Weak Solutions for a 𝑝-Laplacian Impulsive Differential Equation

Abstract

By the virtue of variational method and critical point theory, we give some existence results of weak solutions for a 𝑝-Laplacian impulsive differential equation with Dirichlet boundary conditions.

1. Introduction

In this paper, we shall consider the following problem: −𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′=𝑓(𝑡,𝑢(𝑡)),a.e[],𝑢.𝑡∈0,𝑇ΔΦ^′𝑡_𝑗=𝑢^′𝑡⁺_𝑗^𝑝−2𝑢^′𝑡⁺_𝑗−𝑢^′𝑡⁻_𝑗^𝑝−2𝑢^′𝑡⁻_𝑗=𝐼_𝑗𝑢𝑡_𝑗𝑢(0)(𝑇),𝑗=1,2,…,𝑚,=𝑢=0, (1) where 𝑝>1, 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑚<𝑡_𝑚+1=𝑇, 𝑓∶[0,𝑇]×ℝ→ℝ and 𝐼_𝑗∶ℝ→ℝ, (𝑗=1,2,…,𝑚) are continuous.

Many evolution processes are characterized by the fact that, at certain moments of time, they experience a change of state abruptly. These processes are subject to short-term perturbations whose duration is negligible in comparison with the duration of the process. Consequently, it is natural to assume that these perturbations act instantaneously, that is, in the form of impulses. Thus impulsive differential equations appear as a natural description of observed evolution phenomena of several real world problems.

Recently, there have been many papers to study impulsive problems by variational method and critical point theory, such as [1–11] and the references therein.

In [7], Nieto and O’Regan studied the linear Dirichlet impulsive problem −𝑢^′′(𝑡)+𝜆𝑢(𝑡)=𝜎(𝑡),𝑡≠𝑡_𝑗,𝑡∈[0,𝑇],Δ𝑢^′𝑡_𝑗=𝑑_𝑗𝑢(0)(𝑇),𝑗=1,2,…,𝑝,=𝑢=0 (2) and the nonlinear Dirichlet impulsive problem −𝑢^′′(𝑡)+𝜆𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑡≠𝑡_𝑗,𝑡∈[0,𝑇],Δ𝑢^′𝑡_𝑗=𝐼_𝑗𝑢𝑡⁻_𝑗𝑢(0)(𝑇),𝑗=1,2,…,𝑝,=𝑢=0. (3) In the paper, they have shown that the impulsive problem minimizes some (energy) functional, and the critical points of that functional are indeed solutions of the impulsive problem.

In [3, 4], Sun et al. utilized some variant fountain theorems by [12] to consider the existence of infinitely many solutions for the following two impulsive problems: −𝑢^′′(𝑡)+𝑔(𝑡)𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),a.e.𝑡∈[0,𝑇],Δ𝑢^′𝑡_𝑗=𝐼_𝑗𝑢𝑡_𝑗𝑢(0)(𝑇)−̈(𝑡)(),,𝑗=1,2,…,𝑝,=𝑢=0,𝑢+𝐴𝑢=∇𝑊𝑡,𝑢a.e[],Δ𝑢̇𝑡.𝑡∈0,𝑇_𝑗=𝐼_𝑖𝑗𝑢^𝑖𝑡_𝑗𝑢(0)(𝑇)=𝑢̇(0)−𝑢̇(𝑇),𝑖=1,2,…,𝑁,𝑗=1,2,…,𝑙,−𝑢=0. (4) Admittedly, they obtained many perfect results.

We also note that some people begin to study 𝑝-Laplacian differential equations with impulsive effects; for example, see [1, 2, 8–11].

In [1], Chen and Tang considered the 𝑝-Laplacian impulsive problem −𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′+𝑔(𝑡)|𝑢(𝑡)|^𝑝−2𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),a.e.𝑡∈[0,𝑇],Δ𝑢^′𝑡_𝑗=𝐼_𝑗𝑢𝑡_𝑗𝑢(0)(𝑇),𝑗=1,2,…,𝑚,=𝑢=0. (5) They established some existence theorems for one or infinitely many solutions under more relaxed assumptions on their nonlinearity 𝑓, which satisfies a kind of new superquadratic and subquadratic condition.

In [8], Bogun discussed the existence of weak solutions for the 𝑝-Laplacian problem with superlinear impulses by the virtue of mountain pass theorem and symmetric mountain pass theorem −𝑢^′𝑝−2𝑢^′′=𝑓(𝑡,𝑢),in𝑢(0)(1)𝑢𝑡Ω,=𝑢=0,_+𝑗−𝑢𝑡⁻_𝑗Δ𝑢=0,𝑗=1,2,…,𝑛,_′𝑡_𝑗^𝑝−2𝑢^′𝑡_𝑗=𝐼_𝑗𝑢𝑡_𝑗,𝑗=1,2,…,𝑛. (6)

In [9], Xu et al. reconsidered the previous problem by topological degree theory and Fountain theorem under Cerami condition.

In [11], by the virtue of three critical points theorem obtained by Bai and Dai is studied the existence of at least three solutions for the following 𝑝-Laplacian boundary value problem: 𝜌(𝑡)𝜙_𝑝𝑢^′(𝑡)^′−𝑠(𝑡)𝜙_𝑝(𝑢(𝑡))+𝜆𝑓(𝑡,𝑢(𝑡))=0,a.e(),𝛼.𝑡∈𝑎,𝑏₁𝑢^′𝑎⁺−𝛼₂𝑢(𝑎)=0,𝛽𝑢^′(𝑏⁻)+𝛽₂𝑢(𝑏)Δ𝜌𝑡=0,_𝑗𝜙_𝑝𝑢^′𝑡_𝑗=𝐼_𝑗𝑢𝑡_𝑗,𝑗=1,2,…,𝑙. (7)

Motivated by the previous facts, in this paper, our aim is to study the existence and multiplicity of weak solutions for impulsive problem (1) by using variational method and critical point theory. It is well known that the Ambrosetti-Rabinowitz type condition is to ensure the boundedness of all (PS) sequences of the corresponding functional. However, without it, it will become more complicated. Therefore, we will use new variant fountain theorems due to Zou [12] to overcome this difficulty and obtain infinitely many weak solutions for (1). On the other hand, for the superlinear at +∞ and asymptotically linear at −∞, we obtain a weak solution for (1) by the mountain pass theorem. The results obtained here improve some existing results in the literature.

