Existence of Homoclinic Orbits for Hamiltonian Systems with Superquadratic Potentials

Abstract

This paper concerns solutions for the Hamiltonian system: 𝑧̇=𝒥𝐻_𝑧(𝑡,𝑧). Here 𝐻(𝑡,𝑧)=(1/2)𝑧⋅𝐿𝑧+𝑊(𝑡,𝑧), 𝐿 is a 2𝑁×2𝑁 symmetric matrix, and 𝑊∈𝐶¹(ℝ×ℝ^2𝑁,ℝ). We consider the case that 0∈𝜎_𝑐(−(𝒥(𝑑/𝑑𝑡)+𝐿)) and 𝑊 satisfies some superquadratic condition different from the type of Ambrosetti-Rabinowitz. We study this problem by virtue of some weak linking theorem recently developed and prove the existence of homoclinic orbits.

1. Introduction and the Main Results

In this paper, we consider the existence of homoclinic orbits for the following Hamiltonian system: 𝑧̇=𝒥𝐻_𝑧(𝑡,𝑧),(1.1) where 𝐻(𝑡,𝑧)=(1/2)𝑧⋅𝐿𝑧+𝑊(𝑡,𝑧), 𝐿 is a 2𝑁×2𝑁 symmetric matrix-valued function, and 𝑊∈𝐶¹(ℝ×ℝ^2𝑁,ℝ) is superquadratic both around 0 and at infinity in 𝑧∈ℝ^2𝑁.

A solution of (1.1) is called to be homoclinic to 0 if 𝑧(𝑡)≢0 and 𝑧(𝑡)→0 as |𝑡|→∞.

In recent years, the existence and multiplicity of homoclinic orbits for Hamiltonian systems have been investigated in many papers via variational methods. See, for example, [1–7] for the second-order systems and [8–12] for the first-order systems. We note that in most of the papers on the first order system (1.1) it was assumed that

(◊)𝐿 is constant such that sp(𝒥𝐿)∩𝑖ℝ=∅, where sp(𝒥𝐿) denotes the set of all eigenvalues of 𝒥𝐿.

Thus, if we let 𝜎(𝐴) denote the spectrum of 𝐴, (◊) means that 𝐿 is independent of 𝑡 and there is 𝛼>0 such that (−𝛼,𝛼)∩𝜎(𝐴)=∅. Consequently, the operator 𝐴∶=−(𝒥(𝑑/𝑑𝑡)+𝐿)∶𝑊^1,𝑝(ℝ,ℝ^2𝑁)→𝐿^𝑝(ℝ,ℝ^2𝑁) is a homeomorphism for all 𝑝>1. This is important for the variational arguments. Later in [13], Ding considered the case that 𝐿 depends periodically on 𝑡. He made assumptions on 𝐿 such that 0 lies in a gap of 𝜎(𝐴). If additionally 𝑊(𝑡,𝑧) is periodic in 𝑡 and satisfies some superquadratic or asymptotically quadratic conditions in 𝑧 at infinity, then infinitely many homoclinic orbits were obtained.

If 0∈𝜎_𝑐(𝐴), then the problem is quite different in nature since the operator 𝐴 cannot lead the behavior at 0 of the equation. Ding and Willem considered this case in [14]. They assumed that

(𝐴₀)𝐿(𝑡)∈𝐶(ℝ,ℝ^4𝑁2) is 1-periodic. There exists 𝛼>0 such that (0,𝛼)∩𝜎(𝐴)=∅.

Under (𝐴₀), 0 may belong to continuous spectrum of 𝐴. The authors managed to construct an appropriate Banach space, on which some embedding results necessary for variational arguments were obtained. Using a generalized linking theorem developed by Kryszewski and Szulkin in [15], they got one homoclinic orbit of (1.1). Later, Ding and Girardi obtained infinitely many homoclinic orbits in [16] under the conditions of [14] with an additional evenness assumption on 𝑊. Note that in both papers 𝑊 satisfies a condition of the type of Ambrosetti-Rabinowitz (see [17]), that is,

∃𝜇>2suchthat0<𝜇𝑊(𝑡,𝑧)≤𝑊_𝑧(𝑡,𝑧)𝑧,𝑡∈ℝ,𝑧∈ℝ^2𝑁⧵{0}.(A-R) The (A-R) condition is essential to prove the Palais-Smale condition since the variational functional Φ is strongly indefinite and 0∈𝜎_𝑐(−𝒥(𝑑/𝑑𝑡)+𝐿). The argument of Palais-Smale condition is rather technical and not standard without the (A-R) condition. In this paper, we consider the existence of solutions of (1.1) under (𝐴₀) without the (A-R) condition on 𝑊.

We observed that just recently some abstract linking theorems were developed by Bartsch and Ding in [18]. These theorems are impactful to study the existence and multiplicity of solutions for the strongly indefinite problem. Many new results have been obtained by these theorems based on the use of (𝐶)_𝑐 sequence. See [19–21] for applications of these ideas. Note that in [19–21] 0 either is not a spectral point or is at most an isolated eigenvalue of finite multiplicity. Thus (𝐶)_𝑐 condition was checked by virtue of some very technical analysis. However, if 0∈𝜎_𝑐(𝐴), then we can find a sequence {𝑧_𝑛}⊂𝐻¹ with |𝑧_𝑛|_𝐿²=1 and |𝐴𝑧_𝑛|_𝐿²→0. Thus the operator 𝐴 cannot lead the behavior at 0 of the equation. Consequently, besides (𝐶)_𝑐 condition, it seems also hard to check the following condition necessary for the linking theorems in [19–21]:

(Φ₁)for any 𝑐>0, there exists 𝜁>0 such that ‖𝑧‖<𝜁‖𝑃_𝑌𝑧‖ for all 𝑧∈Φ_𝑐.

Our work benefits from [14] and some weak linking theorem recently developed by Schechter and Zou in [22]. This theorem permits us first to study a sequence of approximating problems Φ_𝜆 for 𝜆∈[1,2] (the initial problem corresponds to 𝜆=1) for which a bounded Palais-Smale sequence of Φ_𝜆 is given for almost each 𝜆∈[1,2]. Then by monotonicity, we find a sequence of {𝜆_𝑛} and {𝑤_𝑛} such that 𝜆_𝑛→1, Φ^′_𝜆𝑛(𝑤_𝑛)=0, and Φ_𝜆𝑛(𝑤_𝑛)≤𝑑. Since the sequence {𝑤_𝑛} consists of critical points of Φ_𝜆𝑛, then its boundedness can be checked. Consequently one solution of (1.1) is obtained. The idea of first studying approximating problems for which the existence of a bounded Palais-Smale sequence is given freely and then proving that the sequence of approximated critical points is bounded was originally introduced in [23]. See also [24].

