯_𝑟 be the set of functions (𝑓,𝑔)∈ℱ²₁ such that 𝐺(𝑓,𝑔) is transverse on 𝐴_𝑟+𝛿 for some sufficiently small 𝛿>0.

We first prove that each set 𝒯_𝑟 is open in ℱ²₁. For any (𝑓,𝑔)∈ℱ²₁, 𝜌>1, the set 𝒩(𝑓,𝑔)∩𝐴_𝜌 is compact. Thus (𝑓,𝑔)∉𝒯_𝑟 if and only if there exists 𝑧∈𝒩(𝑓,𝑔)∩𝐴_𝑟 such that the operator 𝐷𝐺(𝑓,𝑔)(𝑧) is not surjective (here the derivative is with respect to (𝜆,𝜇,𝑢,𝑣)). Thus, in deciding whether (𝑓,𝑔)∈𝒯_𝑟, the values of 𝑓 and 𝑔 on |𝑢|>𝑟, |𝑣|>𝑟 are irrelevant. Therefore, it suffices to show that if (𝑓^𝑖,𝑔^𝑖)∈ℱ²₁⧵𝒯_𝑟, 𝑖=1,2,…, is a sequence converging to (𝑓⁰,𝑔⁰)∈ℱ²₁ with respect to the 𝐶^𝑘 topology on [0,1]×[−𝑟,𝑟]², then (𝑓⁰,𝑔⁰)∉𝒯_𝑟. Now, for each 𝑖, there exists 𝑧^𝑖∈𝒩(𝑓^𝑖,𝑔^𝑖)∩𝐴_𝑟 such that 𝐷𝐺(𝑓^𝑖,𝑔^𝑖)(𝑧^𝑖) is not surjective. Furthermore, it can be shown that the sequence 𝑧^𝑖 converges (in 𝐸) to a point 𝑧⁰ (using standard Sobolev embedding and regularity arguments; see for instance, the proof of Theorem  2.1 in [4] or the proof of Theorem  3.a.1 in [5]). Then, by continuity, 𝑧⁰∈𝒩(𝑓⁰,𝑔⁰)∩𝐴_𝑟, and the operator 𝐷𝐺(𝑓⁰,𝑔⁰)(𝑧⁰)=lim_𝑖→∞𝐷𝐺(𝑓^𝑖,𝑔^𝑖)(𝑧^𝑖) cannot be surjective since this would contradict standard perturbation results for Fredholm operators (see for instance, Theorem  13.6 in [6]). Thus (𝑓⁰,𝑔⁰)∉𝒯_𝑟, which completes the proof that 𝒯_𝑟 is open in ℱ²₁.

We will now show that the sets 𝒯_𝑟 are dense in ℱ²₁. Choose an arbitrary, fixed (𝑓,𝑔)∈ℱ²₁ and 𝑟≥1. Let 𝜃_𝑟∶𝑅→𝑅 be a decreasing 𝐶^∞ function such that 𝜃_𝑟(𝑠)=1 if 𝑠≤𝑟+1 and 𝜃_𝑟(𝑠)=0 if 𝑠≥𝑟+2. For any set of functions 𝛾=(𝛾_𝑖𝑗)²_𝑖,𝑗=1∈Γ∶=𝐶^𝑘([0,1])⁴ we define 𝑓^𝛾∶[0,1]×𝑅²→𝑅 by 𝑓^𝛾𝑡,𝜂,𝜁=𝑓𝑡,𝜂,𝜁+𝜃_𝑟𝜂²+𝜁^21/2𝛾₁₁(𝑥)𝜂+𝛾₁₂(𝑥)𝜁, (3.6) for (𝑡,𝜂,𝜁)∈[0,1]×𝑅²; we define 𝑔^𝛾∶[0,1]×𝑅²→𝑅 similarly, using 𝛾₂₁,𝛾₂₂. Clearly, for any given positive functions 𝜀₁,𝜀₂ (see the definition of the topology on ℱ) we have 𝑓^𝛾∈𝑈_𝜖1(𝑓), 𝑔^𝛾∈𝑈_𝜖2(𝑔), for all sufficiently small 𝛾∈Γ. We now define a function 𝐻∶𝐴_𝑟+1×Γ→𝑍×𝑍 by 𝐻𝜆,𝜇,𝑢,𝑣,𝛾=𝐺𝑓^𝛾,𝑔^𝛾𝜆,𝜇,𝑢,𝑣,𝜆,𝜇,𝑢,𝑣,𝛾∈𝐴_𝑟+1×Γ. (3.7) Since Γ is a Banach space (with norm ‖𝛾‖²=∑²_𝑖,𝑗=1‖𝛾_𝑖,𝑗‖²_𝑘), we can differentiate 𝐻 with respect to 𝛾 and apply transversality results. Clearly, 𝐻 is 𝐶^𝑘.

Lemma 3.3. The derivative 𝐷𝐻(𝑧⁰) is surjective at any point 𝑧⁰∈𝐻⁻¹(0).