2. Preliminaries

In this section, we recall some fundamental facts of critical point theory which will be used in the proofs of our main results. Let 𝑊 be the Sobolev space 𝑊₀^1,𝑝(0,𝑇) with the usual norm ‖𝑢‖=^𝑇₀𝑢^′(𝑡)^𝑝d𝑡^1/𝑝,∀𝑢∈𝑊₀^1,𝑝(0,𝑇). (8) It is clear that 𝑊₀^1,𝑝(0,𝑇) is a reflexive Banach space. Next, we make a finite dimensional decomposition for 𝑊. In order to do this, we first need to consider the eigenvalue problem 𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′+𝜆|𝑢(𝑡)|^𝑝−2𝑢(𝑡)[],𝑢(0)(𝑇)=0,𝑡∈0,𝑇=𝑢=0. (9) It is well known that the set of all eigenvalues of the problem (9) is given by the sequence of positive numbers (see [1, 13–15]) 𝜆_𝑘∶=𝑝−1𝑘𝜋_𝑝𝑇^𝑝,for𝑘=1,2,…,where𝜋_𝑝∶=2𝜋.𝑝sin𝜋/𝑝 (10) We denote by 𝜑_𝑘 the corresponding eigenfunctions associated with 𝜆_𝑘 for all 𝑘, and 𝜑_𝑘∈𝑊. Moreover, the first eigenvalue 𝜆₁ is simple and isolated, and 𝜑₁ is positive in [0,𝑇]. Furthermore, the Poincaré inequality ^𝑇₀𝑢^′(𝑡)^𝑝d𝑡≥𝜆₁^𝑇₀|𝑢(𝑡)|^𝑝d𝑡,∀𝑢∈𝑊₀^1,𝑝(0,𝑇) (11) holds. Note that we can normalize 𝜑_𝑘 such that ^𝑇₀𝜑_𝑘(𝑡)^𝑝d𝑡=1,∀𝑘. (12) Fix any 𝑘≥1 define 𝑌_𝑘=span{𝜑₁,…,𝜑_𝑘} and 𝑍_𝑘=^𝑘_𝑗=1kerℒ_𝑗=𝑢∈𝑊∶ℒ₁(𝑢)=⋯=ℒ_𝑘(𝑢)=0, (13) where ℒ_𝑗𝑢=^𝑇₀𝜑_𝑗(𝑡)^𝑝−2𝜑_𝑗(𝑡)𝑢(𝑡)d𝑡. (14) By [16, Section 5], the conclusions are 𝑊=𝑌_𝑘⊕𝑍_𝑘,dim𝑌_𝑘𝜆=𝑘,_𝑘+1𝑇0|𝑢(𝑡)|^𝑝d𝑡≤^𝑇₀𝑢^′(𝑡)^𝑝d𝑡,∀𝑢∈𝑍_𝑘,𝑘≥1. (15) Note the definitions of 𝜆_𝑘 and 𝜑_𝑘; by (9), we have 𝜑^′_𝑘(𝑡)^𝑝−2𝜑^′_𝑘(𝑡)^′+𝜆_𝑘𝜑_𝑘(𝑡)^𝑝−2𝜑_𝑘(𝑡)=0for𝑡∈[0,𝑇]. (16) For each 𝑣∈𝑊, multiply by 𝑣 on both sides of (16) to obtain ^𝑇₀𝜑^′_𝑘(𝑡)^𝑝−2𝜑^′_𝑘(𝑡)𝑣^′(𝑡)d𝑡=𝜆_𝑘^𝑇₀𝜑_𝑘(𝑡)^𝑝−2𝜑_𝑘(𝑡)𝑣(𝑡)d𝑡. (17) In particular, choosing 𝑣=𝜑_𝑘, we see ∫^𝑇₀|𝜑^′_𝑘(𝑡)|^𝑝d𝑡=𝜆_𝑘 for all 𝑘.

We denote the norms in 𝐿^𝑟(0,𝑇)(1<𝑟<∞) and 𝐶[0,𝑇] as follows: ‖𝑢‖_𝑟∶=^𝑇₀|𝑢(𝑡)|^𝑟d𝑡^1/𝑟,∀𝑢∈𝐿^𝑟(),‖𝑢‖0,𝑇_∞∶=max_{𝑡∈[0,𝑇]}|𝑢(𝑡)|,∀𝑢∈𝐶[0,𝑇]. (18) By the Sobolev embedding theorem, the embeddings 𝑊↪𝐿^𝑟(0,𝑇) and 𝑊↪𝐶[0,𝑇] are compact. Consequently, we also find that there are two constants 𝐶_emb1>0 and 𝐶_emb2>0 such that ‖𝑢‖_𝑟≤𝐶_emb1‖𝑢‖,‖𝑢‖_∞≤𝐶_emb2‖𝑢‖,∀𝑢∈𝑊. (19)

For 𝑢∈𝑊^1,𝑝(0,𝑇), we have that 𝑢 and 𝑢^′ are both absolutely continuous. Hence, Δ𝑢^′(𝑡)=𝑢^′(𝑡⁺)−𝑢^′(𝑡⁻)=0 for any 𝑡∈[0,𝑇]. If 𝑢∈𝑊₀^1,𝑝(0,𝑇), then 𝑢 is absolutely continuous. In this case, the one-sided derivatives 𝑢^′(𝑡⁺), 𝑢^′(𝑡⁻) may not exist. It leads to the impulsive effects. As a result, we need to introduce a different concept of solution. Suppose that 𝑢∈𝐶[0,𝑇] satisfies the Dirichlet condition 𝑢(0)=𝑢(𝑇)=0. Assume that, for every 𝑗=1,2,…,𝑚, 𝑢_𝑗=𝑢|_{(𝑡𝑗,𝑡𝑗+1)} and 𝑢_𝑗∈𝑊^1,𝑝(0,𝑇). Let 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑚<𝑡_𝑚+1=𝑇.

Take 𝑣∈𝑊 and multiply the two sides of the equality −𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′=𝑓(𝑡,𝑢(𝑡)) (20) by 𝑣 and integrate from 0 to 𝑇: ^𝑇₀−𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′𝑣(𝑡)d𝑡=^𝑇₀𝑓(𝑡,𝑢(𝑡))𝑣(𝑡)d𝑡. (21) For the left term, in view of impulsive effects, we find ^𝑇₀−𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′𝑣(𝑡)d𝑡=^𝑚_𝑗=0^𝑡𝑗+1_𝑡𝑗−𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′𝑣(𝑡)d𝑡=^𝑚_𝑗=1𝐼_𝑗𝑢𝑡_𝑗𝑣𝑡_𝑗−𝑣(𝑇)𝑢^′(𝑇)^𝑝−2𝑢^′(𝑇)+𝑣(0)𝑢^′(0)^𝑝−2𝑢^′(0)+^𝑇₀𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)𝑣^′(𝑡)d𝑡=^𝑚_𝑗=1𝐼_𝑗𝑢𝑡_𝑗𝑣𝑡_𝑗+^𝑇₀𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)𝑣^′(𝑡)d𝑡. (22) Consequently, ^𝑇₀𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)𝑣^′(𝑡)d𝑡+^𝑚_𝑗=1𝐼_𝑗𝑢𝑡_𝑗𝑣𝑡_𝑗=^𝑇₀𝑓(𝑡,𝑢(𝑡))𝑣(𝑡)d𝑡. (23) Considering the previous, we introduce the following concept for the solution for (1).

