We consider a nonlinear equation 𝐹(𝜀,𝜆,𝑢)=0, where the parameter 𝜀 is a perturbation parameter, 𝐹 is a differentiable mapping from R×R×𝑋 to 𝑌, and 𝑋, 𝑌 are Banach spaces. We obtain an abstract bifurcation theorem by using the generalized saddle-node bifurcation theorem.

1. Introduction

In [1, 2], Crandall and Rabinowitz proved two celebrated theorems which are now regarded as foundation of the analytical bifurcation theory in infinite-dimensional spaces and both results are based on the implicit function theorem. In [3], we obtained the generalized saddle-node bifurcation theorem by the generalized inverse. In [4], we proved a perturbed problem using Morse Lemma. For a more general introduction to bifurcation theory and other related methods in nonlinear analysis, see, for example, [5–7]. On the other hand, [8–11] provide a more detailed introduction to mathematical models in some recent new results in the application of bifurcation theory including chemical reactions, population ecology, and nonautonomous differential equations.

In this paper, we continue the work of [3] and obtain an abstract bifurcation theorem under the opposite condition in [4]. We consider the solution set of 𝐹(𝜀,𝜆,𝑢)=0,(1.1) where 𝜀 indicates the perturbation. Fix 𝜀=𝜀0; let (𝜆0,𝑢0) be a solution of 𝐹(𝜀0,⋅,⋅)=0. From the implicit function theorem, a necessary condition for bifurcation is that 𝐹𝑢(𝜀0,𝜆0,𝑢0) is not invertible; we call (𝜀0,𝜆0,𝑢0) a degenerate solution. In [12], Shi shows the persistence and the bifurcation of degenerate solutions when 𝜀 varies near 𝜀0 by the implicit function theorem and the saddle-node bifurcation theorem. In this paper, we prove a new perturbed bifurcation theorem by the generalized saddle-node bifurcation theorem.

In the paper, we use ‖⋅‖ as the norm of Banach space 𝑋 and ⟨⋅,⋅⟩ as the duality pair of a Banach space 𝑋 and its dual space 𝑋∗. For a nonlinear operator 𝐹, we use 𝐹𝑢 as the partial derivative of 𝐹 with respect to argument 𝑢. For a linear operator 𝐿, we use 𝑁(𝐿) as the null space of 𝐿 and 𝑅(𝐿) as the range of 𝐿.

2. Preliminaries

Definition 2.1 (see [13]). Let 𝑋, 𝑌 be Banach spaces, and let 𝐴∈ℒ(𝑋,𝑌) be a linear operator. Then, 𝐴+∈ℒ(𝑌,𝑋) is called the generalized inverse of 𝐴 if it satisfies(i)𝐴𝐴+𝐴=𝐴,(ii)𝐴+𝐴𝐴+=𝐴+.

Definition 2.2 (see [13]). Let 𝑋,𝑌, and 𝐴 be the same as in Definition 2.1. If 𝐴∈ℒ(𝑋,𝑌) has the bounded linear generalized inverse 𝐴+, then 𝐴 is called a generalized regular operator.

Lemma 2.3 (see [13]). Let 𝐴∈ℒ(𝑋,𝑌), then 𝐴 is a generalized regular operator if and only if 𝑁(𝐴),𝑅(𝐴) are topologically complemented in 𝑋,𝑌, respectively. In this case, 𝐼−𝐴+𝐴, 𝐴𝐴+ are bounded linear projectors from 𝑋, 𝑌 into 𝑁(𝐴), 𝑅(𝐴), respectively.

We recall the generalized saddle-node bifurcation in [3] and give an alternate proof here using the generalized Lyapunov-Schmidt reduction.

Theorem 2.4 (generalized saddle-node bifurcation). Let 𝑉⊂𝐑×𝑋 be a neighborhood of (𝜆0,𝑢0),𝐹∈𝐶1(𝑉,𝑌). Suppose that (i)𝐹(𝜆0,𝑢0)=0;(ii)𝐹𝑢(𝜆0,𝑢0)∶𝑋→𝑌 is a generalized regular operator, and 𝐹dim𝑁𝑢𝜆0,𝑢0𝐹≥codim𝑅𝑢𝜆0,𝑢0=1,(2.1)(iii)𝐹𝜆(𝜆0,𝑢0)∉𝑅(𝐹𝑢(𝜆0,𝑢0)).
Let 𝑍=𝑅((𝐹𝑢(𝜆0,𝑢0))+), then the subset {(𝜆,𝑢)|𝐹(𝜆,𝑢)=0} contains the curve (𝜆(𝑠),𝑢(𝑠))=(𝜆(𝑠),𝑢0+𝑠𝑤0+𝑧(𝑠)) near (𝜆0,𝑢0), where 𝑤0∈𝑁(𝐹𝑢(𝜆0,𝑢0))⧵{𝜃}, the mapping 𝑧(𝑠) is continuously differentiable near 𝑠=0, and 𝜆(0)=𝜆0,ğœ†î…ž(0)=0,ğ‘§î…ž(0)=𝑧(0)=𝜃.

