Weak Solutions for a 𝑝-Laplacian Antiperiodic Boundary Value Problem with Impulsive Effects

Abstract

By virtue of variational method and critical point theory, we will investigate the existence of weak solutions for a 𝑝-Laplacian impulsive differential equation with antiperiodic boundary conditions.

1. Introduction

In this work, we will study the existence of weak solutions for the 𝑝-Laplacian impulsive differential equation with antiperiodic boundary conditions −𝑢^′𝑝−2𝑢^′′=𝑓(𝑥,𝑢),inΔ𝑢Ω,_′𝑥_𝑘^𝑝−2𝑢^′𝑥_𝑘=𝐼_𝑘𝑢𝑥_𝑘𝑢(0)(𝑇),𝑘=1,2,…,𝑚,=−𝑢,𝑢^′(0)=−𝑢^′(𝑇), (1) where 𝑝>1, 𝑇>0, 0=𝑥₀<𝑥₁<𝑥₂<⋯<𝑥_𝑚<𝑥_𝑚+1=𝑇, Ω∶=[0,𝑇]⧵{𝑥₁,…,𝑥_𝑚}, 𝑓∈𝐶([0,𝑇]×ℝ,ℝ), and 𝐼_𝑘∈𝐶(ℝ,ℝ),  𝑘=1,2,…,𝑚. Moreover, Δ|𝑢^′(𝑥_𝑘)|^𝑝−2𝑢^′(𝑥_𝑘)=|𝑢^′(𝑥⁺_𝑘)|^𝑝−2𝑢^′(𝑥⁺_𝑘)−|𝑢^′(𝑥⁻_𝑘)|^𝑝−2𝑢^′(𝑥⁻_𝑘), where 𝑢^′(𝑥⁺_𝑘) and 𝑢^′(𝑥⁻_𝑘) denote the right and left limits, respectively, of 𝑢^′(𝑥) at 𝑥=𝑥_𝑘, for 𝑘=1,2,…,𝑚.

Impulsive differential equations have become more important in recent years in some mathematical models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics. We mention, for instance, the books [1–3] dealing with impulsive differential equations.

Recently, many researchers pay their attention to impulsive differential equations by variational method and critical point theory, to the best our knowledge; we refer the reader to [4–8] and references cited therein. Meanwhile, some people begin to study 𝑝-Laplacian differential equations with impulsive effects; for example, see [9–14].

Chen and Tang [9] adopted the least action principle and the saddle point theorem to obtain some existence theorems for a second-order 𝑝-Laplacian system with or without impulsive effects under weak sublinear growth conditions. In [10], they also considered that a second-order impulsive differential equation with Dirichlet problems has one or infinitely many solutions under more relaxed assumptions on their nonlinearity 𝑓, which satisfies a kind of new superquadratic and subquadratic conditions.

The problem of finding infinitely many large energy solutions is a very classical problem; there is an extensive literature concerning the existence of infinitely many large energy solutions of a plethora of problems via the symmetric mountain pass theorem and fountain theorem; for instance, see [15–22].

Motivated by the works cited above, in this paper, we shall discuss the problem (1). Firstly, we adopt Browder theorem, introduced in [23], to prove that the problem (1) has only one weak solution. Secondly, we shall utilize Fountain theorem under Cerami condition (C), which is introduced in [24], to prove that the problem (1) has infinitely many weak solutions. The results obtained here improve some existing results in the literature.

2. Variational Structure

We first introduce the Banach space 𝑋∶={𝑢∈𝑊^1,𝑝([0,𝑇])∶𝑢(0)=−𝑢(𝑇)}, endowed with the norm ‖𝑢‖=^𝑇₀𝑢^′(𝑥)^𝑝d𝑥^1/𝑝,for𝑢∈𝑋. (2) Then we see that (𝑋,‖⋅‖) is a reflexive Banach space. We denote by ‖𝑢‖_𝑝 the norm in 𝐿^𝑝([0,𝑇]) and ‖𝑢‖_∞ in 𝐶[0,𝑇].