Definition 1. One says that a function 𝑢∈𝑊 is a weak solution for (1) if the identity ^𝑇₀𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)𝑣^′(𝑡)d𝑡+^𝑚_𝑗=1𝐼_𝑗𝑢𝑡_𝑗𝑣𝑡_𝑗=^𝑇₀𝑓(𝑡,𝑢(𝑡))𝑣(𝑡)d𝑡,∀𝑣∈𝑊. (24)
Consider the functional 𝐽∶𝑊→ℝ defined by 𝐽(𝑢)=1𝑝‖𝑢‖^𝑝+^𝑚_𝑗=1^{𝑢(𝑡𝑗)}₀𝐼_𝑗(𝑡)d𝑡−^𝑇₀𝐹(𝑡,𝑢(𝑡))d𝑡, (25) where 𝐹(𝑡,𝑢)=∫^𝑢₀𝑓(𝑡,𝑠)d𝑠. Note that for the continuity of 𝑓 and 𝐼_𝑗(𝑗=1,2,…,𝑚), we see 𝐽∈𝐶¹(𝑊,ℝ). Furthermore, the derivative of 𝐽 is 𝐽^′(𝑢),𝑣=^𝑇₀𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)𝑣^′(𝑡)d𝑡+^𝑚_𝑗=1𝐼_𝑗𝑢𝑡_𝑗𝑣𝑡_𝑗−^𝑇₀𝑓(𝑡,𝑢(𝑡))𝑣(𝑡)d𝑡,∀𝑢,𝑣∈𝑊. (26) Thus, we easily know that weak solutions of (1) coincide with the critical points of the 𝐶¹-functional 𝐽.

For the reader’s convenience, we now present some critical point theorems; one can refer to [12, 17–22] for more details.

Definition 2. Let 𝑋 be a real Banach space and 𝐽∈𝐶¹(𝑋,ℝ). For any sequence {𝑢_𝑛}⊂𝑋, if {𝐽(𝑢_𝑛)} is bounded and 𝐽^′(𝑢_𝑛)→0 as 𝑛→∞ possesses a convergent subsequence, then we say that 𝐽 satisfies the Palais-Smale condition (PS condition for short).

Definition 3. One says that 𝐽 satisfies (PS)_𝑐 condition if the existence of a sequence {𝑢_𝑛}⊂𝑋 such that 𝐽(𝑢_𝑛)→𝑐 and 𝐽^′(𝑢_𝑛)→0 as 𝑛→∞ implies that {𝑢_𝑛} has a convergent subsequence.

Lemma 4 (see [17]). Let 𝐽∈𝐶¹(𝑋,ℝ) satisfy (PS) condition. Suppose that(i)𝐽(0)=0,(ii)there exist 𝜌>0 and 𝛼>0 such that 𝐽(𝑢)≥𝛼 for all 𝑢∈𝑋, ‖𝑢‖=𝜌,(iii)there exists 𝑢₁∈𝑋 with ‖𝑢₁‖>𝜌 such that 𝐽(𝑢₁)<𝛼.
Then 𝐽 has a critical value 𝑐≥𝛼. Moreover, 𝑐 can be characterized as inf_𝑔∈Γmax_{𝑢∈𝑔([0,1])}𝐽(𝑢), where Γ∶={𝑔∈𝐶([0,1],𝑋)∶𝑔(0)=0,𝑔(1)=𝑢₁}.
Let 𝑋 be a Banach space equipped with the norm ‖⋅‖ and 𝑋=⨁_𝑗∈ℕ𝑋_𝑗, where dim𝑋_𝑗<∞ for any 𝑗∈ℕ. Set 𝑌_𝑘=⨁^𝑘_𝑗=1𝑋_𝑗 and 𝑍_𝑘=⨁^∞_𝑗=𝑘𝑋_𝑗. In the following, one will introduce variant fountain theorems by Zou [12]. Let 𝑋 and the subspaces 𝑌_𝑘 and 𝑍_𝑘 be defined as previously. Consider the following 𝐶¹-functional 𝐽_𝜆∶𝑋→ℝ defined by 𝐽_𝜆(𝑢)∶=𝐴(𝑢)−𝜆𝐵(𝑢),𝜆∈[1,2]. (27)

Lemma 5. If the functional 𝐽_𝜆 satisfies the following:(T1)𝐽_𝜆 maps bounded sets to bounded sets uniformly for 𝜆∈[1,2]. Moreover, 𝐽_𝜆(−𝑢)=𝐽_𝜆(𝑢) for all (𝜆,𝑢)∈[1,2]×𝑋,(T2)𝐵(𝑢)≥0 for all 𝑢∈𝑋; moreover, 𝐴(𝑢)→∞ or 𝐵(𝑢)→∞ as ‖𝑢‖→∞,(T3) there exists 𝑟_𝑘>𝜌_𝑘>0 such that 𝑎_𝑘(𝜆)∶=inf_{𝑢∈𝑍𝑘,‖𝑢‖=𝜌𝑘}𝐽_𝜆(𝑢)>𝑏_𝑘(𝜆)∶=max_{𝑢∈𝑌𝑘,‖𝑢‖=𝑟𝑘}𝐽_𝜆(𝑢),∀𝜆∈[1,2], (28)then 𝑎_𝑘(𝜆)≤𝜁_𝑘(𝜆)=inf_{𝛾∈Γ𝑘}max_{𝑢∈𝐵𝑘}𝐽_𝜆𝛾(𝑢),∀𝜆∈[1,2], (29) where 𝐵_𝑘={𝑢∈𝑌_𝑘∶‖𝑢‖≤𝑟_𝑘} and Γ_𝑘={𝛾∈𝐶(𝐵_𝑘,𝑋)∶𝛾isodd,𝛾|_𝜕𝐵𝑘=id}. Moreover, for a.e. 𝜆∈[1,2], there exists a sequence {𝑢^𝑘_𝑛(𝜆)}^∞_𝑛=1 such that sup_𝑛𝑢^𝑘_𝑛(𝜆)<∞,𝐽^′_𝜆𝑢^𝑘_𝑛(𝜆)𝐽⟶0,_𝜆𝑢^𝑘_𝑛(𝜆)→𝜁_𝑘(𝜆)as𝑛⟶∞. (30)

Now, we list our assumptions on 𝑓 and 𝐼_𝑗(𝑗=1,2,…,𝑚).(H1) There exist 𝜇₁,𝜇₂>0 such that |𝑓(𝑡,𝑢)|≤𝜇₁+𝜇₂|𝑢|^𝑞−1(1<𝑞<∞), ∀𝑢∈ℝ,𝑡∈[0,𝑇],(H2)lim_𝑢→0(𝑓(𝑡,𝑢)/|𝑢|^𝑝−2𝑢)=𝑓₀∈[0,+∞) uniformly for 𝑡∈[0,𝑇],(H3)lim_{𝑢→−∞}(𝑓(𝑡,𝑢)/|𝑢|^𝑝−2𝑢)=𝑓_∞∈[0,+∞) uniformly for 𝑡∈[0,𝑇],(H4) There exist 𝛿>0 and 𝜃∈(0,1/𝑝) such that 0<𝐹(𝑡,𝑢)≤𝜃𝑢𝑓(𝑡,𝑢),𝑡∈[0,𝑇],|𝑢|≥𝛿, (31)(H5)𝐹(𝑡,𝑢)≥0 for all (𝑡,𝑢)∈[0,𝑇]×ℝ and lim_|𝑢|→∞(𝐹(𝑡,𝑢)/|𝑢|^𝑝)=+∞, uniformly on 𝑡∈[0,𝑇],(H6) there is a positive constant 𝑏>0 such that lim_|𝑢|→∞((−𝑝𝐹(𝑡,𝑢)+𝑓(𝑡,𝑢)𝑢)/|𝑢|^𝑞)≥𝑏, uniformly on 𝑡∈[0,𝑇],(H7)∫^𝑢₀𝐼_𝑗(𝑠)d𝑠≥0, ∀𝑢∈ℝ, 𝑗=1,2,…,𝑚,(H8)𝑝∫^𝑢₀𝐼_𝑗(𝑠)d𝑠−𝐼_𝑗(𝑢)𝑢≥0, ∀𝑢∈ℝ, 𝑗=1,2,…,𝑚,(H9) There exist 𝜎_𝑗,𝜏_𝑗>0 and 𝛾_𝑗∈[1,𝑝) such that |𝐼_𝑗(𝑢)|≤𝜎_𝑗+𝜏_𝑗|𝑢|^𝛾𝑗−1, ∀𝑢∈ℝ and 𝑗=1,2,…,𝑚,(H10)𝑓(𝑡,𝑢) and 𝐼_𝑗(𝑢)(𝑗=1,2,…,𝑚) are odd functions about 𝑢, for all 𝑡∈[0,𝑇].