Proof. We must show that for any 𝜁=(𝜁₁,𝜁₂)∈𝑍², the following equation can be solved for (𝜆,𝜇,𝑢,𝑣,𝛾)∈𝐸×Γ: 𝐻⁰_()𝑢,𝑣(𝑢,𝑣)+𝐻⁰_𝜆,𝜇𝜆,𝜇+𝐻⁰_𝛾𝛾=𝜁, (3.8) where 𝐻⁰_(𝑢,𝑣),𝐻⁰_(𝜆,𝜇),𝐻⁰_𝛾 denote the derivatives at 𝑧⁰=(𝜆⁰,𝜇⁰,𝑢⁰,𝑣⁰,𝛾⁰) with respect to (𝑢,𝑣), (𝜆,𝜇) and 𝛾, respectively; these operators have the form 𝐻⁰_()𝑢,𝑣(𝑢,𝑣)=𝐿𝑢𝐿𝑣−𝜆⁰+𝑓⁰+𝑢⁰𝑓⁰_𝑢𝑢+𝜆⁰+𝑢⁰𝑓⁰_𝑣𝑣𝜇⁰+𝑔⁰+𝑣⁰𝑔⁰_𝑣𝑣+𝜇⁰+𝑣⁰𝑔⁰_𝑢𝑢,𝐻⁰_𝜆,𝜇𝜆,𝜇=−𝜆𝑢⁰+𝑣⁰−𝜇𝑢⁰+𝑣⁰,𝐻⁰_𝛾𝛾=𝛾₁₁𝑢⁰𝑢⁰+𝛾₁₂𝑢⁰𝑣⁰𝛾₂₁𝑢⁰𝑣⁰+𝛾₂₂𝑣⁰𝑣⁰, (3.9) (where 𝑓⁰ denotes 𝑓^𝛾0(⋅,𝑢⁰,𝑣⁰), etc.). From [4, Page 301] we know that the operator 𝐻⁰_(𝑢,𝑣) is Fredholm with index 0 and, if 𝜉=(𝜉₁,𝜉₂)∈𝑅(𝐻⁰_(𝑢,𝑣))^⟂⊂𝑍², then 𝜉 is a solution of a homogeneous coupled pair of linear ordinary differential equations. Let 𝑑⁰=dim𝑅(𝐻⁰_(𝑢,𝑣))^⟂. If 𝑑⁰=0 there is nothing further to prove, so suppose that 𝑑⁰≥1 and let {𝜉_𝑖∈𝑍²∶𝑖=1,…,𝑑⁰} be a basis for 𝑅(𝐻⁰_(𝑢,𝑣))^⟂. Since the functions 𝑢⁰, 𝑣⁰ and 𝜉_𝑖, 𝑖=1,…,𝑑⁰ are all nonzero and can be regarded as solutions of homogeneous linear ordinary differential equations (or systems), it follows from the uniqueness of the solution of the initial value problem for such equations that these functions cannot be identically zero on any open set. Thus there must exist an open interval 𝐼⁰⊂(0,1) and a number 𝛿>0, such that for 𝑥∈𝐼⁰ we have |𝑢⁰(𝑥)|>𝛿, |𝑣⁰(𝑥)|>𝛿, and the functions 𝜉_𝑖, 𝑖=1,…,𝑑⁰ are linearly independent on 𝐼⁰.
Now, since 𝑢⁰ and 𝑣⁰ are bounded away from 0 on 𝐼⁰, for any 𝜓=(𝜓₁,𝜓₂)∈𝐶^∞₀(𝐼⁰)², the equation 𝐻⁰_𝛾𝛾=𝜓 has a solution 𝛾∈𝐶^𝑘₀(𝐼⁰)⁴. Thus, we can choose 𝛾^𝑗∈𝐶^𝑘₀(𝐼⁰)⁴, 𝑗=1,…,𝑑⁰ such that ⟨𝐻⁰_𝛾𝛾^𝑗,𝜉^𝑖⟩=𝛿_𝑖𝑗, 𝑖,𝑗=1,…,𝑑⁰ (𝛿_𝑖𝑗 is the Kronecker delta); thus the set {𝐻⁰_𝛾𝛾^𝑗∶𝑗=1,…,𝑑⁰} spans a complement of 𝑅(𝐻⁰_(𝑢,𝑣))^⟂ in 𝑍², and so the operator (𝑢,𝑣,𝛾)→𝐻⁰_(𝑢,𝑣)(𝑢,𝑣)+𝐻⁰_𝛾𝛾∈𝑍² is surjective. This proves Lemma 3.3.
It now follows from Lemma 3.3 and Theorem 3.1 (it can readily be verified that 𝐻 satisfies the necessary Fredholm conditions) that there exists a residual set Θ⊂Γ such that if 𝛾∈Θ then the mapping 𝐻(⋅,𝛾)∶𝐴_𝑟+1→𝑍² is transverse, that is, (𝑓^𝛾,𝑔^𝛾)∈𝒯_𝑟. Since (𝑓,𝑔)∈ℱ²₁ was arbitrary, it follows from this that the set 𝒯_𝑟 is dense in ℱ²₁. Now let 𝒢=⋂^∞_𝑟=1𝒯_𝑟. By construction, 𝒢 is a residual subset of ℱ²₁ and, for any (𝑓,𝑔)∈𝒢, the mapping 𝐺(𝑓,𝑔)∶𝐸₀→𝑍² is transverse. It follows from 𝐻⁰_(𝑢,𝑣), and the results in [7], that 𝐷𝐺(𝑓,𝑔)(𝑧) is Fredholm with index 2. So from Theorem 3.1 the zero set of this mapping is a 2-dimensional, 𝐶^𝑘 manifold. Thus we have proved Theorem 3.2.

Remark 3.4. For more information about generic result or nodal solutions, see [8–12].

Acknowledgments

This paper is supported by the NNSF of China (nos. 11101335, 11201378, and 11261052), Gansu Provincial Department of Education Fund (no. 1101-02), and Project of Science and Technology Bureau of Lanzhou (no. 2011-2-72).