In what follows, we shall convert the problem (1) into an integral equation. For all 𝑣∈𝑋, multiply (1) by 𝑣 and integrate between 0 and 𝑇 to obtain ^𝑇₀−𝑢^′(𝑥)^𝑝−2𝑢^′(𝑥)^′𝑣(𝑥)d𝑥=^𝑇₀𝑓(𝑥,𝑢(𝑥))𝑣(𝑥)d𝑥. (3) Noting the impulsive effects, for the left integral of (3), we have ^𝑇₀−𝑢^′(𝑥)^𝑝−2𝑢^′(𝑥)^′𝑣(𝑥)d𝑥=^𝑚_𝑗=0^{𝑥−𝑗+1}_𝑥+𝑗−𝑢^′(𝑥)^𝑝−2𝑢^′(𝑥)^′𝑣(𝑥)d𝑥=−^𝑚_𝑗=0𝑢^′(𝑥)^𝑝−2𝑢^′(𝑥)𝑣(𝑥)^{𝑥−𝑗+1}_𝑥+𝑗+^𝑚_𝑗=0^{𝑥−𝑗+1}_𝑥+𝑗𝑢^′(𝑥)^𝑝−2𝑢^′(𝑥)𝑣^′(𝑥)d𝑥=^𝑚_𝑗=1𝑢^′𝑥⁺_𝑗^𝑝−2𝑢^′𝑥⁺_𝑗−𝑢^′𝑥⁻_𝑗^𝑝−2𝑢^′𝑥⁻_𝑗𝑣𝑥_𝑗−𝑢^′(𝑇)^𝑝−2𝑢^′(𝑇)𝑣(𝑇)+𝑢^′(0)^𝑝−2𝑢^′(0)𝑣(0)+^𝑇₀𝑢^′(𝑥)^𝑝−2𝑢^′(𝑥)𝑣^′(𝑥)d𝑥=^𝑚_𝑗=1𝐼_𝑗𝑢𝑥_𝑗𝑣𝑥_𝑗+^𝑇₀𝑢^′(𝑥)^𝑝−2𝑢^′(𝑥)𝑣^′(𝑥)d𝑥. (4) Substituting (4) into (3), we easily find that ^𝑇₀𝑢^′(𝑥)^𝑝−2𝑢^′(𝑥)𝑣^′(𝑥)d𝑥+^𝑚_𝑗=1𝐼_𝑗𝑢𝑥_𝑗𝑣𝑥_𝑗−^𝑇₀𝑓(𝑥,𝑢(𝑥))𝑣(𝑥)d𝑥=0. (5)

Definition 1. If for each 𝑣∈𝑋 there is a 𝑢∈𝑋 such that (5) holds, then 𝑢∈𝑋 is called a weak solution for (1).
From (5), we can obtain the weak solutions for (1) that coincide with critical points of the energy functional 𝐽(𝑢)=1𝑝^𝑇₀𝑢^′(𝑥)^𝑝d𝑥+^𝑚_𝑗=1^{𝑢(𝑥𝑗)}₀𝐼_𝑗(𝑥)d𝑥−^𝑇₀𝐹(𝑥,𝑢(𝑥))d𝑥, (6) where 𝐹(𝑥,𝑢)=∫^𝑢₀𝑓(𝑥,𝑡)d𝑡. Clearly, 𝐽 is class of 𝐶¹ and its derivative is 𝐽^′(𝑢),𝑣=^𝑇₀𝑢^′(𝑥)^𝑝−2𝑢^′(𝑥)𝑣^′(𝑥)d𝑥+^𝑚_𝑗=1𝐼_𝑗𝑢𝑥_𝑗𝑣𝑥_𝑗−^𝑇₀𝑓(𝑥,𝑢(𝑥))𝑣(𝑥)d𝑥,∀𝑢,𝑣∈𝑋. (7)

Lemma 2. Consider 𝑋↪↪𝐶[0,𝑇] and 𝑋↪↪𝐿^𝑝([0,𝑇]).

