Multiple Solutions for Generalized Asymptotical Linear Hamiltonian Systems Satisfying Bolza Boundary Conditions

Abstract

This paper is devoted to multiple solutions of generalized asymptotical linear Hamiltonian systems satisfying Bolza boundary conditions. We classify the linear Hamiltonian systems by the index theory and obtain the existence and multiplicity of solutions for the Hamiltonian systems, based on an application of the classical symmetric mountain pass lemma.

1. Introduction

This paper is concerned with a classical problem on the existence of solutions for Hamiltonian systems when a more general form of twist condition between the origin and infinity holds for the Hamiltonian function. More precisely consider the system 𝑥̇=𝐽𝐻^′(),(0)(1),𝑡,𝑥𝑃𝑥=0=𝑃𝑥 (1) where 𝐻∈𝐶(𝐑×𝐑^2𝑛,𝐑), and 𝐻^′∈𝐶(𝐑×𝐑^2𝑛,𝐑^2𝑛)and 𝐻^′  satisfies the following:(𝐻₁)𝐻^′(𝑡,𝑥)=𝐵₀(𝑡,𝑥)𝑥+𝑜(|𝑥|), as  |𝑥|→0 uniformly in 𝑡∈[0,1],(𝐻₂)𝐻^′(𝑡,𝑥)=𝐵_∞(𝑡,𝑥)𝑥+𝑜(|𝑥|), as  |𝑥|→∞ uniformly in 𝑡∈[0,1], for 𝐵₀(𝑡,𝑥),  𝐵_∞∈𝐿^∞((0,1)×𝐑^2𝑛;GLs(𝐑^2𝑛)). Here and below, we use 𝐻^′ to denote the first derivative of 𝐻 with respect to 𝑥∈𝐑^2𝑛.

A quantitative way to measure the twisting is given by the Maslov-type index. As in [1, 2], an index for the second-order and first-order linear Hamiltonian systems was defined and developed in [3] for the study of linear operator equation with infinite Morse index. In [4, 5], by Conley and Zehnder and Long, an index theory for symplectic path was defined. We refer to two excellent books [6, 7] for systematical treatment.

In [2], Dong discussed the classification of the linear Hamiltonian systems with Bolza boundary conditions as follows: ̇(𝑡)(0)(1),𝑥=𝐽𝐵𝑥,𝑃𝑥=0=𝑃𝑥 (2) where 𝐵∈𝐿^∞((0,1);GLs(𝐑^2𝑛)) and 𝑃𝑥=(𝑥₁,…𝑥_𝑛)∈𝐑^𝑛, for any 𝑥=(𝑥₁,…,𝑥_2𝑛). That is, for any 𝐵∈𝐿^∞((0,1);GLs(𝐑^2𝑛)), he associated it with a pair of numbers (𝑖(𝐵),𝜈(𝐵))∈𝑍×{0,1,…,𝑛}. This pair of integers is called index and nullity of 𝐵, respectively. And he defined the nullity 𝜈(𝐵) as the dimension of the solution space of (1). Let 𝐸[𝛼] be the integer 𝑎∈𝐙 such that 𝑎<𝛼≤𝑎+1. So that 𝐸[𝑎]=𝑎−1 for all integer 𝑎 defined as in [6]. In order to process the definition of 𝑖(𝐵), he defined 𝑖(𝜆𝐼_2𝑛)=𝑛𝐸[𝜆/𝜋]. In particular, when 𝜆∈(𝑘𝜋,(𝑘+1)𝜋), 𝑖(𝜆𝐼_2𝑛)=𝑛𝑘. We will introduce the definitions and properties of 𝑖(𝐵) in detail in Section 2. After the discussion of the index theory, we will prove our main result in Section 3.