We make the following assumptions.

(𝐴₁)𝑊(𝑡,𝑧)∈𝐶¹(ℝ×ℝ^2𝑁,ℝ) is 1-periodic in 𝑡. 𝑊(𝑡,0)=0 for all 𝑡∈ℝ. There exist constants 𝑐₁>0 and 𝜇>2 such that 𝑊_𝑧(𝑡,𝑧)𝑧≥𝑐₁|𝑧|^𝜇 for (𝑡,𝑧)∈ℝ×ℝ^2𝑁.(𝐴₂)there exist 𝑐₂, 𝑟>0 such that |𝑊_𝑧(𝑡,𝑧)|≤𝑐₂|𝑧|^𝜇−1 for 𝑡∈ℝ and |𝑧|≤𝑟.(𝐴₃)there exist 𝑐₃, 𝑅≥𝑟 and 𝑝≥𝜇 such that |𝑊_𝑧(𝑡,𝑧)|≤𝑐₃|𝑧|^𝑝−1 for 𝑡∈ℝ and |𝑧|≥𝑅.(𝐴₄)there exists 𝑏₀>2 such that liminf_𝑧→0(𝑊_𝑧(𝑡,𝑧)𝑧/𝑊(𝑡,𝑧))≥𝑏₀ uniformly for 𝑡∈ℝ;(𝐴₅)𝑊(𝑡,𝑧)∶=(1/2)𝑊_𝑧(𝑡,𝑧)𝑧−𝑊(𝑡,𝑧)>0 for all𝑡∈ℝ, 𝑧∈ℝ^2𝑁⧵{0}. There exist constants 𝑏_∞>0 and 𝛽>𝑝(𝑝−2)/(𝑝−1) such that liminf_|𝑧|→∞𝑊(𝑡,𝑧)/|𝑧|^𝛽≥𝑏_∞ uniformly for 𝑡∈ℝ.

Theorem 1.1. Let (𝐴₀), (𝐴₁)-(𝐴₅) be satisfied, then (1.1) has at least one homoclinic orbit.

Remark 1.2. We can easily check that the (A-R) condition implies (𝐴₄) and (𝐴₅). But the converse proposition is not true. See the following example: 𝑊(𝑡,𝑧)=|𝑧|^𝜇+(𝜇−2)|𝑧|^𝜇−𝜖sin²(|𝑧|^𝜖𝜖),(1.2) where 2<𝜇<∞, 0<𝜖<min{𝜇−2,𝜇/(𝜇−1)} (see [25] or [26] for details).

If 𝑊_𝑧(𝑡,𝑧)=𝑎|𝑧|^𝜇−2𝑧+𝑅_𝑧(𝑡,𝑧), 𝑎>0, 𝜇∈(2,∞) with 𝑅 satisfying

(𝐵₁)𝑅∈𝐶¹(ℝ×ℝ^2𝑁,ℝ) is 1-periodic in 𝑡 and 𝑅_𝑧(𝑡,𝑧)=𝑜(|𝑧|^𝜇−1𝑅)as|𝑧|→0,_𝑧(𝑡,𝑧)=𝑜(|𝑧|^𝜇−1)as|𝑧|→∞,(1.3) uniformly in 𝑡∈ℝ, then(𝐵₂)0<𝑅_𝑧(𝑡,𝑧)𝑧≤(𝑎(𝜇−2)/2)|𝑧|^𝜇 for all𝑡∈ℝ, 𝑧∈ℝ^2𝑁⧵{0}.

Theorem 1.3. Let (𝐴₀),(𝐵₁), and (𝐵₂) be satisfied, then (1.1) has at least one homoclinic orbit.

This paper is organized as follows. In Section 2 we will construct some appropriate variational space and obtain some embedding results necessary for our variational arguments. In Section 3 we will recall a weak linking theorem, by which we will give the proof of Theorems 1.1 and 1.3 in Section 4.

2. Some Embedding Results

In what follows, by |⋅|_𝑝 we denote the usual 𝐿^𝑝-norm and by (⋅)₂ the usual 𝐿²-inner product. A standard Floquet reduction argument shows that 𝜎(𝐴)=𝜎_𝑐(𝐴) (see [14]).

Let {𝐸(𝜆);𝜆∈ℝ} be the spectral family of 𝐴. 𝐴 possesses the polar decomposition 𝐴=𝑈|𝐴| with 𝑈=𝐼−𝐸(0)−𝐸(−0). By (𝐴₀), 0 is at most a continuous spectrum of 𝐴. 𝐿² has an orthogonal decomposition

𝐿²=𝐿²⁻⊕𝐿²⁺,(2.1) where 𝐿^2±∶={𝑢∈𝐿²;𝑈𝑢=±𝑢}.

Let 𝒟(|𝐴|^1/2) denote the domain of |𝐴|^1/2 and let 𝐸 be the space of the completion of 𝒟(|𝐴|^1/2) under the norm

‖𝑢‖_𝐸∶=||𝐴|^1/2𝑢|₂.(2.2)𝐸 becomes a Hilbert space under the inner product

(𝑢,𝑣)_𝐸∶=(|𝐴|^1/2𝑢,|𝐴|^1/2𝑣)₂.(2.3)𝐸 possesses an orthogonal decomposition

𝐸=𝐸⁻⊕𝐸⁺,(2.4) where 𝐸^±⊇𝐿^2±∩𝒟(|𝐴|^1/2).

Under (𝐴₀), it is easy to check

𝐸⁺=𝐿²⁺∩𝒟(|𝐴|^1/2),‖⋅‖_𝐸∼‖⋅‖_𝐻^1/2on𝐸⁺.(2.5) Therefore, 𝐸⁺ can be embedded continuously into 𝐿^𝑝(ℝ,ℝ^2𝑁) for any 𝑝≥2 and compactly into 𝐿^𝑝_loc(ℝ,ℝ^2𝑁) for any 𝑝∈[1,∞).