Proof. For 𝑢∈𝑋, by mean value theorem, 𝑢(𝑥)=𝑢(0)+∫^𝑥₀𝑢^′(𝑠)d𝑠 and 𝑢(𝑥)=𝑢(𝑇)−∫^𝑇_𝑥𝑢^′(𝑠)d𝑠. It follows from Hölder inequality that 𝑢(𝑥)=12𝑢(0)+^𝑥₀𝑢^′(𝑠)d𝑠+𝑢(𝑇)−^𝑇_𝑥𝑢^′(𝑠)d𝑠≤12^𝑇₀𝑢^′(𝑠)d𝑠≤12𝑇^{(𝑝−1)/𝑝}‖𝑢‖. (8) Consequently, we have ‖𝑢‖_∞≤12𝑇^{(𝑝−1)/𝑝}‖𝑢‖,for𝑢∈𝑋. (9) Furthermore, for 𝑢∈𝑋, we have 𝑡,𝑠∈[0,𝑇], |𝑢(𝑡)−𝑢(𝑠)|=^𝑡_𝑠𝑢^′(𝜏)d𝜏≤^𝑇₀𝑢^′(𝜏)d𝜏≤𝑇^{(𝑝−1)/𝑝}‖𝑢‖. (10) Hence every bounded set in 𝑋 is equicontinuous in 𝐶[0,𝑇]. By Arzela-Ascoli theorem, we claim that 𝑋↪↪𝐶[0,𝑇]. By the same manner, we can prove that 𝑋↪↪𝐿^𝑝([0,𝑇]). Furthermore, we can obtain by (9) ‖𝑢‖_𝑝≤𝑇2‖𝑢‖,for𝑢∈𝑋. (11) This completes the proof.

Definition 3 (see [23, page 303]). Let 𝑋 be a reflexive real Banach space and 𝑋^∗ its dual. The operator ℒ∶𝑋→𝑋^∗ is said to be demicontinuous if ℒ maps strongly convergent sequences in 𝑋 to weakly convergent sequences in 𝑋^∗.

Lemma 4 (Browder theorem, see [23, Theorem 5.3.22]). Let 𝑋 be a reflexive real Banach space. Moreover, let ℒ∶𝑋→𝑋^∗ be an operator satisfying the following conditions:(i)ℒ is bounded and demicontinuous;(ii)ℒ is coercive, that is, lim_{‖𝑥‖→∞}(ℒ(𝑥),𝑥)/‖𝑥‖=+∞;(iii)ℒ is monotone on the space 𝑋; that is, for all 𝑥,𝑦∈𝑋, one has ℒ(𝑥)−ℒ𝑦,𝑥−𝑦≥0. (12)
Then the equation ℒ(𝑥)=𝑓^∗ has at least one solution 𝑥∈𝑋 for every 𝑓^∗∈𝑋^∗. If, moreover, the inequality (12) is strict for all 𝑥,𝑦∈𝑋, 𝑥≠𝑦, then the equation ℒ(𝑥)=𝑓^∗ has precisely one solution 𝑥∈𝑋 for all 𝑓^∗∈𝑋^∗.

In what follows, we shall introduce Fountain theorem under Cerami condition (C). We first give the definition of Cerami condition (C).

Definition 5 (see [24, Definition 1.1]). Assume that 𝑋 is a Banach space with norm ‖⋅‖; we say that 𝐽∈𝐶¹(𝑋,ℝ) satisfies Cerami condition (C), if for all 𝑑∈ℝ,(i)any bounded sequence {𝑢_𝑛}⊂𝑋 satisfying 𝐽(𝑢_𝑛)→𝑑, 𝐽^′(𝑢_𝑛)→0 possesses a convergent subsequence;(ii)there exist 𝛿,𝜉,𝜌>0 such that for any 𝑢∈𝐽⁻¹([𝑑−𝛿,𝑑+𝛿]) with ‖𝑢‖≥𝜉, ‖𝐽^′(𝑢)‖⋅‖𝑢‖≥𝜌.
As 𝑋 is a reflexive Banach space, there exist (see [25, Section 17]) {𝑒_𝑛}^∞_𝑛=1⊂𝑋 and {𝑓_𝑛}^∞_𝑛=1⊂𝑋^∗ such that 𝑓_𝑛(𝑒_𝑚)=𝛿_𝑛,𝑚, 𝑋=span{𝑒_𝑛∶𝑛=1,2,…} and 𝑋^∗=span^𝑊∗{𝑓_𝑛∶𝑛=1,2,…}. For 𝑗,𝑘∈ℕ, denote 𝑋_𝑗∶=span{𝑒_𝑗}, 𝑌_𝑘∶=⊕^𝑘_𝑗=1𝑋_𝑗, and 𝑍_𝑘∶=⊕^∞_𝑗=𝑘𝑋_𝑗. Clearly, 𝑋=⊕_𝑗∈ℕ𝑋_𝑗 with dim𝑋_𝑗<∞ for all 𝑗∈ℕ. Denote 𝑆_𝜌∶={𝑢∈𝑋∶‖𝑢‖=𝜌}.