Proof. Since 𝐴=𝐹𝑢(𝜆0,𝑢0) is a generalized regular operator, there exist closed subspaces 𝑍 in 𝑋, 𝑌1 in 𝑌 satisfing 𝑋=𝑍⊕𝑁(𝐴), 𝑌=𝑅(𝐴)⊕𝑌1.
Taking an arbitrary 𝑤0∈𝑁(𝐴)⧵{𝜃}, from Lemma 2.3, 𝐹(𝜆,𝑢)=0 is equivalent to 𝐼−𝐴𝐴+𝐹𝜆,𝑢0+𝑠𝑤0+𝑧=0,𝐴𝐴+𝐹𝜆,𝑢0+𝑠𝑤0+𝑧=0,(2.2) where 𝑠∈𝐑, 𝑧∈𝑍.
Define 𝐺∶𝐑×𝐑×𝑍→𝑅(𝐴) as 𝐺(𝑠,𝜆,𝑧)=𝐴𝐴+𝐹𝜆,𝑢0+𝑠𝑤0,𝐺+𝑧(𝜆,𝑧)0,𝜆0[(],0𝜏,𝜓)=𝐴𝐴+𝜏𝐹𝜆𝜆0,𝑢0+𝐹𝑢𝜆0,𝑢0[𝜓],=𝐴𝐴+𝐴[𝜓][𝜓],=𝐴(2.3) because of (iii), then 𝐺(𝜆,𝑧)(0,𝜆0,0)∶𝑅×𝑍→𝑅(𝐴) is an isomorphism.
For the equation 𝐺(𝑠,𝜆,𝑧)=0, by the implicit function theorem, there exist 𝜀>0 and (𝜆(𝑠),𝑧(𝑠))∈𝐶1(−𝜀,𝜀), with 𝜆(0)=𝜆0, 𝑧(0)=0 satisfying 𝐺(𝑠,𝜆(𝑠),𝑧(𝑠))=0.(2.4) From (2.2), we have 𝐹𝜆(𝑠),𝑢0+𝑠𝑤0+𝑧(𝑠)=0,𝑠∈(−𝜀,𝜀).(2.5) Differentiating (2.5) with respect to 𝑠, we have 𝐹𝜆𝜆(𝑠),𝑢0+𝑠𝑤0𝜆+𝑧(𝑠)(𝑠)+𝐹𝑢𝜆(𝑠),𝑢0+𝑠𝑤0𝑤+𝑧(𝑠)0+ğ‘§î…žî€»(𝑠)=0.(2.6) Setting 𝑠=0, 𝐹𝜆𝜆0,𝑢0î€¸ğœ†î…ž(0)+𝐹𝑢𝜆0,𝑢0𝑤0+ğ‘§î…žî€»(0)=0.(2.7) Thus, ğœ†î…ž(0)=0 since (iii) and we have 𝐹𝑢𝜆0,𝑢0ğ‘§î€¸î€ºî…žî€»(0)=0,(2.8) that is, ğ‘§î…ž(0)∈𝑁(𝐴)∩𝑍, we have ğ‘§î…ž(0)=0.

Corollary 2.5. Assume the conditions in Theorem 2.4 are satisfied and 𝐹dim𝑁𝑢𝜆0,𝑢0𝐹=𝑛,𝑁𝑢𝜆0,𝑢0𝑤=span1,𝑤2,…,𝑤𝑛,(2.9) then the direction of the solution curves is determined by ğœ†ğ‘–î…žî…žî«(0)=−𝑙,𝐹𝑢𝑢𝜆0,𝑢0𝑤𝑖,𝑤𝑖𝑙,𝐹𝜆𝜆0,𝑢0,(2.10) where 𝑙∈𝑅(𝐹𝑢(𝜆0,𝑢0))⟂, 𝑖=1,2,…,𝑛. Furthermore, when 𝐹𝑢𝑢𝜆0,𝑢0𝑤𝑖,𝑤𝑖𝐹∉𝑅𝑢𝜆0,𝑢0(2.11) is satisfied, ğœ†ğ‘–î…žî…ž(0)≠0, and the solution curve {(𝜆𝑖(𝑠),𝑢𝑖(𝑠))∶|𝑠|<𝛿} is a parabola-like curve which reaches an extreme point at (𝜆0,𝑢0).

We illustrate our result by a simple example.

Example 2.6. Define 𝐹𝑥𝑦𝜆,=𝜆−𝑥2−𝑦2=0,(2.12) where 𝑈=𝑥𝑦∈𝐑2, 𝜆∈𝐑. From simple calculations, we obtain 𝐹𝑈=−2𝑥,−2𝑦,𝐹𝜆=1,𝐹𝑈𝑈=−200−2.(2.13) We analyze the bifurcation at (0,00). It is easy to see that 𝑁(𝐹𝑈)=span{𝑤1,𝑤2}, where 𝑤1=10, 𝑤2=01, 𝑅(𝐹𝑈)={0}. So, obviously, dim𝑁(𝐹𝑈)=2, codim𝑅(𝐹𝑈)=1, and 𝐹𝜆∉𝑅(𝐹𝑈). From the above calculation, 𝐹𝑈𝑈𝑤1,𝑤1=−2,𝐹𝑈𝑈𝑤2,𝑤2=−2.(2.14) Obviously, 𝐹𝑈𝑈(0,00)[𝑤𝑖,𝑤𝑖]∉𝑅(𝐹𝑈(0,00)) and ğœ†ğ‘–î…žî…ž(0)=−2, 𝑖=1,2. Thus, we can apply Corollary 2.5 to (2.12). In fact, all solution curves for all 𝑤𝑖∈𝑁(𝐹𝑈) form a surface (see Figure 1).