For any 𝜖>0, set 𝐿_𝜖²⁻∶=𝐸(−𝜖)𝐿² and 𝐸⁻_𝜖∶=𝐿_𝜖²⁻∩𝒟(|𝐴|^1/2)=𝐿_𝜖²⁻∩𝐸⁻. Then on 𝐸⁻_𝜖, we also have ‖⋅‖_𝐸∼‖⋅‖_𝐻^1/2 and the same embedding conclusion as that of 𝐸⁺.

Let 𝐿_𝜖²⁻∶=𝐿²⁻∩(cl_𝐿²(⋃_𝜆<−𝜖𝐸(𝜆)𝐿²))^⟂ where cl_𝐿²(⋅) stands for the closure of ⋅ in 𝐿².

For 𝜇>2, let 𝐸⁻_𝜖,𝜇 be the completion of 𝐿_𝜖²⁻ under the norm ‖𝑢‖_𝜇∶=(||𝐴|^1/2𝑢|²₂+|𝑢|²_𝜇)^1/2,(2.6) and let 𝐸⁻_𝜇 denote the completion of 𝒟(𝐴)∩𝐿²⁻ with respect to the norm ‖⋅‖_𝜇. Then 𝐸⁻_𝜖 is a closed subspace of 𝐸⁻_𝜇, and 𝐸⁻_𝜇 possesses the following decomposition:

𝐸⁻_𝜇=𝐸⁻_𝜖⊕𝐸⁻_𝜖,𝜇.(2.7) Moreover, 𝐸⁻_𝜖 is orthogonal to 𝐸⁻_𝜖,𝜇 with respect to (⋅)_𝐸.

Let 𝐸_𝜇 be the completion of 𝒟(𝐴) under the norm ‖⋅‖_𝜇. The following result holds true.

Lemma 2.1 (see [14]). Under (𝐴₀), 𝐸_𝜇 has the direct sum decomposition 𝐸_𝜇=𝐸⁻_𝜇⊕𝐸⁺,(2.8) and 𝐸_𝜇 is embedded continuously in 𝐿^𝜈 for any 𝜈∈[𝜇,∞) and compactly in 𝐿^𝜈_loc  for any 𝜈∈[2,∞).

3. A Weak Linking Theorem

In this section we state some weak linking theorem due to [22] which was first built in a Hilbert space. This theorem is still true in a reflexive Banach space (cf. Willem and Zou [25]).

Let 𝐸 be a reflexive Banach space with norm ‖⋅‖ and possess a direct sum decomposition 𝐸=𝑁⊕𝑀, where 𝑁⊂𝐸 is a closed and separable subspace. Since 𝑁 is separable, we can define a new norm |𝑧|_𝑤 satisfying |𝑧|_𝑤≤‖𝑧‖, forall𝑧∈𝑁 such that the topology induced by this norm is equivalent to the weak topology of 𝑁 on bounded subsets of 𝑁. For 𝑧=𝑣+𝑤∈𝐸 with 𝑣∈𝑁 and 𝑤∈𝑀, we define |𝑧|²_𝑤=|𝑣|²_𝑤+‖𝑤‖², then |𝑧|_𝑤≤‖𝑧‖, forall𝑧∈𝐸. In particular, if 𝑧_𝑛=𝑣_𝑛+𝑤_𝑛 is |⋅|_𝑤-bounded and 𝑧_𝑛→𝑧 under the norm |⋅|_𝑤 in 𝐸, then 𝑣_𝑛⇀𝑣 weakly in 𝑁, 𝑤_𝑛→𝑤 strongly in 𝑀, and 𝑧_𝑛⇀𝑣+𝑤 weakly in 𝐸. Let 𝑄⊂𝑁 be a ‖⋅‖-bounded open convex subset and let 𝑝₀∈𝑄 be a fixed point. Let 𝐹 be a |⋅|_𝑤-continuous map from 𝐸 onto 𝑁 satisfying the following.

(i)𝐹|_𝑄=id; 𝐹 maps bounded sets to bounded sets.(ii)there exists a fixed finite-dimensional subspace 𝐸₀ of 𝐸 such that 𝐹(𝑢−𝑣)−(𝐹(𝑢)−𝐹(𝑣))⊂𝐸₀, forall𝑢,𝑣∈𝐸.(iii)𝐹 maps finite-dimensional subspaces of 𝐸 into finite-dimensional subspaces of 𝐸.

Set 𝐴∶=𝜕𝑄, 𝐵∶=𝐹⁻¹(𝑝₀), where 𝜕𝑄 denotes the ‖⋅‖-boundary of 𝑄. For Φ∈𝐶¹(𝐸,ℝ), we introduce the class Γ of mappings ℎ∶[0,1]×𝑄→𝐸 with the following properties.

(a)ℎ∶[0,1]×𝑄→𝐸 is |⋅|_𝑤-continuous.(b)for any (𝑠₀,𝑢₀)∈[0,1]×𝑄, there is a |⋅|_𝑤-neighborhood 𝑈_{(𝑠0,𝑢0)} such that {𝑢−ℎ(𝑡,𝑢)∶(𝑡,𝑢)∈𝑈_{(𝑠0,𝑢0)}∩([0,1]×𝑄)}⊂𝐸_𝑓𝑖𝑛, where 𝐸_𝑓𝑖𝑛 denotes some finite-dimensional subspaces of 𝐸.(c)ℎ(0,𝑢)=𝑢,Φ(ℎ(𝑠,𝑢))≤Φ(𝑢),forall𝑢∈𝑄.

The following is a variant weak linking theorem in [22].

Theorem 3.1. Let the family of 𝐶¹-functionals (Φ_𝜆) have the form Φ_𝜆(𝑢)∶=𝐼(𝑢)−𝜆𝐽(𝑢),∀𝜆∈[1,2].(3.1) Assume that the following conditions hold.(1)𝐽(𝑢)≥0,forall𝑢∈𝐸;Φ₁∶=Φ.(2)𝐼(𝑢)→∞ or 𝐽(𝑢)→∞ as ‖𝑢‖→∞.(3)Φ_𝜆 is |⋅|_𝑤-upper semicontinuous; Φ^′_𝜆 is weakly sequentially continuous on 𝐸. Moreover, Φ_𝜆 maps bounded sets into bounded sets.(4)sup_𝐴Φ_𝜆<inf_𝐵Φ_𝜆,forall𝜆∈[1,2].Then for almost all 𝜆∈[1,2], there exists a sequence {𝑢_𝑛} such that sup_𝑛‖𝑢_𝑛‖<∞,Φ^′_𝜆(𝑢_𝑛)→0,Φ_𝜆(𝑢_𝑛)→𝐶_𝜆,(3.2) where 𝐶_𝜆∶=inf_ℎ∈Γsup_𝑢∈𝑄Φ_𝜆(ℎ(1,𝑢))∈[inf_𝐵Φ_𝜆,sup_𝑄Φ_𝜆].