Lemma 6 (see [24, Proposition 1.2]). Let 𝑋, 𝑌_𝑘, 𝑍_𝑘 be defined as above. Assume that 𝐽∈𝐶¹(𝑋,ℝ) satisfies condition (C), and 𝐽(−𝑢)=𝐽(𝑢). For each 𝑘∈ℕ, there exist 𝜌_𝑘>𝑟_𝑘>0 such that(i)𝑏_𝑘∶=inf_{𝑢∈𝑍𝑘∩𝑆𝑟𝑘}𝐽(𝑢)→+∞, 𝑘→∞,(ii)𝑎_𝑘∶=max_{𝑢∈𝑌𝑘∩𝑆𝜌𝑘}𝐽(𝑢)≤0.
Then 𝐽 has a sequence of critical points 𝑢_𝑛, such that 𝐽(𝑢_𝑛)→+∞ as 𝑛→∞.

3. Main Results

Now, we list our assumptions on 𝑓 and 𝐼_𝑗, 𝑗=1,2,…,𝑚.(H1)𝑓(𝑥,𝑢) is a decreased function about 𝑢, uniformly in 𝑥∈[0,𝑇], and 𝐼_𝑗(𝑢)(𝑗=1,2,…,𝑚) are increased functions with 𝑢.(H2) There exist 𝑎_𝑗,𝑏_𝑗>0 and 𝛾_𝑗∈[1,𝑝) such that |𝐼_𝑗(𝑢)|≤𝑎_𝑗+𝑏_𝑗|𝑢|^𝛾𝑗−1, for all 𝑢∈ℝ and 𝑗=1,2,…,𝑚.(H3) There exist 𝑐₁,𝑐₂>0 such that 𝑓(𝑥,𝑢)≤𝑐₁+𝑐₂|𝑢|^𝑝−1, for all 𝑢∈ℝ,  𝑥∈[0,𝑇].(H4) There is a positive constant 𝑎>0 such that lim_|𝑢|→∞(−𝑝𝐹(𝑥,𝑢)+𝑓(𝑥,𝑢)𝑢)/|𝑢|≥𝑎, uniformly in 𝑥∈[0,𝑇].(H5)𝑝∫^𝑢₀𝐼_𝑗(𝑠)d𝑠−𝐼_𝑗(𝑢)𝑢≥0,  ∫^𝑢₀𝐼_𝑗(𝑠)d𝑠≥0,  for all 𝑢∈ℝ, 𝑗=1,2,…,𝑚.(H6)lim_|𝑢|→∞𝐹(𝑥,𝑢)/|𝑢|^𝑝=+∞, uniformly on 𝑥∈[0,𝑇].(H7)𝐹(𝑥,𝑢) is an even function about 𝑢 and 𝐼_𝑗(𝑢)(𝑗=1,2,…,𝑚) are odd functions about 𝑢, for all 𝑥∈[0,𝑇].

Theorem 7. Let 0<𝑐₂<(𝑇/2)^−𝑝 and (H1)–(H3) hold. Then (1) has precisely a weak solution.