Throughout this paper, ‖𝑥‖_𝐶 denotes the usual norm in 𝐶[0,1]. For any 𝐴₁, 𝐴₂∈GLs(𝐑^2𝑛), we write 𝐴₁≤𝐴₂ if 𝐴₂−𝐴₁ is positively semidefinite, and we write 𝐴₁<𝐴₂ if 𝐴₂−𝐴₁ is positive definite. For any 𝐴₁, 𝐴₂∈𝐿^∞((0,1);GLs(𝐑^2𝑛)), we write 𝐴₁≤𝐴₂ if 𝐴₁(𝑡)≤𝐴₂(𝑡) for a.e. 𝑡∈(0,1), and we write 𝐴₁<𝐴₂ if 𝐴₁≤𝐴₂ and 𝐴₁(𝑡)<𝐴₂(𝑡) on a subset of (0,1) with a nonzero measure.

Remark 1. Let 𝐶₁, 𝐶₂∈(𝑘𝜋,(𝑘+1)𝜋), 𝐶₃, 𝐶₄∈((𝑘+1)𝜋,(𝑘+2)𝜋) with 𝐶₁<𝐶₂, 𝐶₃<𝐶₄. Assume that 𝐴_𝑖=𝐶_𝑖𝐼_2𝑛, 𝑖=1,2, and 𝐵_𝑖=𝐶_𝑖𝐼_2𝑛, 𝑖=3,4. Then, we have 𝑖(𝐴₁)=𝑖(𝐴₂)=𝑛𝑘, 𝑖(𝐵₁)=𝑖(𝐵₂)=𝑛(𝑘+1), and 𝜈(𝐴_𝑖)=𝜈(𝐵_𝑖)=0, for 𝑖=1,2; (1) has at least 𝑛 pairs of solutions.

We make use of the critical point theory [8, 9] to prove Theorem 8. The novelty of our result is that it suffices to assume that 𝐻(𝑡,𝑥)+(1/2)𝜇|𝑥|² is convex. The twisting between origin and infinity is reflected in (𝐻^∗₁) and (𝐻^∗₂). Thus, our results complement with Theorem 3.8 in [2] and Theorem 1.1 in [10]. For other results, we refer to [11–15].

This paper is organized as follows. In Section 2, we introduce some preliminaries including index theory, and establish the 𝜇-index theory which is needed in the proofs. In Section 3, we present the proofs of the results.

2. Index Theory for Linear Hamiltonian Systems Satisfying Bolza Boundary Value Conditions

First, we recall some definitions and propositions in [2]. We consider the following system: ̇(𝑡)(0)(1).𝑥=𝐽𝐵𝑥,𝑃𝑥=0=𝑃𝑥 (3)(4)Let 𝐻={𝑥∶[0,1]→𝐑^2𝑛|𝑥 is continuous on [0,1], satisfies (4), and 𝑥̇∈𝐿²} with the norm ‖𝑥‖_𝐻=(∫¹₀(|𝑥^′(𝑡)|²+|𝑥(𝑡)|²)𝑑𝑡)^1/2, (Λ₁𝑥)(𝑡)=𝐽𝑥^′(𝑡), (𝐵𝑥)(𝑡)=𝐵(𝑡)𝑥(𝑡), where 𝐿²∶=𝐿²((0,1);𝐑^2𝑛), and let |𝑥|=(∑^2𝑛_𝑖=1|𝑥_𝑖|)^1/2 for any 𝑥=(𝑥₁,…,𝑥_2𝑛)∈𝐑^2𝑛.    Then, Λ₁ and 𝐵 are self-adjoint operators, and 𝐵 is bounded.

Definition 2 (see [2, Definitions 2.1, 2.3, 2.4, and 2.7]). For any 𝐵₁,𝐵₂,𝐵∈𝐿^∞((0,1),GLs(𝐑^2𝑛)), one defines that(1)𝜈(𝐵)=dimker(Λ₁+𝐵), (2)𝐼(𝐵₁,𝐵₂)=∑_𝜆∈[0,1)𝜈((1−𝜆)𝐵₁+𝜆𝐵₂) for 𝐵₁<𝐵₂,(3)𝑖(𝐵)=𝑖(𝜆𝐼_2𝑛)−𝐼(𝐵,𝜆𝐼_2𝑛) for 𝜆∈𝐑 satisfies 𝐵<𝜆𝐼_2𝑛.