Remark 3.2. Consider 𝐸_𝜇=𝐸⁻_𝜇⊕𝐸⁺ defined as in Section 2. Obviously, 𝐸_𝜇 is reflexive. For 𝑧₀∈𝐸⁺ with ‖𝑧₀‖_𝜇=1,𝐸⁺=ℝ𝑧₀⊕𝐸⁺₁. Let 𝑁=𝐸⁻_𝜇⊕ℝ𝑧₀ and 𝑀=𝐸⁺₁, then 𝐸_𝜇=𝑁⊕𝑀. It is easy to see that 𝑁 is a closed and separable subspace of 𝐸_𝜇. For any 𝑢∈𝐸_𝜇,𝑢 can be written as 𝑢=𝑢⁻+𝑠𝑧₀+𝑤⁺ with 𝑢⁻∈𝐸⁻_𝜇 and 𝑤⁺∈𝐸⁺₁. For 𝑅>0, let 𝑄∶={𝑢∶=𝑢⁻+𝑠𝑧₀,𝑠>0,𝑢⁻∈𝐸⁻_𝜇,‖𝑢‖_𝜇<𝑅}, then 𝑝₀∶=𝑠₀𝑧₀∈𝑄 for 0<𝑠₀<𝑅. Define 𝐹∶𝐸_𝜇→𝑁 as 𝐹𝑢∶=𝑢⁻+‖𝑠𝑧₀+𝑤⁺‖_𝜇𝑧₀. Then it is easy to check that 𝐹,𝑄,and𝑝₀ satisfy (i), (ii), and (iii). If we let 𝐴∶=𝜕𝑄 and 𝐵∶=𝐹⁻¹(𝑠₀𝑧₀)={𝑢∶=𝑠𝑧₀+𝑤⁺,𝑠≥0,𝑤⁺∈𝐸⁺₁,‖𝑠𝑧₀+𝑤⁺‖_𝜇=𝑠₀}, then 𝐴 links 𝐵 (see Lemmas 4.2 and 4.3 in Section 4).

4. The Proof of the Main Results

Consider the functional

1Φ(𝑧)∶=2‖𝑧⁺‖²_𝐸−12‖𝑧⁻‖²_𝐸−_ℝ𝑊(𝑡,𝑧),(4.1) for 𝑧=𝑧⁺+𝑧⁻∈𝐸_𝜇. Then by assumptions (𝐴₁)-(𝐴₃) and Lemma 2.1, Φ∈𝐶¹(𝐸_𝜇,ℝ). A standard argument shows that any critical point of Φ is a homoclinic orbit of (1.1) (cf. [14]).

Set

Φ_𝜆1(𝑧)∶=2‖𝑧⁺‖²_𝐸1−𝜆(2‖𝑧⁻‖²_𝐸+_ℝ𝑊(𝑡,𝑧))=𝐼(𝑧)−𝜆𝐽(𝑧),𝜆∈[1,2].(4.2) Then Φ₁=Φ and 𝐽(𝑧)≥0. By (𝐴₂) and (𝐴₃),

|𝑊_𝑧(𝑡,𝑧)|≤𝐶(|𝑧|^𝜇−1+|𝑧|^𝑝−1),(4.3) where, as below,𝐶 stands for some generic positive constant.

Together with (𝐴₁), one has

𝑐₁𝜇|𝑧|^𝜇≤𝑊(𝑡,𝑧)≤𝐶(|𝑧|^𝜇+|𝑧|^𝑝).(4.4) Thus 𝐼(𝑧)→∞ or 𝐽(𝑧)→∞ if ‖𝑧‖²_𝜇=‖𝑧⁺‖²_𝐸+‖𝑧⁻‖²_𝐸+|𝑧|²_𝜇→∞.

Lemma 4.1. Φ_𝜆 is |⋅|_𝑤-upper semicontinuous and Φ^′_𝜆 is weakly sequentially continuous.

Proof. For any 𝑐∈ℝ, assume that 𝑧_𝑛∈{𝑧∈𝐸_𝜇,Φ_𝜆(𝑧)≥𝑐} with 𝑧_𝑛⇀𝑧. Let 𝑧_𝑛=𝑧⁺_𝑛+𝑧⁻_𝑛 with 𝑧⁺_𝑛∈𝐸⁺ and 𝑧⁻_𝑛∈𝐸⁻_𝜇. Then 𝑧⁺_𝑛→𝑧⁺ in 𝐸_𝜇 and hence sup‖𝑧⁺_𝑛‖<∞. Since Φ_𝜆(𝑧_𝑛)≥𝑐 and 𝑊(𝑡,𝑧_𝑛)≥0, we have sup‖𝑧⁻_𝑛‖<∞.
By (4.4),
𝐶|𝑧_𝑛|^𝜇_𝜇≤_ℝ𝑊(𝑡,𝑧_𝑛1)d𝑡≤𝜆(12‖𝑧⁺_𝑛‖²_𝐸−𝜆2‖𝑧⁻_𝑛‖²_𝐸−𝑐)<∞.(4.5) Then sup‖𝑧_𝑛‖_𝜇<∞. By Lemma 2.1, 𝑧_𝑛⇀𝑧 in 𝐸_𝜇,𝑧_𝑛→𝑧 in 𝐿^𝜇_loc, and 𝑧_𝑛(𝑡)→𝑧(𝑡) a.e. for 𝑡∈ℝ. By Fatou's Lemma, Φ_𝜆(𝑧)≥𝑐. Therefore, Φ_𝜆 is |⋅|_𝑤-upper semicontinuous.
Let 𝑧_𝑛⇀𝑧 in 𝐸_𝜇, then 𝑧_𝑛→𝑧 in 𝐿^𝑝_loc,2≤𝑝<∞. By (4.3), 𝑊_𝑧(𝑡,𝑧_𝑛)→𝑊_𝑧(𝑡,𝑧) in 𝐿^{𝜇/(𝜇−1)}_loc and ∫_ℝ𝑊_𝑧(𝑡,𝑧_𝑛)𝑣→∫_ℝ𝑊_𝑧(𝑡,𝑧)𝑣 for any 𝑣∈𝐸_𝜇. Therefore, Φ^′_𝜆(𝑧_𝑛)→Φ^′_𝜆(𝑧).