3. Main Theorems

Applying Theorem 2.4, we discuss the bifurcation of solutions of the perturbed problem. We consider the solution set of 𝐹(𝜀,𝜆,𝑢)=0,(3.1) where the parameter 𝜀 indicates the perturbation, 𝐹∈𝐶1(𝑀,𝑌), 𝑀≡𝐑×𝐑×𝑋, and 𝑋, 𝑌 are Banach spaces. Let 𝐹𝐻(𝜀,𝜆,𝑢,𝑤)=𝐹(𝜀,𝜆,𝑢)𝑢[𝑤](𝜀,𝜆,𝑢).(3.2)

Suppose that (𝜀0,𝜆0,𝑢0,𝑤0) is a solution of 𝐻(𝜀,𝜆,𝑢,𝑤)=0. For (𝜀0,𝜆0,𝑢0)∈𝑀 and 𝑤0∈𝑋1≡{𝑥∈𝑋∶‖𝑥‖=1},(3.3) by Hahn-Banach theorem, there exists a closed subspace 𝑋3 of 𝑋 with codimension 1 such that 𝑋=𝐿(𝑤0)⊕𝑋3, where 𝐿(𝑤0)=span{𝑤0} and 𝑑(𝑤0,𝑋3)=inf{||𝑤0−𝑥||∶𝑥∈𝑋3}>0. Let 𝑋2=𝑤0+𝑋3={𝑤0+𝑥∶𝑥∈𝑋3}. Then, 𝑋2 is a closed hyperplane of 𝑋 with codimension 1. Since 𝑋3 is a closed subspace of 𝑋 and 𝑋3 is also a Banach space in the subspace topology, Hence we can regard 𝑀1=𝑀×𝑋2 as a Banach space with product topology. Moreover, the tangent space of 𝑀1 is homeomorphic to 𝑀×𝑋3 (see [12] for more on the setting).

In the following, we will still use the conditions (𝐹𝑖) on 𝐹 defined in [12].(F1)dim𝑁(𝐹𝑢(𝜀0,𝜆0,𝑢0))=codim𝑅(𝐹𝑢(𝜀0,𝜆0,𝑢0))=1, and 𝑁(𝐹𝑢(𝜀0,𝜆0,𝑢0))=span{𝑤0};(F2)𝐹𝜆(𝜀0,𝜆0,𝑢0)∉𝑅(𝐹𝑢(𝜀0,𝜆0,𝑢0));(F3)𝐹𝜆𝑢(𝜀0,𝜆0,𝑢0)[𝑤0]∉𝑅(𝐹𝑢(𝜀0,𝜆0,𝑢0));(F4)𝐹𝑢𝑢(𝜀0,𝜆0,𝑢0)[𝑤0,𝑤0]∉𝑅(𝐹𝑢(𝜀0,𝜆0,𝑢0));(F5)𝐹𝜀(𝜀0,𝜆0,𝑢0)∉𝑅(𝐹𝑢(𝜀0,𝜆0,𝑢0)).

We use the convention that (ğ¹ğ‘–î…ž) means that the condition defined in (𝐹𝑖) does not hold.