Remark 6. (1) As known to all, (H4) implies that lim_𝑢→∞(𝐹(𝑡,𝑢)/|𝑢|^𝑝)=+∞; that is, 𝑓(𝑡,𝑢) is superlinear at ∞ with respect to |𝑢|^𝑝−2𝑢. In view of (H3) and (H4), we see that 𝑓(𝑡,𝑢) is superlinear at +∞ and asymptotically linear at −∞; this is a new case. However, the nonlinearity 𝑓 in [9] is asymptotically linear at ±∞.
(2) Condition (H6) is weaker than the well-known Ambrosetti-Rabinowitz condition (H4); also see condition (𝑝₂) in [8]. Indeed, by (H6), there is a 𝛿>0 such that −𝑝𝐹(𝑡,𝑢)+𝑢𝑓(𝑡,𝑢)≥𝑏|𝑢|^𝑞≥0,∀|𝑢|≥𝛿. (32) Consequently, 𝑝𝐹(𝑡,𝑢)≤𝑢𝑓(𝑡,𝑢) for all 𝑡∈[0,𝑇] and |𝑢|≥𝛿, which is weaker than condition (H4).

3. Main Results

Theorem 7. Suppose that (H1)–(H4), (H7), and (H9) hold, 𝑞∈(𝑝,+∞), and 𝑓₀<𝜆₁<𝑓_∞ with 𝑓_∞≠𝜆_𝑘 for all 𝑘. Then (1) has a weak solution.