Using spectral theory, a Morse-type index 𝑖_𝜇(𝐵) was established in [2]. More precisely, for any 𝐵∈𝐿^∞((0,1),GLs(𝐑^2𝑛)), let 𝜇∈𝐑⧵𝜋𝐙 with 𝐵+𝜇𝐼_2𝑛≥𝐼_2𝑛. Then, 𝜈(−𝜇𝐼_2𝑛=0), Λ𝑥∶=𝐽𝑥̇(𝑡)−𝜇𝑥(𝑡) is invertible, and the inverse Λ⁻¹∶𝐿²→𝐿² is self-adjoint and compact. He put a quadratic form: 𝑞_𝜇,𝐵()=1𝑢,𝑢2¹₀Λ⁻¹𝑢(𝑡),𝑢(𝑡)+𝐶_𝜇(𝑡)𝑢(𝑡),𝑢(𝑡)𝑑𝑡∀𝑢∈𝐿²,𝐶_𝜇(𝑡)𝑢(𝑡),𝑢(𝑡)∶=¹₀𝐶_𝜇(𝑡)𝑢(𝑡),𝑢(𝑡), (5)(6) where 𝐶_𝜇(𝑡)∶=(𝜇𝐼_2𝑛+𝐵(𝑡))⁻¹. Then, (𝐶_𝜇(𝑡)𝑢(𝑡),𝑢(𝑡)) defines a Hilbert space structure on 𝐿². 𝐶_𝜇⁻¹Λ⁻¹ is a self-adjoint and compact operator under this interior product. By the spectral theory, there is a basis 𝑒_𝑗, 𝑗∈𝐍 of 𝐿², and a sequence 𝜆_𝑗→0 in 𝐑 such that 𝐶_𝜇(𝑡)𝑒_𝑖(𝑡),𝑒_𝑗(𝑡)=𝛿_𝑖𝑗,Λ⁻¹𝑒_𝑗,𝑢=𝐶_𝜇𝜆_𝑗𝑒_𝑗,𝑢∀𝑢∈𝐿². (7) For any 𝑢∈𝐿² as 𝑢=∑^∞_𝑗=1𝜉_𝑗𝑒_𝑗, (5) can be rewritten as follows: 𝑞_𝜇,𝐵()=1𝑢,𝑢2¹₀Λ⁻¹𝑢,𝑢+𝐶_𝜇(𝑡)=1𝑢,𝑢𝑑𝑡2^∞_𝑗=11+𝜆_𝑗𝜉²_𝑗. (8) Define that 𝐸⁻_𝜇(𝐵)∶=^∞_𝑗=1𝜉_𝑗𝑒_𝑗∣𝜉_𝑗=0if1+𝜆_𝑗,𝐸≥0_0𝜇(𝐵)∶=^∞_𝑗=1𝜉_𝑗𝑒_𝑗∣𝜉_𝑗=0if1+𝜆_𝑗,𝐸≠0_+𝜇(𝐵)∶=^∞_𝑗=1𝜉_𝑗𝑒_𝑗∣𝜉_𝑗=0if1+𝜆_𝑗≤0. (9)𝐸⁻_𝜇(𝐵), 𝐸⁰_𝜇(𝐵), and 𝐸⁺_𝜇(𝐵) are 𝑞_𝜇,𝐵-orthogonal, and 𝐸⁻_𝜇(𝐵)⊕𝐸⁰_𝜇(𝐵)⊕𝐸⁺_𝜇(𝐵)=𝐿². Since 𝜆_𝑗→0 as 𝑗→∞, 𝐸⁻_𝜇(𝐵) and 𝐸⁰_𝜇(𝐵) are two finite-dimensional subspaces.

Definition 3 (see [2, Definition 2.9]). For any 𝐵∈𝐿^∞((0,1),GLs(𝐑^2𝑛)), 𝜇∈𝐑 with 𝜇𝐼_2𝑛+𝐵≥𝐼_2𝑛, one defines that 𝜈_𝜇(𝐵)∶=dim𝐸⁰_𝜇(𝐵),𝑖_𝜇(𝐵)∶=dim𝐸⁻_𝜇(𝐵). (10)

One calls 𝜈_𝜇(𝐵) and 𝑖_𝜇(𝐵)  𝜇-nullity and 𝜇-index of 𝐵, respectively.