Theorem 3.1. Let 𝐹∈𝐶2(𝑀,𝑌), 𝑇0=(𝜀0,𝜆0,𝑢0,𝑤0)∈𝑀1 such that 𝐻(𝑇0)=(0,0). Suppose that the operator 𝐹 satisfies (𝐹1), (𝐹2), (𝐹3), (𝐹4), and (𝐹5) at 𝑇0. One also assumes that 𝐹𝑢𝑢𝜀0,𝜆0,𝑢0𝑣1,𝑤0+𝐹𝜆𝑢𝜀0,𝜆0,𝑢0𝑤0𝐹∈𝑅𝑢𝜀0,𝜆0,𝑢0,(3.4) where 𝑣1∈𝑋3⧵{0} is the unique solution of 𝐹𝜆𝜀0,𝜆0,𝑢0+𝐹𝑢𝜀0,𝜆0,𝑢0[𝑣]=0.(3.5) Then, there exists 𝛿>0 such that all the solutions of 𝐻(𝜀,𝜆,𝑢,𝑤)=(0,0) near 𝑇0 form two 𝐶2 curves: 𝑇1𝜀(𝑠)=1(𝑠),𝜆1(𝑠),𝑢1(𝑠),𝑤1,𝑇(𝑠),𝑠∈𝐼=(−𝛿,𝛿)2(𝜀𝑠)=2(𝑠),𝜆2(𝑠),𝑢2(𝑠),𝑤2(,𝑠),𝑠∈𝐼=(−𝛿,𝛿)(3.6) where 𝜀𝑖(𝑠)=𝜀0+𝜏𝑖(𝑠), 𝑠∈𝐼; 𝜏𝑖(⋅)∈𝐶2(𝐼,𝐑); 𝜏𝑖(0)=ğœî…žğ‘–(0)=0, and 𝜆1(𝑠)=𝜆0+𝑧11(𝑠),𝜆2(𝑠)=𝜆0+𝑠+𝑧21𝑢(𝑠),𝑠∈𝐼,1(𝑠)=𝑢0+𝑠𝑤0+𝑧12(𝑠),𝑢2(𝑠)=𝑢0+𝑠𝑣1+𝑧22𝑤(𝑠),𝑠∈𝐼,1(𝑠)=𝑤0+𝑠𝜓0+𝑧13(𝑠),𝑤2(𝑠)=𝑤0+𝑠𝜓1+𝑧23(𝑠),𝑠∈𝐼,(3.7) where 𝑧𝑖𝑗(⋅)∈𝐶2(𝐼,𝑍), 𝑧𝑖𝑗(0)=ğ‘§î…žğ‘–ğ‘—(0)=0  (𝑖=1,2, 𝑗=1,2,3), 𝜓0∈𝑋3, 𝜓1∈𝑋3 are the unique solution of 𝐹𝑢𝑢𝜀0,𝜆0,𝑢0𝑤0,𝑤0+𝐹𝑢𝜀0,𝜆0,𝑢0[𝜓]𝐹=0,(3.8)𝑢𝑢𝜀0,𝜆0,𝑢0𝑣1,𝑤0+𝐹𝜆𝑢𝜀0,𝜆0,𝑢0𝑤0+𝐹𝑢𝜀0,𝜆0,𝑢0[𝜓]=0,(3.9) respectively.

Remark 3.2. Theorem 2.4 complements Theorem  3.2 in [4], where the opposite condition (3.4) is imposed.