Proof. From (H1)–(H4), for all 𝜀>0, there exist 𝜇₃>0,𝜇₄>0 such that 𝐹()≤1𝑡,𝑢𝑝𝑓₀+𝜀|𝑢|^𝑝+𝜇₃|𝑢|^𝑞,∀𝑡∈[0,𝑇],𝑢∈ℝ,𝐹()≥1𝑡,𝑢𝑝𝑓_∞−𝜀|𝑢|^𝑝−𝜇₄,∀𝑡∈[0,𝑇],𝑢∈ℝ. (33)(34) Choose 𝜀>0 such that (𝑓₀+𝜀)<𝜆₁, together with (33), (11), (19), and (H7); we obtain 𝐽(𝑢)=1𝑝‖𝑢‖^𝑝+^𝑚_𝑗=1^{𝑢(𝑡𝑗)}₀𝐼_𝑗(𝑡)d𝑡−^𝑇₀𝐹(𝑡,𝑢(𝑡))d𝑡≥1𝑝‖𝑢‖^𝑝−^𝑇₀𝐹(𝑡,𝑢(𝑡))d𝑡≥1𝑝‖𝑢‖^𝑝−^𝑇₀1𝑝𝑓₀+𝜀|𝑢|^𝑝+𝜇₃|𝑢|^𝑞d𝑡≥1𝑝‖𝑢‖^𝑝−1𝑝𝑓₀+𝜀𝜆₁‖𝑢‖^𝑝−𝜇₃𝐶^𝑞_emb1‖𝑢‖^𝑞. (35) If 𝜌 is small enough, (ii) of Lemma 4 can be proved.
On the other hand, we can take 𝜀>0 such that 𝑓_∞−𝜀>𝜆₁; by (34), (19), and (H9), noting that 𝛾_𝑗∈[1,𝑝), we find 𝐽𝑠𝜑₁=1𝑝𝑠𝜑₁^𝑝+^𝑚_𝑗=1^{𝑠𝜑1(𝑡𝑗)}₀𝐼_𝑗(𝑡)d𝑡−^𝑇₀𝐹𝑡,𝑠𝜑₁(𝑡)d𝑡≤1𝑝𝑠𝜑₁^𝑝+^𝑚_𝑗=1^{𝑠𝜑1(𝑡𝑗)}₀𝜎_𝑗+𝜏_𝑗|𝑡|^𝛾𝑗−1d𝑡−^𝑇₀1𝑝𝑓_∞−𝜀𝑠𝜑₁^𝑝−𝜇₄d𝑡≤1𝑝𝑠𝜑₁^𝑝−1𝑝𝑓_∞−𝜀𝜆₁𝑠𝜑₁^𝑝+^𝑚_𝑗=1𝜎_𝑗𝑠𝜑_1∞+𝜏_𝑗𝛾_𝑗𝑠𝜑₁^𝛾𝑗_∞+𝜇₄𝑇≤1𝑝𝑠𝜑₁^𝑝−1𝑝𝑓_∞−𝜀𝜆₁𝑠𝜑₁^𝑝+^𝑚_𝑗=1𝜎_𝑗𝐶_emb2𝑠𝜑₁+𝜏_𝑗𝛾_𝑗𝐶^𝛾𝑗_emb2𝑠𝜑₁^𝛾𝑗+𝜇₄𝑇⟶−∞as𝑠⟶−∞. (36) Therefore, (iii) of Lemma 4 is also proved, as required.
Now, we only claim that 𝐽 satisfies (PS) condition. Supposing that {𝑢_𝑛}⊂𝑊 is a (PS) sequence, for all 𝑛∈ℕ, we have 1𝑝𝑢_𝑛^𝑝+^𝑚_𝑗=1^{𝑢𝑛(𝑡𝑗)}₀𝐼_𝑗(𝑡)d𝑡−^𝑇₀𝐹𝑡,𝑢_𝑛(𝑡)d𝑡≤𝜁,_𝑇0𝑢^′_𝑛(𝑡)^𝑝−2𝑢^′_𝑛(𝑡)𝑣^′(𝑡)d𝑡+^𝑚_𝑗=1𝐼_𝑗𝑢_𝑛𝑡_𝑗𝑣𝑡_𝑗−^𝑇₀𝑓𝑡,𝑢_𝑛(𝑡)𝑣(𝑡)d𝑡≤𝜀_𝑛‖𝑣‖,∀𝑣∈𝑊, (37)(38) where 𝜁>0 is a constant and 𝜀_𝑛→0⁺ as 𝑛→∞. Next, we will show that {𝑢_𝑛} is a bounded sequence in 𝑊. If not, there is a subsequence of {𝑢_𝑛}, still denoted by {𝑢_𝑛}, such that 𝑢_𝑛⟶∞,𝑛⟶∞. (39) Define 𝑧_𝑛=𝑢_𝑛/‖𝑢_𝑛‖, then ‖𝑧_𝑛‖=1,∀𝑛∈ℕ, and thus it has a subsequence, still denoted {𝑧_𝑛}, such that 𝑧_𝑛⇀𝑧₀ weakly in 𝑊, 𝑧_𝑛→𝑧₀ strongly in 𝐿^𝑝(0,𝑇), 𝑧_𝑛(𝑡)→𝑧₀(𝑡)a.e.𝑡∈[0,𝑇], and |𝑧_𝑛(𝑡)|≤𝜂(𝑡), a.e.𝑡∈[0,𝑇], where 𝑧₀∈𝑊, 𝜂∈𝐿^𝑝(0,𝑇).
Divide (38) by ‖𝑢_𝑛‖^𝑝−1 to get ^𝑇₀𝑧^′_𝑛(𝑡)^𝑝−2𝑧^′_𝑛(𝑡)𝑣^′(𝑡)d𝑡+^𝑚_𝑗=1𝐼_𝑗𝑢_𝑛𝑡_𝑗𝑢_𝑛^𝑝−1𝑣𝑡_𝑗−^𝑇₀𝑓𝑡,𝑢_𝑛(𝑡)𝑢_𝑛^𝑝−1𝑣(𝑡)d𝑡≤𝜀_𝑛𝑢_𝑛^𝑝−1‖𝑣‖,∀𝑣∈𝑊. (40) Passing to the limit in (40), we see lim_𝑛→∞^𝑇₀𝑓𝑡,𝑢_𝑛(𝑡)𝑢_𝑛^𝑝−1𝑣(𝑡)d𝑡=^𝑇₀𝑧^′₀(𝑡)^𝑝−2𝑧^′₀(𝑡)𝑣^′(𝑡)d𝑡,∀𝑣∈𝑊, (41) with the fact that ^𝑚_𝑗=1𝐼_𝑗𝑢_𝑛𝑡_𝑗𝑢_𝑛^𝑝−1𝑣𝑡_𝑗≤^𝑚_𝑗=1𝐼_𝑗𝑢_𝑛𝑡_𝑗𝑢_𝑛^𝑝−1𝑣𝑡_𝑗≤^𝑚_𝑗=1𝜎_𝑗+𝜏_𝑗𝐶^𝛾𝑗−1_emb2𝑢_𝑛^𝛾𝑗−1𝑢_𝑛^𝑝−1𝐶_emb2‖𝑣‖⟶0,∀𝑣∈𝑊. (42) Now, we claim that 𝑧₀(𝑡)≤0, a.e.𝑡∈[0,𝑇]. Indeed, in (41), taking 𝑣=𝑧⁺₀=max{𝑧₀,0}, we arrive at lim_{𝑛→∞Ω+}𝑓𝑡,𝑢_𝑛(𝑡)𝑢_𝑛^𝑝−1𝑧₀(𝑡)d𝑡=_Ω+𝑧^′₀(𝑡)^𝑝d𝑡<+∞, (43) where Ω⁺∶={𝑡∈[0,𝑇]∶𝑧₀(𝑡)>0}. However, by (H3) and (H4), there is 𝜇₅>0 such that 𝑓𝑡,𝑢_𝑛(𝑡)𝑢_𝑛^𝑝−1𝑧₀(𝑡)≥−𝑓_∞𝜂(𝑡)^𝑝−2𝜂(𝑡)−𝜇₅𝑧₀(𝑡),a.e.𝑡∈[0,𝑇]. (44) In addition, note that lim_𝑛→∞𝑢_𝑛(𝑡)=+∞a.e.𝑡∈Ω⁺ and the super-linearity of 𝑓 we have lim_𝑛→∞𝑓𝑡,𝑢_𝑛(𝑡)𝑢_𝑛^𝑝−1𝑧₀(𝑡)=lim_𝑛→∞𝑓𝑡,𝑢_𝑛(𝑡)𝑢_𝑛^𝑝−1𝑧_𝑛^𝑝−1(𝑡)𝑧₀(𝑡)=+∞,a.e.𝑡∈Ω⁺. (45) Consequently, if |Ω⁺|>0, by the Fatou theorem, we get lim_{𝑛→∞Ω+}𝑓𝑡,𝑢_𝑛(𝑡)𝑢_𝑛^𝑝−1𝑧₀(𝑡)d𝑡=+∞, (46) which contradicts the fact of (43).
Obviously, 𝑧₀(𝑡)≢0. From (H2) and (H3), there exists 𝜇₆>0 such that |𝑓(𝑡,𝑢_𝑛)|/|𝑢_𝑛|^𝑝−1≤𝜇₆a.e.𝑡∈[0,𝑇]. By (41), Lebesgue’s dominated convergence theorem enables us to see ^𝑇₀𝑧^′₀(𝑡)^𝑝−2𝑧^′₀(𝑡)𝑣^′(𝑡)d𝑡=𝑓_∞^𝑇₀𝑧₀(𝑡)^𝑝−2𝑧₀(𝑡)𝑣(𝑡)d𝑡,∀𝑣∈𝑊. (47) This contradicts 𝑓_∞≠𝜆_𝑘 for all 𝑘. Therefore, {𝑢_𝑛} is bounded, as required. Going, if necessary, to a subsequence, we can assume that 𝑢_𝑛⇀𝑢 weakly in 𝑊; then 𝐽^′𝑢_𝑛−𝐽^′(𝑢),𝑢_𝑛−𝑢=^𝑇₀𝑢^′_𝑛^𝑝−2𝑢^′_𝑛−𝑢^′𝑝−2𝑢^′𝑢^′_𝑛−𝑢^′d𝑡+^𝑚_𝑗=1𝐼_𝑗𝑢_𝑛𝑡_𝑗−𝐼_𝑗𝑢𝑡_𝑗𝑢_𝑛𝑡_𝑗−𝑢𝑡_𝑗−^𝑇₀𝑓𝑡,𝑢_𝑛−𝑓(𝑡,𝑢)𝑢_𝑛−𝑢d𝑡. (48)𝑊↪↪𝐶[0,𝑇] enables us to obtain that ^𝑚_𝑗=1𝐼_𝑗𝑢_𝑛𝑡_𝑗−𝐼_𝑗𝑢𝑡_𝑗𝑢_𝑛𝑡_𝑗−𝑢𝑡_𝑗⟶0,_𝑇0𝑓𝑡,𝑢_𝑛−𝑓(𝑡,𝑢)𝑢_𝑛−𝑢d𝑡⟶0as𝑛⟶∞. (49) It follows from 𝑢_𝑛⇀𝑢 weakly in 𝑊 and (𝐽^′(𝑢_𝑛)−𝐽^′(𝑢),𝑢_𝑛−𝑢)→0 that ^𝑇₀𝑢^′_𝑛^𝑝−2𝑢^′_𝑛−𝑢^′𝑝−2𝑢^′𝑢^′_𝑛−𝑢^′d𝑡⟶0as𝑛⟶∞. (50) Note that ^𝑇₀𝑢^′_𝑛^𝑝−2𝑢^′_𝑛−𝑢^′𝑝−2𝑢^′𝑢^′_𝑛−𝑢^′d𝑡≥𝑢_𝑛^𝑝−1−‖𝑢‖^𝑝−1𝑢_𝑛−‖𝑢‖, (51) and thus ‖𝑢_𝑛−𝑢‖→0 as 𝑛→∞. So, 𝐽 satisfies (PS) condition. This completes the proof.