Proposition 4 (see [2, Propositions 2.10, 2.13]). The index, relative Morse index, and 𝜇-index have the following properties.(1) For any 𝐵₁, 𝐵₂, 𝐵∈𝐿^∞((0,1),𝐺𝐿𝑠(𝐑^2𝑛)), with 𝐵₁<𝐵₂, one has 𝜈_𝜇(𝐵)=𝜈(𝐵),𝐼𝐵₁,𝐵₂=𝑖_𝜇𝐵₂−𝑖_𝜇𝐵₁,𝑖_𝜇𝐵₁−𝑖_𝜇𝐵₂=𝑖𝐵₁−𝑖𝐵₂. (11)(2) There exists 𝜖₀>0, such that, for any 𝜖∈(0,𝜖₀], one gets 𝜈𝐵+𝜖𝐼_2𝑛=0=𝜈𝐵−𝜖𝐼_2𝑛,𝑖𝐵−𝜖𝐼_2𝑛=𝑖(𝐵),𝑖𝐵+𝜖𝐼_2𝑛=𝑖(𝐵)+𝜈(𝐵). (12) In particular, if 𝜈(𝐵)=0, one obtains 𝑖(𝐵+𝜖𝐼_2𝑛)=𝑖(𝐵) for 𝜖∈(0,𝜖₀].(3)𝑖_𝜇(𝐵)−𝑖(𝐵) is a constant for 𝐵 satisfying 𝐵+𝜇𝐼_2𝑛≥𝐼_2𝑛, that is, 𝑖_𝜇(𝐵)−𝑖(𝐵)=𝑖_𝜇(𝐵₁)−𝑖(𝐵₁) for any other 𝐵₁∈𝐿^∞((0,1),GLs(𝐑^2𝑛)) with 𝐵₁+𝜇𝐼_2𝑛≥𝐼_2𝑛. For any 𝜇>1, one has 𝑖_𝜇(0)𝜇=𝑛𝐸𝜋,𝑖_𝜇(𝐵)𝜇=𝑛𝐸𝜋+𝑛+𝑖(𝐵), (13) for any 𝐵∈𝐿^∞((0,1),GLs(𝐑^2𝑛)) with 𝐵+𝜇𝐼_2𝑛≥𝐼_2𝑛.

Remark 5. From this proposition, there is 𝜖₀>0, 𝜖∈(0,𝜖₀] such that 𝜈_𝜇(𝐵+𝜖𝐼_2𝑛)=0=𝜈_𝜇(𝐵−𝜖𝐼_2𝑛) and 𝑖_𝜇(𝐵−𝜖𝐼_2𝑛)=𝑖_𝜇(𝐵),𝑖_𝜇(𝐵+𝜖𝐼_2𝑛)=𝑖_𝜇(𝐵)+𝜈_𝜇(𝐵).

In order to prove Theorem 8, we make use of minimax arguments for the multiplicity of critical points in the presence of symmetry. We state two results of this type from [8, 9].

Lemma 6 (cf. Chang [9, Theorem 4.3.4]). Assume that  𝑓∈   𝐶¹(𝑋,𝐑¹) satisfies the (PS) condition, 𝑓(𝜃)=0, 𝑓(−𝑥)=𝑓(𝑥), and (1) there are an m-dimensional subspace 𝑋₁ and a constant 𝜌>0 such that sup_{𝑥∈𝑋1∩𝑆𝜌}𝑓(𝑥)<0, (14) (2) there is a j-dimensional subspace 𝑋₂ such that inf_{𝑥∈𝑋⟂2}𝑓(𝑥)>−∞, (15) where 𝑋^⟂₂ is a subset of 𝑋 such that 𝑋^⟂₂⊕𝑋₂=𝑋. Then, 𝑓 has at least 𝑚−𝑗 pairs of critical points if 𝑚−𝑗>0.