Proof. We apply Theorem 2.4 to the operator 𝐻, so we need to verify all the conditions. We define a differential operator 𝐾∶𝐑×𝑋×𝑋3→𝑌×𝑌,  𝐾[]𝜏,𝑣,𝜓=𝐻(𝜆,𝑢,𝑤)𝜀0,𝜆0,𝑢0,𝑤0[]=𝜏,𝑣,𝜓𝜏𝐹𝜆𝜀0,𝜆0,𝑢0+𝐹𝑢𝜀0,𝜆0,𝑢0[𝑣]𝜏𝐹𝜆𝑢𝜀0,𝜆0,𝑢0𝑤0+𝐹𝑢𝑢𝜀0,𝜆0,𝑢0𝑣,𝑤0+𝐹𝑢𝜀0,𝜆0,𝑢0[𝜓].(3.10)
(1) dim𝑁(𝐾)=2. Suppose that (𝜏,𝑣,𝜓)∈𝑁(𝐾) and (𝜏,𝑣,𝜓)≠0. If 𝜏=0, from 𝐹𝑢(𝜀0,𝜆0,𝑢0)[𝑣]=0 and (𝐹1), then we have 𝑣=𝑘𝑤0 and 𝑘𝐹𝑢𝑢𝜀0,𝜆0,𝑢0𝑤0,𝑤0+𝐹𝑢𝜀0,𝜆0,𝑢0[𝜓]=0.(3.11) From (𝐹4), we can define 𝜓0∈𝑋3 is the unique solution of (3.8). Thus, (0,𝑤0,𝜓0)∈𝑁(𝐾) and (𝜏,𝑣,𝜓)=𝑘(0,𝑤0,𝜓0).
Next, we consider 𝜏≠0. Without loss of generality, we assume that 𝜏=1. Notice that 𝐹𝜆(𝜀0,𝜆0,𝑢0)∈𝑅(𝐹𝑢(𝜀0,𝜆0,𝑢0)) from (𝐹2), we can define that 𝑣1∈𝑋3⧵{0} is unique solution of (3.5). Substituting 𝜏=1, 𝑣=𝑣1 into (3.10), we have 𝐹𝜆𝑢𝜀0,𝜆0,𝑢0𝑤0+𝐹𝑢𝑢𝜀0,𝜆0,𝑢0𝑣1,𝑤0+𝐹𝑢𝜀0,𝜆0,𝑢0[𝜓]=0.(3.12) From (3.4), there exists a unique 𝜓1∈𝑋3 satisfies (3.9). Then, 𝑁(𝐾)=span0,𝑤0,𝜓0,1,𝑣1,𝜓1,(3.13) that is, dim𝑁(𝐾)=2.
(2) codim𝑅(𝐾)=1. We only claim that 𝑅𝐹(𝐾)=𝑅𝑢𝜀0,𝜆0,𝑢0×𝑌.(3.14) Let (ℎ,𝑔)∈𝑅(𝐾) and (𝜏,𝑣,𝜓)∈𝐑×𝑋×𝑋3 satisfy 𝜏𝐹𝜆𝜀0,𝜆0,𝑢0+𝐹𝑢𝜀0,𝜆0,𝑢0[𝑣]=ℎ,(3.15)𝜏𝐹𝜆𝑢𝜀0,𝜆0,𝑢0𝑤0+𝐹𝑢𝑢𝜀0,𝜆0,𝑢0𝑣,𝑤0+𝐹𝑢𝜀0,𝜆0,𝑢0[𝜓]=𝑔.(3.16) Using (3.15) and (𝐹2), then (ℎ,𝑔)∈𝑅(𝐹𝑢(𝜀0,𝜆0,𝑢0))×𝑌 and 𝑅(𝐾)⊂𝑅(𝐹𝑢(𝜀0,𝜆0,𝑢0))×𝑌.
Conversely, for any (ℎ,𝑔)∈𝑅(𝐹𝑢(𝜀0,𝜆0,𝑢0))×𝑌, from (𝐹3), set 𝜏1=⟨𝑙,𝑔⟩𝑙,𝐹𝜆𝑢𝜀0,𝜆0,𝑢0𝑤0,(3.17) where 𝑙∈𝑅(𝐹𝑢(𝜀0,𝜆0,𝑢0))⟂⊂𝑌∗. From (𝐹2), we have â„Žâˆ’ğœ1𝐹𝜆𝜀0,𝜆0,𝑢0𝐹∈𝑅𝑢𝜀0,𝜆0,𝑢0.(3.18) Set 𝑣2=[𝐹𝑢|𝑋3]−1[â„Žâˆ’ğœ1𝐹𝜆(𝜀0,𝜆0,𝑢0)]∈𝑋3, we obtain that 𝜏1𝐹𝜆𝜀0,𝜆0,𝑢0+𝐹𝑢𝜀0,𝜆0,𝑢0𝑣2=ℎ.(3.19) Substituting 𝜏=𝜏1, 𝑣=𝑣2 into (3.16), we have 𝜏1𝐹𝜆𝑢𝜀0,𝜆0,𝑢0𝑤0+𝐹𝑢𝑢𝜀0,𝜆0,𝑢0𝑣2,𝑤0+𝐹𝑢𝜀0,𝜆0,𝑢0[𝜓]=𝑔.(3.20) Using (𝐹1), (𝐹3), then there exists 𝑣3∈𝑋3, 𝜏2∈𝐑 satisfies 𝐹𝑢𝑢𝜀0,𝜆0,𝑢0𝑣2,𝑤0=𝜏2𝐹𝜆𝑢𝜀0,𝜆0,𝑢0𝑤0+𝐹𝑢𝜀0,𝜆0,𝑢0𝑣3.(3.21) Substituting (3.21) into (3.20), we have 𝜏1+𝜏2𝐹𝜆𝑢𝜀0,𝜆0,𝑢0𝑤0+𝐹𝑢𝜀0,𝜆0,𝑢0𝜓+𝑣3=𝑔.(3.22) Applying 𝑙 to (3.22), we have 𝜏2=0 because of the definition of 𝜏1 and 𝑔−𝜏1𝐹𝜆𝑢𝜀0,𝜆0,𝑢0𝑤0𝐹∈𝑅𝑢𝜀0,𝜆0,𝑢0.(3.23) Thus we can define 𝜓2=𝐹𝑢∣𝑋3−1𝑔−𝜏1𝐹𝜆𝑢𝜀0,𝜆0,𝑢0𝑤0−𝑣3∈𝑋3.(3.24) Therefore, 𝐾(𝜏1,𝑣2,𝜓2)=(ℎ,𝑔), that is, 𝑅(𝐹𝑢(𝜀0,𝜆0,𝑢0))×𝑌⊂𝑅(𝐾). Hence, 𝑅(𝐾)=𝑅(𝐹𝑢(𝜀0,𝜆0,𝑢0))×𝑌. That is, codim𝑅(𝐾)=1.
(3) 𝐻𝜀(𝜀0,𝜆0,𝑢0,𝑤0)∉𝑅(𝐾). Since 𝑅(𝐾)=𝑅(𝐹𝑢(𝜀0,𝜆0,𝑢0))×𝑌, we need only to show that 𝐹𝜀(𝜀0,𝜆0,𝑢0)∉𝑅(𝐹𝑢(𝜀0,𝜆0,𝑢0)) but that is exactly assumed in (𝐹5). So, the statement of the theorem follows from Theorem 2.4.

4. Calculations of Bifurcation Directions

In Theorem 3.1, we have 𝜀1(0)=𝜀2(0)=𝜀0, ğœ€î…ž1(0)=ğœ€î…ž2(0)=0, 𝜆1(0)=𝜆2(0)=𝜆0, 𝑢1(0)=𝑢2(0)=𝑢0, 𝑤1(0)=𝑤2(0)=𝑤0, ğœ†î…ž1(0)=0, ğ‘¢î…ž1(0)=𝑤0, ğ‘¤î…ž1(0)=𝜓0, ğœ†î…ž2(0)=1, ğ‘¢î…ž2(0)=𝑣1, ğ‘¤î…ž2(0)=𝜓1.