Proof. For 𝑢,𝑣∈𝑋, we define 𝐿₁(𝑢),𝑣∶=^𝑇₀𝑢^′(𝑥)^𝑝−2𝑢^′(𝑥)𝑣^′(𝑥)d𝐿𝑥,₂(𝑢),𝑣∶=^𝑚_𝑗=1𝐼_𝑗𝑢𝑥_𝑗𝑣𝑥_𝑗,𝐿₃(𝑢),𝑣∶=^𝑇₀𝑓(𝑥,𝑢(𝑥))𝑣(𝑥)d𝑥. (13) Let (ℒ(𝑢),𝑣)=𝐿₁(𝑢),𝑣+𝐿₂(𝑢)−𝐿,𝑣₃(𝑢),𝑣,∀𝑢,𝑣∈𝑋. (14) Then, to find a weak solution of (1) is equivalent to finding a solution for the operator equation ℒ(𝑢)=0. In what follows, we shall sketch the properties of operators 𝐿_𝑖(𝑖=1,2,3). By Hölder inequality, we have ^𝑇₀𝑢^′(𝑥)^𝑝−2𝑢^′(𝑥)𝑣^′(𝑥)d𝑥≤^𝑇₀𝑢^′(𝑥)^𝑝−2𝑢^′(𝑥)^{𝑝/(𝑝−1)}d𝑥^{(𝑝−1)/𝑝}×^𝑇₀𝑣^′(𝑥)^𝑝d𝑥^1/𝑝=^𝑇₀𝑢^′(𝑥)^𝑝d𝑥^{(𝑝−1)/𝑝𝑇}₀𝑣^′(𝑥)^𝑝d𝑥^1/𝑝=‖𝑢‖^𝑝−1‖𝑣‖<∞,∀𝑢,𝑣∈𝑋. (15) Consequently, 𝐿₁ is bounded: 𝐿₁𝑢_𝑛−𝐿₁(𝑢)=,𝑣_𝑇0𝑢^′_𝑛(𝑥)^𝑝−2𝑢^′_𝑛(𝑥)−𝑢^′(𝑥)^𝑝−2𝑢^′(𝑥)𝑣^′(𝑥)d𝑥≤^𝑇₀𝑢^′_𝑛(𝑥)^𝑝−2𝑢^′_𝑛(𝑥)−𝑢^′(𝑥)^𝑝−2𝑢^′(𝑥)^{𝑝/(𝑝−1)}d𝑥^{(𝑝−1)/𝑝}‖𝑣‖. (16) The last integral tends to zero as ‖𝑢_𝑛−𝑢‖→0 due to the continuity of the Nemytskii operator 𝜑(𝑠)=|𝑠|^𝑝−2𝑠. Hence, 𝐿₁ is continuous. We assume that 𝑢_𝑛⇀𝑢 in 𝑋; by Lemma 2, we know that there is a subsequence, still denoted by {𝑢_𝑛}, which strongly converges to 𝑢 in 𝐶[0,𝑇]. Therefore, utilizing the continuity of 𝑓, we have 𝐿₃𝑢_𝑛−𝐿₃(𝑢)=,𝑣_𝑇0𝑓𝑥,𝑢_𝑛(𝑥)−𝑓(𝑥,𝑢(𝑥))𝑣(𝑥)d𝑥⟶0as𝑛⟶∞. (17) So, 𝐿₃ is continuous. In the same methods, we see that 𝐿₂ is also continuous. On the other hand, by (9) and (11), for 𝑢,𝑣∈𝑋, it follows from (H2) and (H3) that 𝐿₂(𝑢)≤,𝑣_𝑚𝑗=1𝐼_𝑗𝑢𝑥_𝑗𝑣𝑥_𝑗≤^𝑚_𝑗=1𝑎_𝑗+𝑏_𝑗𝑢(𝑥_𝑗)^𝛾𝑗−1𝑣𝑥_𝑗≤^𝑚_𝑗=1𝑎_𝑗+𝑏_𝑗‖𝑢‖^𝛾𝑗_∞⁻¹‖𝑣‖_∞≤𝑚2𝑇^{(𝑝−1)/𝑝}𝑎_𝑗+𝑏_𝑗12𝑇^{(𝑝−1)/𝑝𝛾𝑗−1}‖𝑢‖^𝛾𝑗−1𝑣‖𝐿‖<∞,₃(𝑢)=,𝑣_𝑇0𝑓(𝑥,𝑢(𝑥))𝑣(𝑥)d𝑥≤^𝑇₀𝑓(𝑥,𝑢(𝑥))|𝑣(𝑥)|d𝑥≤^𝑇₀𝑐₁+𝑐₂|𝑢(𝑥)|^𝑝−1|𝑣(𝑥)|d𝑥≤𝑐₁𝑇^{(𝑝−1)/𝑝}‖𝑣‖_𝑝+𝑐₂‖𝑢‖_𝑝^𝑝−1‖𝑣‖_𝑝≤𝑐₁2𝑇^{(2𝑝−1)/𝑝}‖𝑣‖+𝑐₂𝑇2^𝑝‖𝑢‖^𝑝−1‖𝑣‖<∞. (18) Therefore, 𝐿₂ and 𝐿₃ are bounded. Up to now, we have proved that ℒ is bounded and continuous, so (i) of Lemma 4 holds. Finally, we shall show the monotonicity and coercivity of ℒ. Indeed, (H1) implies that for 𝑢,𝑣∈𝑋 and 𝑢≠𝑣, (ℒ(𝑢)(𝑣))=−ℒ,𝑢−𝑣_𝑇0𝑢^′(𝑥)^𝑝−2𝑢^′(𝑥)−𝑣^′(𝑥)^𝑝−2𝑣^′(𝑥×𝑢_′)(𝑥)−𝑣^′(𝑥)d𝑥+^𝑚_𝑗=1𝐼_𝑗𝑢𝑥_𝑗−𝐼_𝑗𝑣𝑥_𝑗𝑢𝑥_𝑗−𝑣𝑥_𝑗−^𝑇₀𝑓(𝑥,𝑢(𝑥))−𝑓(𝑥,𝑣(𝑥))(𝑢(𝑥)−𝑣(𝑥))d𝑥≥^𝑇₀𝑢^′(𝑥)^𝑝−2𝑢^′(𝑥)−𝑣^′(𝑥)^𝑝−2𝑣^′(𝑥×𝑢_′)(𝑥)−𝑣^′(𝑥)d𝑥≥‖𝑢‖^𝑝−‖𝑢‖^𝑝−1‖𝑣‖−‖𝑣‖^𝑝−1‖𝑢‖+‖𝑣‖^𝑝≥‖𝑢‖^𝑝−1−‖𝑣‖^𝑝−1(‖𝑢‖−‖𝑣‖)>0. (19) (H2) and (H3) enable us to get (ℒ(𝑢))=‖𝑢‖,𝑢_𝑝+^𝑚_𝑗=1𝐼_𝑗𝑢𝑥_𝑗𝑢𝑥_𝑗−^𝑇₀𝑓(𝑥,𝑢(𝑥))𝑢(𝑥)d𝑥≥‖𝑢‖^𝑝−^𝑚_𝑗=1𝑎_𝑗+𝑏_𝑗𝑢𝑥_𝑗^𝛾𝑗−1𝑢𝑥_𝑗−^𝑇₀𝑐₁+𝑐₂|𝑢(𝑥)|^𝑝−1|𝑢(𝑥)|d𝑥≥1−𝑐₂𝑇2^𝑝‖𝑢‖^𝑝−𝑚𝑎_𝑗2𝑇^{(𝑝−1)/𝑝}+𝑐₁2𝑇^{(2𝑝−1)/𝑝}‖𝑢‖−𝑚𝑏_𝑗12𝑇^{(𝑝−1)/𝑝𝛾𝑗}‖𝑢‖^𝛾𝑗. (20) Therefore, by the span of 𝑐₂∈(0,(𝑇/2)^−𝑝), we arrive at lim_{‖𝑢‖→∞}(ℒ(𝑢),𝑢)/‖𝑢‖=+∞. As a result, (ii) and (iii) of Lemma 4 hold. Hence, Lemma 4 implies that (1) has precisely a weak solution. This completes the proof.