Lemma 7 (see [8, Theorem 2.2.8]). Suppose that 𝑓∈   𝐶^′(𝐻,𝐑) satisfies (C) condition, 𝑓(𝜃)≥0, even, and there exist two closed subspaces 𝐻⁺𝐻⁻ of 𝐻, with codim𝐻⁻<+∞, and two constants 𝑐_∞>𝑐₀≥𝑓(0) such that(a)𝑓(𝑢)≥𝑐₀ for all 𝑢∈𝑆_𝜌∩𝐻⁺,(b)𝑓(𝑢)≤𝑐_∞ for all 𝑢∈𝐻⁻.
Then, if dim𝐻⁻≥codim𝐻⁺, 𝑓 possesses at least 𝑚=dim𝐻⁻−codim𝐻⁺ distinct pairs of critical points whose corresponding critical values belong to [𝑐₀, 𝑐_∞].

3. Proof of Main Results

We state the main result in this paper. We further make the following assumptions. (𝐻^∗₁) There exists 𝐴₁, 𝐴₂∈𝐿^∞((0,1);GLs(𝐑^2𝑛)) such that 𝐴₁(𝑡)≤𝐵₀(𝑡,𝑥)≤𝐴₂(𝑡) with 𝑖(𝐴₁)=𝑖(𝐴₂), 𝜈(𝐴₂)=0. (𝐻^∗₂) There exists 𝐵₁, 𝐵₂∈𝐿^∞((0,1);GLs(𝐑^2𝑛)) such that 𝐵₁(𝑡)≤𝐵_∞(𝑡,𝑥)≤𝐵₂(𝑡) with 𝑖(𝐵₁)=𝑖(𝐵₂), 𝜈(𝐵₂)=0. (𝐻₃)𝐻(𝑡,𝜃)=0, 𝐻(𝑡,𝑥)=𝐻(𝑡,−𝑥). (𝐻₄)𝐻(𝑡,𝑥)+(1/2)𝜇|𝑥|² is convex.

Our main result reads as follows.

Theorem 8.  Assume that (𝐻₁), (𝐻^∗₁), (𝐻₂), (𝐻^∗₂), (𝐻₃), (𝐻₄)    are satisfied, then (1) has at least |𝑖(𝐴₁)−𝑖(𝐵₁)| pairs of solutions.

Theorem 9. If 𝑖(𝐴₁)>𝑖(𝐵₁), then (1) has at least 𝑖(𝐴₁)−𝑖(𝐵₁) pairs of solutions.
If 𝑖(𝐴₁)>𝑖(𝐵₁), let 𝜇∈𝐑⧵𝜋𝐙 with 𝐴₁+𝜇𝐼_2𝑛≥𝐼_2𝑛, 𝐵₁+𝜇𝐼_2𝑛≥𝐼_2𝑛. Recalling that Λ𝑥=𝐽𝑥̇−𝜇𝐼_2𝑛,  one denotes   𝐻_𝜇(𝑡,𝑥)=𝐻(𝑡,𝑥)+(1/2)𝜇|𝑥|². Then, one obtains 𝐻^∗_𝜇(𝑡,𝑥)=sup_{𝑦∈𝐑2𝑛}𝑥,𝑦−𝐻_𝜇𝑡,𝑦. (16) Thus, 𝐻^∗_𝜇(𝑡,−𝑥)=𝐻^∗(𝑡,𝑥) and 𝑥=𝐻_𝜇^∗′𝑡,𝑥^∗𝑖𝑓𝑓𝑥^∗=𝐻^′_𝜇(𝑡,𝑥) (17) (by the Fenchel conjugate formula; see [6, Proposition II]). Hence, |𝑥|→∞ if and only if |𝑥^∗|→∞. Consider the functional defined by 𝜓(𝑢)=¹₀12Λ⁻¹𝑢(𝑡),𝑢(𝑡)+𝐻^∗_𝜇(𝑡,𝑢(𝑡))𝑑𝑡∀𝑢∈𝐿². (18)

In order to prove Theorem 9, we need the following lemmas.

Lemma 10. 𝜓 satisfies the (PS) condition.