To completely determine the turning direction of curve of degenerate solutions, we need some calculations.

Let {𝑇𝑖(𝑠)=(𝜀𝑖(𝑠),𝜆𝑖(𝑠),𝑢𝑖(𝑠),𝑤𝑖(𝑠))∶𝑠∈(−𝛿,𝛿)} be a curve of degenerate solutions which we obtain in Theorem 3.1. Differentiating 𝐻(𝜀𝑖(𝑠),𝜆𝑖(𝑠),𝑢𝑖(𝑠),𝑤𝑖(𝑠))=0 with respect to 𝑠, we obtain ğ¹ğœ€ğœ€î…žğ‘–(𝑠)+ğ¹ğœ†ğœ†î…žğ‘–(𝑠)+ğ¹ğ‘¢î€ºğ‘¢î…žğ‘–î€»ğ¹(𝑠)=0,𝜀𝑢𝑤𝑖(𝜀𝑠)î…žğ‘–(𝑠)+𝐹𝜆𝑢𝑤𝑖(𝜆𝑠)î…žğ‘–(𝑠)+𝐹𝑢𝑢𝑤𝑖(𝑠),ğ‘¢î…žğ‘–(𝑠)+ğ¹ğ‘¢î€ºğ‘¤î…žğ‘–(𝑠)=0.(4.1) Setting 𝑠=0 in (4.1), we get exactly 𝐹𝑢[𝑤0]=0, (3.5), (3.8), and (3.9). We differentiate (4.1) again, and we have (omit the subscript 𝑖 in the equation) ğ¹ğœ€ğœ€î€ºğœ€î…žî€»(𝑠)2+ğ¹ğœ€ğœ€î…žî…ž(𝑠)+ğ¹ğœ†ğœ†î€ºğœ†î…žî€»(𝑠)2+ğ¹ğœ†ğœ†î…žî…ž(𝑠)+ğ¹ğ‘¢ğ‘¢î€ºğ‘¢î…ž(𝑠),ğ‘¢î…žî€»(𝑠)+ğ¹ğ‘¢î€ºğ‘¢î…žî…žî€»(𝑠)+2ğ¹ğœ€ğœ†ğœ€î…ž(𝑠)ğœ†î…ž(𝑠)+2ğ¹ğœ€ğ‘¢î€ºğ‘¢î…žî€»ğœ€(𝑠)(𝑠)+2ğ¹ğœ†ğ‘¢î€ºğ‘¢î…žî€»ğœ†(𝑠)î…žğ¹(𝑠)=0,(4.2)𝜀𝜀𝑢[]𝜀𝑤(𝑠)(𝑠)2+𝐹𝜀𝜆𝑢[]𝜀𝑤(𝑠)(𝑠)ğœ†î…ž(𝑠)+ğ¹ğœ€ğ‘¢ğ‘¢î€ºğ‘¢î…žî€»ğœ€(𝑠),𝑤(𝑠)(𝑠)+ğ¹ğœ€ğ‘¢î€ºğ‘¤î…žî€»ğœ€(𝑠)(𝑠)+𝐹𝜀𝑢[]𝜀𝑤(𝑠)(𝑠)+𝐹𝜀𝜆𝑢[]𝜀𝑤(𝑠)(𝑠)ğœ†î…ž(𝑠)+𝐹𝜆𝜆𝑢[]𝜆𝑤(𝑠)(𝑠)2+ğ¹ğœ†ğ‘¢ğ‘¢î€ºğ‘¢î…žî€»ğœ†(𝑠),𝑤(𝑠)(𝑠)+ğ¹ğœ†ğ‘¢î€ºğ‘¤î…žî€»ğœ†(𝑠)(𝑠)+𝐹𝜆𝑢[]𝜆𝑤(𝑠)(𝑠)+ğ¹ğœ€ğ‘¢ğ‘¢î€ºğ‘¢î…žî€»(𝑠),𝑤(𝑠)𝜀′(𝑠)+ğ¹ğœ†ğ‘¢ğ‘¢î€ºğ‘¢î…žî€»ğœ†(𝑠),𝑤(𝑠)(𝑠)+ğ¹ğ‘¢ğ‘¢ğ‘¢î€ºğ‘¢î…ž(𝑠),ğ‘¢î…žî€»(𝑠),𝑤(𝑠)+ğ¹ğ‘¢ğ‘¢î€ºğ‘¤î…ž(𝑠),ğ‘¢î…žî€»(𝑠)+𝐹𝑢𝑢𝑤(𝑠),ğ‘¢î…žî…žî€»(𝑠)+ğ¹ğœ€ğ‘¢î€ºğ‘¤î…žî€»ğœ€(𝑠)(𝑠)+ğ¹ğœ†ğ‘¢î€ºğ‘¤î…žî€»ğœ†(𝑠)(𝑠)+ğ¹ğ‘¢ğ‘¢î€ºğ‘¤î…ž(𝑠),ğ‘¢î…žî€»(𝑠)+ğ¹ğ‘¢î€ºğ‘¤î…žî…žî€»ğ¹(𝑠)=0,(4.3)𝜀𝜀𝑢[]𝜀𝑤(𝑠)(𝑠)2+𝐹𝜀𝑢[]𝜀𝑤(𝑠)(𝑠)+𝐹𝜆𝑢[]𝜆𝑤(𝑠)(𝑠)+𝐹𝜆𝜆𝑢[]𝜆𝑤(𝑠)(𝑠)2+ğ¹ğ‘¢ğ‘¢ğ‘¢î€ºğ‘¢î…ž(𝑠),ğ‘¢î…žî€»(𝑠),𝑤(𝑠)+𝐹𝑢𝑢𝑤(𝑠),ğ‘¢î…žî…žî€»(𝑠)+ğ¹ğ‘¢î€ºğ‘¤î…žî…žî€»(𝑠)+2𝐹𝜀𝜆𝑢[]𝜀𝑤(𝑠)(𝑠)ğœ†î…ž(𝑠)+2ğ¹ğœ€ğ‘¢ğ‘¢î€ºğ‘¢î…žî€»ğœ€(𝑠),𝑤(𝑠)(𝑠)+2ğ¹ğœ†ğ‘¢ğ‘¢î€ºğ‘¢î…žî€»ğœ†(𝑠),𝑤(𝑠)(𝑠)+2ğ¹ğœ€ğ‘¢î€ºğ‘¤î…žî€»ğœ€(𝑠)(𝑠)+2ğ¹ğœ†ğ‘¢î€ºğ‘¤î…žî€»ğœ†(𝑠)(𝑠)+2ğ¹ğ‘¢ğ‘¢î€ºğ‘¤î…ž(𝑠),ğ‘¢î…žî€»(𝑠)=0.(4.4)