Lemma 4.2. There exist 𝑏>0, 𝑟>0 such that Φ_𝜆(𝑧)≥𝑏>0, forall𝑧∈𝐸⁺ with ‖𝑧‖_𝜇=𝑟, forall𝜆∈[1,2].

Proof. By (4.4) and Lemma 2.1, _ℝ𝑊(𝑡,𝑧)≤𝐶(|𝑧|^𝜇_𝜇+|𝑧|^𝑝_𝑝)≤𝐶(‖𝑧‖^𝜇_𝜇+‖𝑧‖^𝑝_𝜇).(4.6) The conclusion is obvious.

Lemma 4.3. There exists 𝑅>𝑟>0 such that Φ_𝜆|_𝜕𝑀=0 and sup_𝑀Φ_𝜆<𝑑<∞ for all 𝜆∈[1,2], where 𝑀∶={𝑧=𝑥+𝑠𝑧₀,𝑥∈𝐸⁻_𝜇,‖𝑧‖_𝜇≤𝑅,𝑠>0} and 𝑧₀∈𝐸⁺, ‖𝑧₀‖_𝜇=1.

Proof. For 𝑧=𝑥+𝑠𝑧₀, by (4.4), Φ_𝜆𝑠(𝑧)≤²2‖𝑧₀‖²_𝐸−12‖𝑥‖²_𝐸−𝐶_ℝ|𝑥+𝑠𝑧₀|^𝜇.(4.7)
Since 𝐸_𝜇 is continuously embedded in 𝐿^𝑡 for 𝜇≤𝑡<∞, there exists a continuous projection from 𝐸⁻_𝜇⊕ℝ𝑧₀ in 𝐿^𝜇 to ℝ𝑧₀. Thus, |𝑠𝑧₀|_𝜇≤𝐶|𝑥+𝑠𝑧₀|_𝜇 for some 𝐶>0 and then
Φ_𝜆(𝑧)≤𝐶𝑠²−𝐶‖𝑥‖²_𝐸−𝐶𝑠^𝜇,(4.8) and thus the lemma follows easily.

Combining Lemmas 4.1–4.3 and Theorem 3.1, we get the following.

Lemma 4.4. Under (𝐴₀) and (𝐴₁)-(𝐴₃), for almost every 𝜆∈[1,2], there exist {𝑧_𝑛}⊆𝐸_𝜇 such that sup‖𝑧_𝑛‖_𝜇<∞,Φ^′_𝜆(𝑧_𝑛)→0,Φ_𝜆(𝑧_𝑛)→𝐶_𝜆∈[𝑏,𝑑].(4.9)

We need the following lemma which is a special case of a more general result due to Lions [27, 28].

Lemma 4.5. Let 𝑎>0 and 2≤𝑝<∞. If {𝑧_𝑛}⊂𝐻¹ is bounded and if sup_{𝑠∈ℝ𝐵(𝑠,𝑎)}|𝑢_𝑛|^𝑝→0,𝑛→∞,(4.10) where 𝐵(𝑠,𝑎)∶=(𝑠−𝑎,𝑠+𝑎), then 𝑢_𝑛→0 in 𝐿^𝑡 for 2<𝑡<∞.

Lemma 4.6. Under (𝐴₀)-(𝐴₃), let 𝜆∈[1,2] be fixed. For the sequence {𝑧_𝑛} in Lemma 4.4, there exist {𝑘_𝑛}⊂ℤ such that, up to a subsequence, 𝑢_𝑛(𝑡)∶=𝑧_𝑛(𝑡+𝑘_𝑛) satisfies 𝑢_𝑛⇀𝑢_𝜆≠0, Φ^′_𝜆(𝑢_𝜆)=0 and Φ_𝜆(𝑢_𝜆)≤𝑑.