In what follows, we will utilize Lemma 5 to study (1). Now, we define a class of functionals on 𝑊 by 𝐽_𝜆(𝑢)=1𝑝‖𝑢‖^𝑝+^𝑚_𝑗=1^{𝑢(𝑡𝑗)}₀𝐼_𝑗(𝑡)d𝑡−𝜆^𝑇₀𝐹(𝑡,𝑢(𝑡))d𝑡=𝐴(𝑢)−𝜆𝐵(𝑢),𝜆∈[1,2]. (52) It is easy to know that 𝐽_𝜆∈𝐶¹(𝑊,ℝ) for all 𝜆∈[1,2] and the critical points of 𝐽₁ correspond to the weak solutions of problem (1). Note that 𝐽₁=𝐽, where 𝐽 is the functional defined in (25).

Theorem 8. Assume that (H1) and (H5)–(H10) hold. Then (1) possesses infinitely many weak solutions.

Proof. We first prove that there is a positive integer 𝑘₁ and two sequences 𝑟_𝑘>𝜌_𝑘→∞ as 𝑘→∞ such that 𝑎_𝑘(𝜆)=inf_{𝑢∈𝑍𝑘,‖𝑢‖=𝜌𝑘}𝐽_𝜆(𝑢)>0,∀𝑘≥𝑘₁,𝑏_𝑘(𝜆)=max_{𝑢∈𝑌𝑘,‖𝑢‖=𝑟𝑘}𝐽_𝜆(𝑢)<0,∀𝑘∈ℕ, (53)(54) where 𝑌_𝑘=span{𝜑₁,…,𝜑_𝑘} and 𝑍_𝑘=⋂^𝑘_𝑗=1ker(ℒ_𝑗).
Step 1. We will show that (53) holds true.
From (H1), we see that there exist 𝜇₇>0, 𝜇₈>0 such that 𝐹(𝑡,𝑢)≤𝜇₇|𝑢|+𝜇₈|𝑢|^𝑞,∀(𝑡,𝑢)∈[0,𝑇]×ℝ. (55) Consequently, by (H5) and (H7), we obtain 𝐽_𝜆(𝑢)=1𝑝‖𝑢‖^𝑝+^𝑚_𝑗=1^{𝑢(𝑡𝑗)}₀𝐼_𝑗(𝑡)d𝑡−𝜆^𝑇₀𝐹(𝑡,𝑢(𝑡))d𝑡≥1𝑝‖𝑢‖^𝑝−2^𝑇₀𝐹(𝑡,𝑢(𝑡))d𝑡≥1𝑝‖𝑢‖^𝑝−2^𝑇₀𝜇₇|𝑢|+𝜇₈|𝑢|^𝑞d𝑡≥1𝑝‖𝑢‖^𝑝−2𝜇₇𝑇^1−1/𝑞‖𝑢‖_𝑞−2𝜇₈‖𝑢‖^𝑞_𝑞. (56) Let 𝜈_𝑞(𝑘)=sup_{𝑢∈𝑍𝑘,‖𝑢‖=1}‖𝑢‖_𝑞, ∀𝑘∈ℕ. Then by [19, Lemma 3.8],  𝜈_𝑞(𝑘)→0 as 𝑘→∞. Since 𝑊↪↪𝐿^𝑞(0,𝑇), we find 𝐽_𝜆(𝑢)≥1𝑝‖𝑢‖^𝑝−2𝜇₇𝑇^1−1/𝑞𝜈_𝑞(𝑘)‖𝑢‖−2𝜇₈𝜈^𝑞_𝑞(𝑘)‖𝑢‖^𝑞. (57) Let 𝜌_𝑘=1/𝜈_𝑞(𝑘)→∞ as 𝑘→∞. Then there exists 𝑘₁ such that (1/𝑝)𝜌^𝑝_𝑘−2𝜇₇𝑇^1−1/𝑞−2𝜇₈>0, ∀𝑘≥𝑘₁. Therefore, 𝑎_𝑘(𝜆)=inf_{𝑢∈𝑍𝑘,‖𝑢‖=𝜌𝑘}𝐽_𝜆(𝑢)≥1𝑝𝜌^𝑝_𝑘−2𝜇₇𝑇^1−1/𝑞−2𝜇₈>0,∀𝑘≥𝑘₁. (58)Step 2. We will show that (54) holds true.
We first prove that there exists 𝜀>0 such that ([]∶|𝑢(𝑡‖𝑢{0}meas𝑡∈0,𝑇)|≥𝜀‖)≥𝜀,∀𝑢∈𝒳⧵,∀𝒳⊂𝑊,dim𝒳<∞. (59) There exists otherwise a sequence {𝑢_𝑛}_𝑛∈ℕ⊂𝒳⧵{0} such that meas𝑡∈[0,𝑇]∶𝑢_𝑛(𝑡≥𝑢)^𝑛𝑛<1𝑛,∀𝑛∈ℕ. (60) For each 𝑛∈ℕ, let 𝑣_𝑛∶=𝑢_𝑛/‖𝑢_𝑛‖∈𝒳⇒‖𝑣_𝑛‖=1,∀𝑛∈ℕ and meas𝑡∈[0,𝑇]∶𝑣_𝑛(𝑡≥1)𝑛<1𝑛,∀𝑛∈ℕ. (61) Passing to a subsequence if necessary, we may assume 𝑣_𝑛→𝑣₀ in 𝑊 for some 𝑣₀∈𝒳 since 𝒳 is of finite dimension. We easily find ‖𝑣₀‖=1. Consequently, there exists a constant 𝜎₀>0 such that meas𝑡∈[0,𝑇]∶𝑣₀(𝑡)≥𝜎₀≥𝜎₀. (62) Indeed, if not, then we have meas𝑡∈[0,𝑇]∶𝑣₀(𝑡≥1)𝑛=0,i.e.,meas𝑡∈[0,𝑇]∶𝑣₀(𝑡<1)𝑛=𝑇,∀𝑛∈ℕ, (63) which implies 0<^𝑇₀𝑣₀(𝑡)^𝑝d𝑇𝑡≤𝑛^𝑝⟶0,as𝑛⟶∞. (64) This leads to 𝑣₀=0, contradicting to ‖𝑣₀‖=1. In view of 𝑊↪↪𝐿^𝑝(0,𝑇) and the equivalence of any two norms on 𝒳, we have ^𝑇₀𝑣_𝑛−𝑣₀^𝑝d𝑡⟶0,as𝑛⟶∞. (65) For every 𝑛∈ℕ, denote 𝒩∶=𝑡∈[0,𝑇]∶𝑣_𝑛(𝑡<1)𝑛,𝒩^𝑐∶=𝑡∈[0,𝑇]∶𝑣_𝑛(𝑡≥1)𝑛, (66) and 𝒩₀∶={𝑡∈[0,𝑇]∶|𝑣₀(𝑡)|≥𝜎₀}, where 𝜎₀ is defined by (62). Then for 𝑛 large enough, by (62), we see meas𝒩∩𝒩₀≥meas𝒩₀−meas𝒩^𝑐≥𝜎₀−1𝑛≥𝜎₀2. (67) Consequently, for 𝑛 large enough, we arrive immediately at ^𝑇₀𝑣_𝑛−𝑣₀^𝑝d𝑡≥_𝒩∩𝒩0𝑣_𝑛−𝑣₀^𝑝d𝑡≥12^𝑝_𝒩∩𝒩0𝑣₀^𝑝d𝑡−_𝒩∩𝒩0𝑣_𝑛^𝑝d𝑡≥𝜎^𝑝₀2^𝑝−1𝑛^𝑝meas𝒩∩𝒩₀≥𝜎₀^𝑝+12^𝑝+2>0. (68) This contradicts (65). Therefore, (59) holds. For the 𝜀 given in (59), we let 𝒩_𝑢∶={𝑡∈[0,𝑇]∶|𝑢(𝑡)|≥𝜀‖𝑢‖},∀𝑢∈𝒳⧵{0}. (69) Then by (59), we find meas𝒩_𝑢≥𝜀,∀𝑢∈𝒳⧵{0}. (70) By (H5), for any 𝑘∈ℕ, there is a constant 𝑆_𝑘>0 such that 𝐹()≥|𝑢|𝑡,𝑢^𝑝𝜀^𝑝+1,∀|𝑢|≥𝑆_𝑘, (71) where 𝜀 is determined in (59). Therefore, 𝐽_𝜆(𝑢)=1𝑝‖𝑢‖^𝑝+^𝑚_𝑗=1^{𝑢(𝑡𝑗)}₀𝐼_𝑗(𝑡)d𝑡−𝜆^𝑇₀𝐹(𝑡,𝑢(𝑡))d𝑡≤1𝑝‖𝑢‖^𝑝+^𝑚_𝑗=1^{𝑢(𝑡𝑗)}₀𝜎_𝑗+𝜏_𝑗|𝑡|^𝛾𝑗−1d𝑡−_𝒩𝑢|𝑢|^𝑝𝜀^𝑝+1d𝑡≤1𝑝−1‖𝑢‖^𝑝+^𝑚_𝑗=1𝜎_𝑗𝐶_emb2‖𝑢‖+𝜏_𝑗𝛾_𝑗𝐶^𝛾𝑗_emb2‖𝑢‖^𝛾𝑗. (72) Now