Setting 𝑠=0 in (4.2), we obtain 𝐹𝜀𝜀1(0)+𝐹𝜆𝜆1(0)+𝐹𝑢𝑢𝑤0,𝑤0+𝐹𝑢𝑢1(0)=0,(4.5)𝐹𝜀𝜀2(0)+𝐹𝜆𝜆+𝐹𝜆𝜆2(0)+𝐹𝑢𝑢𝑣1,𝑣1+𝐹𝑢𝑢2(0)+2𝐹𝜆𝑢𝑣1=0.(4.6) And applying 𝑙 to it, we have 𝜀1(0)=0,(4.7)𝜀2(0)=−𝑙,𝐹𝜆𝜆+𝐹𝑢𝑢𝑣1,𝑣1+2𝐹𝜆𝑢𝑣1⟨𝑙,𝐹𝜀⟩,(4.8) Using (𝐹2), (𝐹4), (𝐹5). From (4.7), (4.5) implies 𝑢1(0)=𝜆1(0)𝑣1+𝜓0+𝑘𝑤0. Setting 𝑠=0 in (4.4), 𝐹𝜆𝑢𝑤0𝜆1(0)+𝐹𝑢𝑢𝑢𝑤0,𝑤0,𝑤0+𝐹𝑢𝑢𝑤0,𝑢1(0)+𝐹𝑢𝑤1(0)+2𝐹𝑢𝑢𝜓0,𝑤0𝐹=0,(4.9)𝜀𝑢𝑤0𝜀2(0)+𝐹𝜆𝑢𝑤0𝜆2(0)+𝐹𝜆𝜆𝑢𝑤0+𝐹𝑢𝑢𝑢𝑣1,𝑣1,𝑤0+𝐹𝑢𝑢𝑤0,𝑢2(0)+𝐹𝑢𝑤2(0)+2𝐹𝜆𝑢𝑢𝑣1,𝑤0+2𝐹𝜆𝑢𝜓1+2𝐹𝑢𝑢𝜓1,𝑣1=0.(4.10) Substituting the expression of 𝑢1(0) into (4.9), we have 𝜆1î…žî…žî€·ğ¹(0)𝜆𝑢𝑤0+𝐹𝑢𝑢𝑣1,𝑤0+3𝐹𝑢𝑢𝜓0,𝑤0+𝐹𝑢𝑢𝑢𝑤0,𝑤0,𝑤0+𝑘𝐹𝑢𝑢𝑤0,𝑤0+𝐹𝑢𝑤1(0)=0.(4.11) And applying 𝑙 to it, we obtain ⟨𝑙,𝐹𝑢𝑢𝑢[𝑤0,𝑤0,𝑤0]+3𝐹𝑢𝑢[𝜓0,𝑤0]⟩=0, that is, 𝐹𝑢𝑢𝑢𝑤0,𝑤0,𝑤0+3𝐹𝑢𝑢𝜓0,𝑤0𝐹∈𝑅𝑢𝜀0,𝜆0,𝑢0.(4.12) Assume 𝐹𝜀𝑢[𝑤0]∈𝑅(𝐹𝑢(𝜀0,𝜆0,𝑢0)) and applying 𝑙 to (4.10), 𝜆2(0)=−𝑙,𝐹𝜆𝜆𝑢𝑤0+𝐹𝑢𝑢𝑢𝑣1,𝑣1,𝑤0+𝐹𝑢𝑢𝑤0,𝑢2(0)+2𝐹𝜆𝑢𝑢𝑣1,𝑤0+2𝐹𝜆𝑢𝜓1+2𝐹𝑢𝑢𝜓1,𝑣1𝑙,𝐹𝜆𝑢𝑤0.(4.13) We differentiate (4.2) again: ğ¹ğœ€ğœ€ğœ€î€ºğœ€î…žî€»(𝑠)3+ğ¹ğœ€ğœ€î…žî…žî…ž(𝑠)+ğ¹ğœ†ğœ†ğœ†î€ºğœ†î…žî€»(𝑠)3+ğ¹ğœ†ğœ†î…žî…žî…ž(𝑠)+ğ¹ğ‘¢ğ‘¢ğ‘¢î€ºğ‘¢î…ž(𝑠),ğ‘¢î…ž(𝑠),ğ‘¢î…žî€»(𝑠)+ğ¹ğ‘¢î€ºğ‘¢î…žî…žî…žî€»(𝑠)+3ğ¹ğœ€ğœ€ğœ€î…ž(𝑠)ğœ€î…žî…ž(𝑠)+3ğ¹ğœ†ğœ†ğœ†î…ž(𝑠)ğœ†î…žî…ž(𝑠)+3ğ¹ğ‘¢ğ‘¢î€ºğ‘¢î…žî…ž(𝑠),ğ‘¢î…žî€»(𝑠)+3ğ¹ğœ€ğœ†ğœ€î…žî…ž(𝑠)ğœ†î…ž(𝑠)+3ğ¹ğœ€ğœ†ğœ€î…ž(𝑠)ğœ†î…žî…ž(𝑠)+3ğ¹ğœ€ğ‘¢î€ºğ‘¢î…žî€»ğœ€(𝑠)(𝑠)+3ğ¹ğœ€ğ‘¢î€ºğ‘¢î…žî…žî€»ğœ€(𝑠)(𝑠)+3ğ¹ğœ†ğ‘¢î€ºğ‘¢î…žî…žî€»ğœ†(𝑠)(𝑠)+3ğ¹ğœ†ğ‘¢î€ºğ‘¢î…žî€»ğœ†(𝑠)(𝑠)+3ğ¹ğœ€ğœ†ğœ†ğœ€î…žî€·ğœ†(𝑠)(𝑠)2+3ğ¹ğœ€ğœ€ğœ†î€·ğœ€î…žî€¸(𝑠)2ğœ†î…ž(𝑠)+3ğ¹ğœ€ğœ€ğ‘¢î€ºğ‘¢î…žğœ€(𝑠)(𝑠)2+3ğ¹ğœ€ğ‘¢ğ‘¢î€ºğ‘¢î…ž(𝑠),ğ‘¢î…žî€»ğœ€(𝑠)(𝑠)+3ğ¹ğœ†ğœ†ğ‘¢î€ºğ‘¢î…žğœ†(𝑠)(𝑠)2+3ğ¹ğœ†ğ‘¢ğ‘¢î€ºğ‘¢î…ž(𝑠),ğ‘¢î…žî€»ğœ†(𝑠)(𝑠)+6ğ¹ğœ€ğœ†ğ‘¢î€ºğ‘¢î…žî€»ğœ€(𝑠)(𝑠)ğœ†î…ž(𝑠)=0.(4.14) Setting 𝑠=0 in (4.14), we obtain 𝐹𝜀𝜀1(0)+𝐹𝜆𝜆1(0)+𝐹𝑢𝑢𝑢𝑤0,𝑤0,𝑤0𝑢+𝐹𝑢1(0)+3𝐹𝑢𝑢𝑢1(0),𝑤0+3𝐹𝜆𝑢𝑤0𝜆1(𝐹0)=0,(4.15)𝜆𝜆2(0)+𝐹𝜀𝜀2(0)+3𝜆2î…žî…žî€·ğ¹(0)𝜆𝜆+𝐹𝜆𝑢𝑣1+3𝜀2î…žî…žî€·ğ¹(0)𝜀𝜆+𝐹𝜀𝑢𝑣1+𝐹𝜆𝜆𝜆+𝐹𝑢𝑢𝑢𝑣1,𝑣1,𝑣1+𝐹𝑢𝑢2(0)+3𝐹𝑢𝑢𝑢2(0),𝑣1+3𝐹𝜆𝑢𝑢2(0)+3𝐹𝜆𝜆𝑢𝑣1+3𝐹𝜆𝑢𝑢𝑣1,𝑣1=0.(4.16) Substituting the expression of 𝑢1(0) into (4.15) and applying 𝑙 to it, we have 𝜀1(0)=0 using (3.4), (4.12), and (𝐹5).


The authors the referee for very careful reading and helpful suggestions on the paper. The paper was partially supported by NSFC (Grant no. 11071051), Youth Science Foundation of Heilongjiang Province (Grant no. QC2009C73), Harbin Normal University academic backbone of youth project, and NCET of Heilongjiang Province of China (1251–NCET–002).