Proof. Assume that {𝑢_𝑗} is a sequence in 𝐿² such that 𝜓(𝑢_𝑗) is bounded and 𝜓^′(𝑢_𝑗)→0 in 𝐿². It suffices to prove that {𝑢_𝑗} has convergent sequence. By (18), we have 𝜓^′(𝑢),𝑣=¹₀Λ⁻¹𝑢(𝑡),𝑣(𝑡)+𝐻_𝜇^∗′(𝑡,𝑢(𝑡)),𝑣(𝑡)𝑑𝑡, (19) for all 𝑣∈𝐿². Hence, we get 𝜓^′𝑢_𝑗=Λ⁻¹𝑢_𝑗+𝐻_𝜇^∗′𝑡,𝑢_𝑗(𝑡)⟶𝜃,in𝐿². (20) Let 𝑥_𝑗=Λ⁻¹𝑢_𝑗, 𝑦_𝑗=𝑢_𝑗/‖𝑥_𝑗‖_𝐶, and 𝑧_𝑗=𝜓^′(𝑢_𝑗)−Λ⁻¹(𝑢_𝑗). If ‖𝑥_𝑗‖_𝐶→∞, then ‖𝑥_𝑗‖_𝐿²→∞. Joining (𝐻₂) with (17) and (20), we have 𝑢_𝑗(𝑡)=𝐻^′_𝜇𝑡,𝑧_𝑗(𝑡)=𝐵_∞𝑡,𝑧_𝑗(𝑡)+𝜇𝐼_2𝑛𝑧_𝑗(𝑡)+𝑜𝑧_𝑗(𝑡𝐵)=−_∞𝑡,𝑧_𝑗(𝑡)+𝜇𝐼_2𝑛Λ⁻¹𝑢_𝑗+𝐵_∞𝑡,𝑧_𝑗(𝑡)+𝜇𝐼_2𝑛𝜓^′𝑢_𝑗+𝑜𝑧_𝑗(𝑡). (21) Using the preceding notations, we have 𝑦_𝑗(𝑡)=−𝐵_∞𝑡,𝑧_𝑗(𝑡)+𝜇𝐼_2𝑛Λ⁻¹𝑦_𝑗+𝐵_∞𝑡,𝑧_𝑗(𝑡)+𝜇𝐼_2𝑛𝜓^′𝑢_𝑗𝑥_𝑗𝐶⁻¹+𝑜𝑧_𝑗(𝑡)𝑥_𝑗𝐶⁻¹. (22) So 𝑦_𝑗(𝑡) is bounded in 𝐿². We assume that 𝑦_𝑗⇀𝑦 in 𝐿², and hence Λ⁻¹𝑦_𝑗→Λ⁻¹𝑦. From (22), there exists 𝐵∈𝐿^∞((0,1);GLs(𝐑^2𝑛)), such that 𝐵_∞𝑡,𝑧_𝑗(𝑡)𝑣(𝑡)⇀𝐵(𝑡)𝑣(𝑡)forany𝑣∈𝐿₂. (23) And we have 𝐵₁(𝑡)≤𝐵(𝑡)≤𝐵₂(𝑡). Taking the limit as 𝑗→∞ in (22), we obtain 𝑦(𝑡)=−(𝐵(𝑡)+𝜇𝐼_2𝑛)Λ⁻¹𝑦. Let 𝑥=Λ⁻¹𝑦. We get 𝐽𝑥+𝐵(𝑡)𝑥=0,𝑃𝑥(0)=0=𝑃𝑥(1). (24)
By assumption (𝐻^∗₂), we have 𝑖(𝐵)=0 and 𝑥=0, which is impossible. Since ‖𝑥‖_𝐶=1, ‖Λ⁻¹𝑢_𝑗‖_𝐶 is bounded. From (17) and (20), ‖𝑢_𝑗‖_𝐿² is bounded. Assume that 𝑢_𝑗⇀𝑢₀ in 𝐿²; then Λ⁻¹𝑢_𝑗→Λ⁻¹𝑢₀. Let 𝜉_𝑗∶=Λ⁻¹𝑢_𝑗+𝐻_𝜇^∗′(𝑡,𝑢_𝑗); then 𝐻_𝜇^∗′(𝑡,𝑢_𝑗)=𝜉_𝑗−Λ⁻¹𝑢_𝑗→−Λ⁻¹𝑢₀. Fenchel conjugate formula gives 𝑢_𝑗=𝐻_𝜇^∗′(𝑡,𝜉_𝑗−Λ⁻¹𝑢_𝑗)→𝐻_𝜇^∗′(𝑡,−Λ⁻¹𝑢₀) in 𝐿²  (by [6, Preposition II, Theorem 4]).

Lemma 11. The assumption (1) of Lemma 6 is valid, where 𝑓 is defined as in (18).

Proof. From (𝐻₁) and (𝐻^∗₁), we have that, for any 𝜖>0, there exists 𝛿>0 such that 𝐴₁(𝑡)−𝜖𝑥,𝑥≤𝐻^′(𝑡,𝑥),𝑥≤𝐴₂(𝑡),|𝑥|+𝜖𝑥,𝑥forany≤𝛿. (25) Thus, 𝐻_𝜇𝑡,𝑦=¹₀𝐻^′_𝜇≥1𝑡,𝜃𝑦𝑑𝜃,𝑦2𝐴₁(𝑡)−𝜖𝐼_2𝑛+𝜇𝐼_2𝑛,𝑦𝑦,𝑦forany≤𝛿. (26)
Let 𝑔(𝑦)=(𝑥,𝑦)−𝐻_𝜇(𝑡,𝑦). By (16) and (𝐻₄), 𝑔(𝑦) is strictly concave. Then, |𝑦|→∞ as 𝑔(𝑦)→−∞. So, 𝑔 achieves its maximum at a unique point. Because of 𝑔^′(𝑦)=𝑥−𝐻^′_𝜇(𝑡,𝑦), we get that 𝑥=𝐻^′_𝜇𝑡,𝑦hasauniquesolution. (27)
And we can easily get 0=𝐻^′_𝜇(𝑡,0). Hence, we have |𝑦|→0 when |𝑥|→0. Otherwise, there exists 𝜖₀>0, 𝑥_𝑛, 𝑦_𝑛∈𝐑^2𝑛, 𝑡_𝑛∈[0,1] s.t. |𝑥_𝑛|→0. But |𝑦_𝑛|≥𝜖₀, 𝑥_𝑛=𝐻^′_𝜇(𝑡_𝑛,𝑦_𝑛). Hence, |𝑦_𝑛| is bounded, and there exists 𝑦₀, 𝑡₀ such that 𝑦_𝑛⇀𝑦₀, 𝑡_𝑛→𝑡₀, 𝑥_𝑛→0. Thus, 0=𝐻^′_𝜇(𝑡₀,𝑦₀). This contradicts the uniqueness, which yields there exists 𝜌(𝜌<𝛿) such that, for any |𝑥|≤𝜌, we have |𝑦|≤𝛿. So, for any |𝑥|≤𝜌, we have 𝐻^∗_𝜇(𝑡,𝑥)=sup_{𝑦∈𝐑2𝑛}𝑥,𝑦−𝐻_𝜇𝑡,𝑦=sup_𝑦≤𝛿𝑥,𝑦−𝐻_𝜇𝑡,𝑦≤sup_𝑦≤𝛿−1𝑥,𝑦2𝐴₁(𝑡)−𝜖𝐼_2𝑛+𝜇𝐼_2𝑛=1𝑦,𝑦2𝐴₁(𝑡)−𝜖𝐼_2𝑛+𝜇𝐼_2𝑛⁻¹𝑥,𝑥. (28) By (18), 𝜓(𝑢)≤¹₀12Λ⁻¹𝑢(𝑡)(𝑡)+1,𝑢2𝐴₁(𝑡)−𝜖𝐼_2𝑛+𝜇𝐼_2𝑛⁻¹𝑢(𝑡)(𝑡)|𝑢|,𝑢𝑑𝑡forany≤𝜌, (29) that is, 𝜓(𝑢)≤𝑞_{𝜇,(𝐴1−𝜖)}(𝑢,𝑢)forany|𝑢|≤𝜌. (30)
Let 𝑋₁=𝐸⁻_𝜇(𝐴₁−𝜖). Then, dim𝑋₁=𝑖_𝜇(𝐴₁−𝜖). For any 𝑢∈𝑋₁∩𝑆_𝜌, we have sup𝜓(𝑢)<0. By Proposition 4, for 𝜖being small enough, 𝑖_𝜇(𝐴₁−𝜖)=𝑖_𝜇(𝐴₁).