Proof. Write 𝑧_𝑛=𝑧⁺_𝑛+𝑧⁻_𝑛 with 𝑧⁺_𝑛∈𝐸⁺ and 𝑧⁻_𝑛∈𝐸⁻_𝜇. Since sup‖𝑧_𝑛‖_𝜇<∞, sup‖𝑧⁺_𝑛‖_𝐸<∞, let 𝐴₊ denote the part of 𝐴 in 𝒟(𝐴)∩𝐿²⁺=𝐻¹∩𝐿²⁺∶=𝐻¹₊. Then by (𝐴₀), 𝐴₊=^∞_𝛼𝜆d𝐸(𝜆).(4.11) Obviously, 𝐴₊∶𝐻¹₊⊂𝐿²⁺→𝐿²⁺ has a bounded inverse 𝐴₊⁻¹. Since |𝐴₊𝑧|²₂=^∞_𝛼𝜆|d𝐸(𝜆)𝑧|²₂≥𝛼|𝑧|²₂,|𝑧|̇₂=|𝐴𝑧+𝐿𝑧|₂≤|𝐴₊𝑧|₂+|𝐿𝑧|₂for𝑧∈𝐻¹₊,(4.12) then we have ‖𝑧‖_𝐻¹≤𝐶|𝐴₊𝑧|₂for𝑧∈𝐻¹₊.(4.13) Set 𝑣_𝑛=𝐴₊⁻¹𝑧⁺_𝑛∈𝐻¹₊, then ‖𝑣_𝑛‖_𝐻1≤𝐶|𝑧⁺_𝑛|₂≤𝐶‖𝑧⁺_𝑛‖_𝐸<∞.(4.14) We claim that 𝑣_𝑛 is nonvanishing, that is, there exist 𝑀>0, 𝑎>0, and 𝑦_𝑛∈ℝ such that liminf_{𝑛→∞𝐵(𝑦𝑛,𝑎)}|𝑣_𝑛|²d𝑡≥𝑀.(4.15) Indeed, if not, by (4.14), {𝑣_𝑛} is bounded in 𝐻¹. Lemma 4.5 shows that 𝑣_𝑛→0 in 𝐿^𝑡 for 2<𝑡<∞. By (4.3), |_ℝ𝑊_𝑧(𝑡,𝑧_𝑛)𝑣_𝑛|≤𝐶_ℝ(|𝑧_𝑛|^𝜇−1+|𝑧_𝑛|^𝑝−1)|𝑣_𝑛|≤𝐶(|𝑧_𝑛|_𝜇^𝜇−1|𝑣_𝑛|_𝜇+|𝑧_𝑛|_𝑝^𝑝−1|𝑣_𝑛|_𝑝)→0.(4.16) Hence (𝑧⁺_𝑛,𝑣_𝑛)_𝐸=Φ^′_𝜆(𝑧_𝑛)𝑣_𝑛+𝜆_ℝ𝑊_𝑧(𝑡,𝑧_𝑛)𝑣_𝑛→0.(4.17) Thus, |𝑧⁺_𝑛|²₂=(𝑧⁺_𝑛,𝐴₊𝑣_𝑛)_𝐿2=(𝑧⁺_𝑛,𝑣_𝑛)_𝐸→0.(4.18) Therefore, for any 2≤𝑡<∞, |𝑧⁺_𝑛|_𝑡≤|𝑧⁺_𝑛|₂^1/𝑡|𝑧⁺_𝑛|^1−1/𝑡_2(𝑡−1)≤𝐶‖𝑧⁺_𝑛‖_𝐸^1−1/𝑡|𝑧⁺_𝑛|₂^1/𝑡→0.(4.19) Thus we obtain ‖𝑧⁺_𝑛‖²_𝐸=Φ^′_𝜆(𝑧_𝑛)𝑧⁺_𝑛+𝜆_ℝ𝑊_𝑧(𝑡,𝑧_𝑛)𝑧⁺_𝑛→0,(4.20) and then Φ_𝜆(𝑧_𝑛1)≤2‖𝑧⁺_𝑛‖²_𝐸→0,(4.21) a contradiction. Choose 𝑘_𝑛∈ℤ such that |𝑘_𝑛−𝑦_𝑛|=min{|𝑠−𝑦_𝑛|,𝑠∈ℤ} and let 𝑢_𝑛∶=𝑘_𝑛∗𝑧_𝑛=𝑧_𝑛(𝑡+𝑘_𝑛)∶=𝑢⁺_𝑛+𝑢⁻_𝑛. In view of the invariance of 𝐸⁺ under the action ∗, 𝑢⁺_𝑛=𝑘_𝑛∗𝑧⁺_𝑛∈𝐸⁺. Since 𝐴 commutes with ∗, then 𝐴₊⁻¹ also does. Therefore 𝑣_𝑛∶=𝑘_𝑛∗𝑣_𝑛=𝐴₊⁻¹𝑢⁺_𝑛. By (4.15),|𝑣_𝑛|²_{𝐿2(𝐵(0,𝑎+1))}≥𝑀2.(4.22) Clearly, ‖𝑢_𝑛‖_𝜇=‖𝑧_𝑛‖_𝜇<∞.(4.23) Thus, up to a subsequence, we assume that 𝑢_𝑛⇀𝑢_𝜆in𝐸_𝜇,𝑢_𝑛→𝑢_𝜆in𝐿^𝑡_locfor𝑡≥2.(4.24) We now establish that 𝑢_𝜆≠0. If not, 𝑢⁺_𝑛⇀0 in 𝐿², and then (𝑣_𝑛,𝑧)_𝐸=(𝑢⁺_𝑛,𝑧)_𝐿2→0forall𝑧∈𝐻^1/2,(4.25) which implies that 𝑣_𝑛⇀0in𝐻^1/2,𝑣_𝑛→0in𝐿^𝑡_locfor𝑡≥2,(4.26) contradicting (4.22). By Lemma 4.1, Φ^′_𝜆 is weakly continuous, hence we have Φ^′_𝜆(𝑢_𝜆)=lim_𝑛→∞Φ^′_𝜆(𝑢_𝑛)=0.(4.27) By Fatou's Lemma, we obtain Φ_𝜆(𝑢_𝜆)=Φ_𝜆(𝑢_𝜆1)−2Φ^′_𝜆(𝑢_𝜆)𝑢_𝜆≤lim_𝑛→∞(Φ_𝜆(𝑢_𝑛1)−2Φ^′_𝜆(𝑢_𝑛)𝑢_𝑛)=lim_𝑛→∞Φ_𝜆(𝑢_𝑛)=lim_𝑛→∞Φ_𝜆(𝑧_𝑛)≤𝑑.(4.28)

As a straightforward consequence of Lemmas 4.4 and 4.6, we have the following.

Lemma 4.7. Under (𝐴₀)-(𝐴₃), there exist {𝜆_𝑛}⊂[1,2], {𝑤_𝑛}⊂𝐸_𝜇⧵{0} such that 𝜆_𝑛→1,Φ^′_𝜆𝑛(𝑤_𝑛)=0, and Φ_𝜆𝑛(𝑤_𝑛)≤𝑑.

Lemma 4.8. {𝑤_𝑛} is bounded in 𝐸_𝜇.