for any 𝑘∈ℕ, if we take 𝑟_𝑘>max{𝜌_𝑘,𝑆_𝑘/𝜀}, noting that 𝑝>1 and 𝑝>𝛾_𝑗 and ‖𝑢‖=𝑟_𝑘 large enough, we have 𝑏_𝑘(𝜆)=max_{𝑢∈𝑌𝑘,‖𝑢‖=𝑟𝑘}𝐽_𝜆(𝑢)<0,∀𝑘∈ℕ. (73)Step 3. The continuity 𝑓 and 𝐼_𝑗(𝑗=1,2,…,𝑚) and 𝐽_𝜆∈𝐶¹(𝑊,ℝ) imply that 𝐽_𝜆 maps bounded sets to bounded sets uniformly for 𝜆∈[1,2]. In view of (H10), 𝐽_𝜆(−𝑢)=𝐽_𝜆(𝑢) for all (𝜆,𝑢)∈[1,2]×𝑊. Thus condition (T1) of Lemma 5 holds. Besides, by (H7), we get 𝐴(𝑢)=1𝑝‖𝑢‖^𝑝+^𝑚_𝑗=1^{𝑢(𝑡𝑗)}₀𝐼_𝑗(𝑡)d𝑡≥1𝑝‖𝑢‖^𝑝⟶∞as‖𝑢‖⟶∞ (74) and 𝐵(𝑢)≥0 since 𝐹(𝑡,𝑢)≥0. Thus the condition (T2) of Lemma 5 holds. For step 1 and step 2, the condition (T3) of Lemma 5 also holds for all 𝑘≥𝑘₁. Consequently, Lemma 5 implies that, for any 𝑘≥𝑘₁ and a.e. 𝜆∈[1,2], there exists a sequence {𝑢^𝑘_𝑛(𝜆)}^∞_𝑛=1 such that sup_𝑛𝑢^𝑘_𝑛(𝜆)<∞,𝐽^′_𝜆𝑢^𝑘_𝑛(𝜆)𝐽⟶0,_𝜆𝑢^𝑘_𝑛(𝜆)⟶𝜁_𝑘(𝜆)as𝑛⟶∞, (75) where 𝐵_𝑘={𝑢∈𝑌_𝑘∶‖𝑢‖≤𝑟_𝑘}, Γ_𝑘={𝛾∈𝐶(𝐵_𝑘,𝑊)∶𝛾isodd,𝛾|_𝜕𝐵𝑘=id} and 𝜁_𝑘(𝜆)=inf_{𝛾∈Γ𝑘}max_{𝑢∈𝐵𝑘}𝐽_𝜆(𝛾(𝑢)), ∀𝜆∈[1,2].
Furthermore, we easily have 𝜁_𝑘(𝜆)∈𝑎_𝑘,𝜁_𝑘,∀𝑘≥𝑘₁, (76) where 𝜁_𝑘∶=max_{𝑢∈𝐵𝑘}𝐽_𝜆(𝛾(𝑢)) and 𝑎_𝑘∶=(1/𝑝)𝜌^𝑝_𝑘−2𝜇₇𝑇^1−1/𝑞−2𝜇₈→∞ as 𝑘→∞.
Claim 1. {𝑢^𝑘_𝑛(𝜆)}^∞_𝑛=1⊂𝑊 possesses a strong convergent subsequence in 𝑊, for all 𝜆∈[1,2] and 𝑘≥𝑘₁. In fact, by the boundedness of {𝑢^𝑘_𝑛(𝜆)}^∞_𝑛=1, passing to a subsequence, as 𝑛→∞, we may assume 𝑢^𝑘_𝑛(𝜆)⇀𝑢^𝑘(𝜆) in 𝑊. By the method of Theorem 7, we easily prove that 𝑢^𝑘_𝑛(𝜆)→𝑢^𝑘(𝜆) strongly in 𝑊.
Thus, for each 𝑘≥𝑘₁, we can choose 𝜆_𝑙→1 such that the sequence {𝑢^𝑘_𝑛(𝜆_𝑙)}^∞_𝑛=1 obtained a convergent subsequence, and passing again to a subsequence, we may assume lim_𝑛→∞𝑢^𝑘_𝑛𝜆_𝑙=𝑢^𝑘_𝑙in𝑊,∀𝑙∈ℕ,𝑘≥𝑘₁. (77) Thus we obtain 𝐽^′_𝜆𝑙𝑢^𝑘_𝑙=0,𝐽_𝜆𝑙𝑢^𝑘_𝑙∈𝑎_𝑘,𝜁_𝑘,∀𝑙∈ℕ,𝑘≥𝑘₁. (78)Claim 2. {𝑢^𝑘_𝑙} is bounded in 𝑊 and has a convergent subsequence with the limit 𝑢^𝑘∈𝑊 for all 𝑘≥𝑘₁. For convenience, we set 𝑢^𝑘_𝑙=𝑢_𝑙 for all 𝑙∈ℕ. If not, {𝑢_l} is unbounded in 𝑊; that is, ‖𝑢_𝑙‖→∞. By (H6), there is 𝜇₉>0 such that −𝑝𝐹(𝑡,𝑢)+𝑓(𝑡,𝑢)𝑢≥𝑏|𝑢|^𝑞−𝜇₉,∀(𝑡,𝑢)∈[0,𝑇]×ℝ. (79) Combining this and (H8), we have 𝑝𝐽_𝜆𝑙𝑢_𝑙−𝐽^′_𝜆𝑙𝑢_𝑙,𝑢_𝑙≥−𝜆_𝑙𝑝^𝑇₀𝐹𝑡,𝑢_𝑙(𝑡)d𝑡+𝜆_𝑙^𝑇₀𝑓𝑡,𝑢_𝑙(𝑡)𝑢_𝑙(𝑡)d𝑡≥𝜆_𝑙^𝑇₀𝑏𝑢_𝑙^𝑞−𝜇₉d𝑡=𝜆_𝑙𝑏^𝑇₀𝑢_𝑙^𝑞d𝑡−𝜆_𝑙𝜇₉𝑇. (80) This implies that ∫^𝑇₀𝑢_𝑙^𝑞d𝑡𝑢_𝑙^𝑝⟶0as𝑙⟶∞. (81) On the other hand, by (H1) and (H9), we see 𝐽^′_𝜆𝑙𝑢_𝑙,𝑢_𝑙=𝑢_𝑙^𝑝+^𝑚_𝑗=1𝐼_𝑗𝑢_𝑙𝑡_𝑗𝑢_𝑙𝑡_𝑗−𝜆_𝑙^𝑇₀𝑓𝑡,𝑢_𝑙(𝑡)𝑢_𝑙(𝑡)d𝑡≥𝑢_𝑙^𝑝+^𝑚_𝑗=1−𝜎_𝑗−𝜏_𝑗𝑢_𝑙𝑡_𝑗^𝛾𝑗−1𝑢_𝑙𝑡_𝑗−𝜆_𝑙^𝑇₀𝜇₁+𝜇₂𝑢_𝑙(𝑡)^𝑞−1𝑢_𝑙(𝑡)d𝑡≥𝑢_𝑙^𝑝−^𝑚_𝑗=1𝜎_𝑗𝐶_emb2𝑢_𝑙+𝜏_𝑗𝐶^𝛾𝑗_emb2𝑢_𝑙^𝛾𝑗−𝜆_𝑙𝜇₁𝑇^{1−1/𝑞𝑇}₀𝑢_𝑙(𝑡)^𝑞d𝑡^1/𝑞−𝜆_𝑙𝜇₂^𝑇₀𝑢_𝑙(𝑡)^𝑞d𝑡. (82) Consequently, noting that 𝑝>1 and 𝑝>𝛾_𝑗, we have 𝑢1=^𝑙𝑝𝑢_𝑙^𝑝≤𝐽^′_𝜆𝑙𝑢_𝑙,𝑢_𝑙𝑢_𝑙^𝑝+∑^𝑚_𝑗=1𝜎_𝑗𝐶_emb2𝑢_𝑙+𝜏_𝑗𝐶^𝛾𝑗_emb2𝑢_𝑙^𝛾𝑗𝑢_𝑙^𝑝+𝜆_𝑙𝜇₁𝑇^1−1/𝑞∫^𝑇₀𝑢_𝑙(𝑡)^𝑞d𝑡^1/𝑞𝑢_𝑙^𝑝+𝜆_𝑙𝜇₂∫^𝑇₀𝑢_𝑙(𝑡)^𝑞d𝑡𝑢_𝑙^𝑝⟶0as𝑙⟶∞. (83) This is a contradiction. Therefore, {𝑢_𝑙}^∞_𝑙=1 is bounded in 𝑊. By claim 1, we see that {𝑢_𝑙}^∞_𝑙=1 has a convergent subsequence, which converges to an element 𝑢^𝑘∈𝑊 for all 𝑘≥𝑘₁.
Hence, passing to the limit in (78), we see 𝐽^′₁𝑢^𝑘=0,𝐽₁𝑢^𝑘∈𝑎_𝑘,𝜁_𝑘,∀𝑙∈ℕ,𝑘≥𝑘₁. (84) Since 𝑎_𝑘→∞ as 𝑘→∞, we get infinitely many nontrivial critical points of 𝐽₁=𝐽. Therefore (1) possesses infinitely many nontrivial solutions by Lemma 5. This completes the proof.