Lemma 12. The assumption (2) of Lemma 6 is also satisfied.

Proof. From (𝐻₂), let ℎ₂(𝑡,𝑥)=𝐻^′(𝑡,𝑥)−𝐵_∞(𝑡,𝑥)𝑥. Then, for any 𝜖>0, there exists a constant 𝑀₁ such that ℎ₂(𝑡,𝑥)≤𝜖|𝑥|+𝑀₁,∀(𝑡,𝑥)∈[0,1]×𝐑^2𝑛. (31) Combining (𝐻₂) and (31), there exists a constant 𝑀₂ such that 𝐻_𝜇()≤1𝑡,𝑥2𝐵₂(𝑡)+4𝜖𝐼_2𝑛+𝜇𝐼_2𝑛𝑥,𝑥+𝑀₂, (32) for all (𝑡,𝑥)∈[0,1]×𝐑^2𝑛. By the definition of 𝐻^∗_𝜇(𝑡,𝑥), we have 𝐻^∗_𝜇(𝑡,𝑥)≥sup_{𝑦∈𝐑2𝑛}−1𝑥,𝑦2𝐵₂(𝑡)+4𝜖𝐼_2𝑛+𝜇𝐼_2𝑛𝑦,𝑦−𝑀₂=12𝐵₂(𝑡)+4𝜖𝐼_2𝑛+𝜇𝐼_2𝑛⁻¹𝑥,𝑥−𝑀₂. (33) Thus, 𝜓(𝑢)≥¹₀12Λ⁻¹𝑢(𝑡)(𝑡)+1,𝑢2𝐵₂(𝑡)+4𝜖𝐼_2𝑛+𝜇𝐼_2𝑛⁻¹(𝑢(𝑡),𝑢(𝑡))𝑑𝑡−𝑀₂. (34) So, we infer 𝜓(𝑢)≥𝑞_{𝜇,(𝐵2+4𝜖)}(𝑢,𝑢)−𝑀₂. (35) Let 𝑋₂=𝐸⁻_𝜇(𝐵₂+4𝜖). We have dim𝑋₂=𝑖_𝜇(𝐵₂+4𝜖) and 𝑋^⟂₂=𝐸⁰_𝜇(𝐵₂(𝑡)+4𝜖)⊕𝐸⁺_𝜇(𝐵₂(𝑡)+4𝜖). So, we have inf_𝑋⟂2𝜓(𝑢)≥−𝑀₂. (36)
By Proposition 4, we have 𝑖_𝜇(𝐴₁)−𝑖_𝜇(𝐵₂)=𝑖(𝐴₁)−𝑖(𝐵₂), and we can also let 𝜖 be small enough such that 𝑖_𝜇(𝐵₂+4𝜖)=𝑖_𝜇(𝐵₂). This completes the proof.

Theorem 13. If  𝑖(𝐴₁)<𝑖(𝐵₁), then (2) has at least 𝑖(𝐵₁)−𝑖(𝐴₁) pairs of solutions.

The argument in Theorem 8 can also be used here by using Lemma 7. More precisely, if 𝐻⁺=𝐸⁺_𝜇(𝐴₂+𝜖), then codim𝐻⁺=𝑖(𝐴₁) and 𝜓(𝑢)>𝑐₀>0 on 𝐻⁺∩𝑆_𝜌. If 𝐻⁻=𝐸⁻_𝜇(𝐵₁−4𝜖), then dim𝐻⁻=𝑖(𝐵₁) and 𝜓(𝑢)≤𝑐_∞ on 𝐻⁻.

Acknowledgments

The author would like to thank the reviewers for their valuable comments which improved the paper. The paper is partially supported by the National Natural Science Foundation of China (10871095) and the Foundation for Innovative Program of Jiangsu Province (CXZZ12_0377, CXZZ12_0386).