Proof. Our argument is motivated by [26]. Write 𝑤_𝑛=𝑤⁺_𝑛+𝑤⁻_𝑛 with 𝑤⁺_𝑛∈𝐸⁺ and 𝑤⁻_𝑛∈𝐸⁻_𝜇.Since Φ^′_𝜆𝑛(𝑤_𝑛)𝑤_𝑛=0, by (𝐴₁), then ‖𝑤⁺_𝑛‖²_𝐸−𝜆_𝑛‖𝑤⁻_𝑛‖²_𝐸=𝜆_𝑛ℝ𝑊_𝑧(𝑡,𝑤_𝑛)𝑤_𝑛≥𝐶|𝑤_𝑛|^𝜇_𝜇.(4.29) Hence, |𝑤_𝑛|^𝜇_𝜇≤𝐶‖𝑤⁺_𝑛‖²_𝐸,‖𝑤⁻_𝑛‖_𝐸≤𝐶‖𝑤⁺_𝑛‖_𝐸,|𝑤⁻_𝑛|_𝜇≤𝐶‖𝑤⁺_𝑛‖_𝐸+𝐶‖𝑤⁺_𝑛‖_𝐸^2/𝜇.(4.30) In the following, we show that ‖𝑤⁺_𝑛‖_𝐸 is bounded. Choose 𝜖₀>0 small enough such that 𝑏₀−𝜖₀>2. By (𝐴₄), there exists 0<𝑟₀≤1 such that 𝑊_𝑧(𝑡,𝑧)𝑧≥(𝑏₀−𝜖₀)𝑊(𝑡,𝑧)(4.31) for all𝑡∈ℝ and |𝑧|≤𝑟₀. By (𝐴₃) and (𝐴₅), for all𝑡∈ℝ and |𝑧|≥𝑟₀, we can choose 𝐶,𝐶^′>0 such that |𝑊_𝑧(𝑡,𝑧)|≤𝐶|𝑧|^𝑝−1,𝑊(4.32)_𝑧(𝑡,𝑧)𝑧−2𝑊(𝑡,𝑧)≥𝐶^′|𝑧|^𝛽.(4.33) Since Φ_𝜆𝑛(𝑤_𝑛)≤𝑑 and Φ^′_𝜆𝑛(𝑤_𝑛)=0, then we have (12−1𝑏₀−𝜖₀)(‖𝑤⁺_𝑛‖²_𝐸−𝜆_𝑛‖𝑤⁻_𝑛‖²_𝐸)+𝜆_𝑛ℝ(1𝑏₀−𝜖₀𝑊_𝑧(𝑡,𝑤_𝑛)𝑤_𝑛−𝑊(𝑡,𝑤_𝑛))≤𝑑.(4.34) Thus, by (4.31), (4.32), and (𝐴₅), we obtain ‖𝑤⁺_𝑛‖²_𝐸−𝜆_𝑛‖𝑤⁻_𝑛‖²_𝐸≤𝐶(_{|𝑤𝑛|<𝑟0}+_{|𝑤𝑛|≥𝑟0})(𝑊(𝑡,𝑤_𝑛1)−𝑏₀−𝜖₀𝑊_𝑧(𝑡,𝑤_𝑛)𝑤_𝑛)d𝑡+𝐶≤𝐶_{|𝑤𝑛|≥𝑟0}(𝑊(𝑡,𝑤_𝑛1)−𝑏₀−𝜖₀𝑊_𝑧(𝑡,𝑤_𝑛)𝑤_𝑛)d𝑡+𝐶≤𝐶_{|𝑤𝑛|≥𝑟0}𝑊_𝑧(𝑡,𝑤_𝑛)𝑤_𝑛d𝑡+𝐶≤𝐶_{|𝑤𝑛|≥𝑟0}|𝑤_𝑛|^𝑝d𝑡+𝐶.(4.35) By (4.33) and (𝐴₅), 𝐶≥Φ_𝜆𝑛(𝑤_𝑛1)−2Φ^′_𝜆𝑛(𝑤_𝑛)𝑤_𝑛=_ℝ(12𝑊_𝑧(𝑡,𝑤_𝑛)𝑤_𝑛−𝑊(𝑡,𝑤_𝑛))d𝑡≥𝐶_{|𝑤𝑛|≥𝑟0}|𝑤_𝑛|^𝛽d𝑡.(4.36) Choose 𝜈>𝑝 sufficiently large such that (𝜈𝑝(𝑝−2))/(𝜈(𝑝−1)−𝑝)<𝛽. Let 𝑡∶=𝜈(𝑝−𝛽)/(𝜈−𝛽)𝑝, then by (𝐴₅), 0<𝑡<1/(𝑝−1) for 𝜈 being large enough. By Hölder's inequality and Lemma 2.1, we have _{|𝑤𝑛|≥𝑟0}|𝑤_𝑛|^𝑝d𝑡≤(_{|𝑤𝑛|≥𝑟0}|𝑤_𝑛|^𝛽)^{(1−𝑡)𝑝/𝛽}(_ℝ|𝑤_𝑛|^𝜈)^{𝑡𝑝/𝜈}≤𝐶|𝑤_𝑛|_𝜈^𝑡𝑝≤𝐶(‖𝑤⁺_𝑛‖_𝐸+‖𝑤⁻_𝑛‖_𝐸+|𝑤_𝑛|_𝜇)^𝑡𝑝≤𝐶(‖𝑤⁺_𝑛‖_𝐸+‖𝑤⁺_𝑛‖_𝐸^2/𝜇)^𝑡𝑝≤𝐶(‖𝑤⁺_𝑛‖_𝐸^𝑡𝑝+‖𝑤⁺_𝑛‖_𝐸^{2𝑡𝑝/𝜇}).(4.37) By (4.29), (4.35), and (4.37), _ℝ|𝑤_𝑛|^𝜇≤𝐶(‖𝑤⁺_𝑛‖_𝐸^𝑡𝑝+‖𝑤⁺_𝑛‖_𝐸^{2𝑡𝑝/𝜇}+1),(4.38) and then |𝑤_𝑛|_𝜇≤𝐶(‖𝑤⁺_𝑛‖_𝐸^{𝑡𝑝/𝜇}+‖𝑤⁺_𝑛‖^{2𝑡𝑝/𝜇2}_𝐸+1).(4.39) Using (4.3), (4.37), and (4.39), from Φ^′_𝜆𝑛(𝑤_𝑛)𝑤⁺_𝑛=0, we obtain ‖𝑤⁺_𝑛‖²_𝐸≤𝐶_ℝ(|𝑤_𝑛|^𝜇−1|𝑤⁺_𝑛|+|𝑤_𝑛|^𝑝−1|𝑤⁺_𝑛|)d𝑡≤𝐶(|𝑤_𝑛|_𝜇^𝜇−1+(_ℝ|𝑤_𝑛|^𝑝)^{(𝑝−1)/𝑝})‖𝑤⁺_𝑛‖_𝐸≤𝐶(|𝑤_𝑛|_𝜇^𝜇−1+(_{{𝑡∣∣𝑤𝑛(𝑡)∣≤𝑟0}}|𝑤_𝑛|^𝑝+_{{𝑡∣∣𝑤𝑛(𝑡)∣≥𝑟0}}|𝑤_𝑛|^𝑝)^{(𝑝−1)/𝑝})‖𝑤⁺_𝑛‖_𝐸≤𝐶(|𝑤_𝑛|_𝜇^𝜇−1+(_ℝ|𝑤_𝑛|^𝜇)^{(𝑝−1)/𝑝}+(_{{𝑡∣∣𝑤𝑛(𝑡)∣≥𝑟0}}|𝑤_𝑛|^𝑝)^{(𝑝−1)/𝑝})‖𝑤⁺_𝑛‖_𝐸≤𝐶(|𝑤_𝑛|_𝜇^𝜇−1+|𝑤_𝑛|_𝜇^{𝜇(𝑝−1)/𝑝}+‖𝑤⁺_𝑛‖_𝐸^{𝑡(𝑝−1)}+‖𝑤⁺_𝑛‖_𝐸^{2𝑡(𝑝−1)/𝜇})‖𝑤⁺_𝑛‖_𝐸≤𝐶(‖𝑤⁺_𝑛‖_𝐸^{𝑡𝑝(𝜇−1)/𝜇}+‖𝑤⁺_𝑛‖^{2𝑡𝑝(𝜇−1)/𝜇2}_𝐸+‖𝑤⁺_𝑛‖_𝐸^{𝑡(𝑝−1)}+‖𝑤⁺_𝑛‖_𝐸^{2𝑡(𝑝−1)/𝜇}+1)‖𝑤⁺_𝑛‖_𝐸,(4.40) which implies sup‖𝑤⁺_𝑛‖_𝐸<∞ since 𝑡(𝑝−1)<1.