Remark 9. (1) Let 𝑓()=𝑔(𝑢𝑢|𝑡,𝑢)|^𝑝−2𝑔(𝑢𝑢|𝑢,𝑢≤0,)|^𝑝−2𝑢+𝑢^𝑞−1,𝑢>0, (85) where 𝑔∈𝐶(ℝ), 𝑔(0)=0, 𝑔(−∞)∈(𝜆₁,+∞) with avoiding 𝜆_𝑘 for all 𝑘. Then (H1)–(H4) hold true with 𝑞∈(𝑝,+∞) and 𝑓₀<𝜆₁<𝑓_∞ with 𝑓_∞≠𝜆_𝑘 for all 𝑘.
(2) Let 𝑝=5, 𝑞=6, 𝑓(𝑡,𝑢)=𝑢⁵, and 𝐼_𝑗(𝑢)=√³𝑢 for all 𝑡∈[0,𝑇] and 𝑢∈ℝ. Then 𝐹(𝑡,𝑢)=∫^𝑢₀𝑓(𝑡,𝑠)d𝑠=(1/6)|𝑢|⁶, ∫^𝑢₀𝐼_𝑗(𝑠)d𝑠=(3/4)√³𝑢⁴. Clearly, (H1), (H7), (H9), and (H10) are satisfied.(1)lim_|𝑢|→∞(𝐹(𝑡,𝑢)/|𝑢|^𝑝)=lim_|𝑢|→∞(|𝑢|⁶/6|𝑢|⁵)=+∞, uniformly on 𝑡∈[0,𝑇], and 𝐹(𝑡,𝑢)≥0 for all (𝑡,𝑢)∈[0,𝑇]×ℝ. So, (H5) holds.(2)lim_|𝑢|→∞((−𝑝𝐹(𝑡,𝑢)+𝑓(𝑡,𝑢)𝑢)/|𝑢|^𝑞)=lim_|𝑢|→∞((−5×(1/6)|𝑢|⁶+𝑢⁵𝑢)/|𝑢|⁶)=1/6, uniformly on 𝑡∈[0,𝑇], and hence (H6) holds.(3)𝑝∫^𝑢₀𝐼_𝑗(𝑠)d𝑠−𝐼_𝑗(𝑢)𝑢=5×(3/4)√³𝑢⁴−√³𝑢×𝑢=(11/4)√³𝑢⁴≥0, ∀𝑢∈ℝ, 𝑗=1,2,…,𝑚. Therefore, (H8) holds.

Acknowledgments

This research is supported by the NNSF, China (10971046), Shandong and Hebei Provincial Natural Science Foundation (ZR2012AQ007, A2012402036), GIIFSDU (yzc12063), and IIFSDU (2012TS020).