Proof of Theorem 1.1. Since {𝑤_𝑛} is bounded, 𝑤_𝑛⇀𝑤 in 𝐸_𝜇 and 𝑤_𝑛→𝑤 in 𝐿^𝑡_loc for 2≤𝑡<∞. We show that 𝑤≠0.
In fact, by (4.3) and (4.30),
|_ℝ𝑊_𝑧(𝑡,𝑤_𝑛)𝑤⁺_𝑛|≤𝐶_ℝ(|𝑤_𝑛|^𝜇−1+|𝑤_𝑛|^𝑝−1)|𝑤⁺_𝑛|≤𝐶(|𝑤_𝑛|_𝜇^𝜇−1+|𝑤_𝑛|_𝑝^𝑝−1)‖𝑤⁺_𝑛‖_𝐸≤𝐶(‖𝑤_𝑛‖_𝜇^𝜇−1+‖𝑤_𝑛‖_𝜇^𝑝−1)‖𝑤⁺_𝑛‖_𝐸≤𝐶(‖𝑤⁺_𝑛‖^𝜇_𝐸+‖𝑤⁺_𝑛‖_𝐸^{2(𝜇−1)/𝜇+1}+‖𝑤⁺_𝑛‖^𝑝_𝐸+‖𝑤⁺_𝑛‖_𝐸^{2(𝑝−1)/𝜇+1}).(4.41)
It follows from Φ^′_𝜆𝑛(𝑤_𝑛)𝑤⁺_𝑛=0 that
‖𝑤⁺_𝑛‖²_𝐸=𝜆_𝑛ℝ𝑊_𝑧(𝑡,𝑤_𝑛)𝑤⁺_𝑛≤𝐶(‖𝑤⁺_𝑛‖^𝜇_𝐸+‖𝑤⁺_𝑛‖_𝐸^{2(𝜇−1)/𝜇+1}+‖𝑤⁺_𝑛‖^𝑝_𝐸+‖𝑤⁺_𝑛‖_𝐸^{2(𝑝−1)/𝜇+1}),(4.42) which implies that there exists 𝐶₀>0 such that ‖𝑤⁺_𝑛‖_𝐸≥𝐶₀.
If {𝑤⁺_𝑛} is vanishing, then
‖𝑤⁺_𝑛‖²_𝐸=𝜆_𝑛ℝ𝑊_𝑧(𝑡,𝑤_𝑛)𝑤⁺_𝑛→0,(4.43) a contradiction. Hence {𝑤⁺_𝑛} is nonvanishing.
Just along the proof of Lemma 4.6, we can see that there exist 𝑀>0 and 𝑎>0 such that
_{𝐵(0,𝑎+1)}|𝑤⁺_𝑛𝑀|d𝑡≥2,(4.44) where 𝑤⁺_𝑛∶=𝑤⁺_𝑛(𝑡+𝑦_𝑛).
Set 𝑤⁻_𝑛∶=𝑤⁻_𝑛(𝑡+𝑦_𝑛) and 𝑤_𝑛=𝑤⁺_𝑛+𝑤⁻_𝑛. Then sup‖𝑤_𝑛‖_𝜇<∞ and then 𝑤_𝑛⇀𝑤, 𝑤⁺_𝑛⇀𝑤⁺, and 𝑤⁻_𝑛⇀𝑤⁻. By Lemma 2.1, 𝑤⁺_𝑛→𝑤⁺ in 𝐿²_loc, and hence
_{𝐵(0,𝑎+1)}|𝑤⁺|²≥𝑀2>0.(4.45) It follows that 𝑤≠0.
Since Φ^′_𝜆𝑛(𝑤_𝑛)=0, using Lebesgue's theorem, then we obtain
−Φ^′(𝑤)𝜙=Φ^′_𝜆𝑛(𝑤_𝑛)𝜙−Φ^′_𝜆𝑛(𝑤)𝜙+Φ^′_𝜆𝑛(𝑤)𝜙−Φ^′(𝑤)𝜙=⟨𝑤⁺_𝑛−𝑤⁺,𝜙⟩_𝐸−𝜆_𝑛⟨𝑤⁻_𝑛−𝑤⁻,𝜙⟩_𝐸−𝜆_𝑛ℝ(𝑊_𝑧(𝑡,𝑤_𝑛)−𝑊_𝑧(𝑡,𝑤))𝜙+(1−𝜆_𝑛)⟨𝑤,𝜙⟩_𝐸+(1−𝜆_𝑛)_ℝ𝑊_𝑧(𝑡,𝑤)𝜙→0,(4.46) for any 𝜙∈𝐶^∞₀, that is, Φ^′(𝑤)=0.

Proof of Theorem 1.3. It is easy to check that 𝑊_𝑧(𝑡,𝑧)=𝑎|𝑧|^𝜇−2𝑧+𝑅_𝑧(𝑡,𝑧) satisfies all the assumptions of Theorem 1.1 with 𝑏₀=𝛽=𝜇.

Acknowledgments

The authors would like to thank the reviewers for the valuable suggestions. This work was supported by the Natural Science Foundation of China.