Lemma 8. Let (H3)–(H5) hold. Then 𝐽 satisfies Cerami condition (C).

Proof. For all 𝑑∈ℝ, we assume that {𝑢_𝑛}^∞_𝑛=1⊂𝑋 is bounded and 𝐽𝑢_𝑛⟶𝑑,𝐽^′𝑢_𝑛⟶0,𝑛⟶∞. (21) Going, if necessary, to a subsequence, we can assume that 𝑢_𝑛⇀𝑢 weakly in 𝑋, and then 𝐽^′𝑢_𝑛−𝐽^′(𝑢),𝑢_𝑛−𝑢=^𝑇₀𝑢^′_𝑛^𝑝−2𝑢^′_𝑛−𝑢^′𝑝−2𝑢^′𝑢^′_𝑛−𝑢^′d𝑥+^𝑚_𝑗=1𝐼_𝑗𝑢_𝑛𝑥_𝑗−𝐼_𝑗𝑢𝑥_𝑗𝑢_𝑛𝑥_𝑗−𝑢𝑥_𝑗−^𝑇₀𝑓𝑥,𝑢_𝑛−𝑓(𝑥,𝑢)𝑢_𝑛−𝑢d𝑥. (22) By Lemma 2, 𝑋↪↪𝐶[0,𝑇] enables us to obtain that ^𝑚_𝑗=1𝐼_𝑗𝑢_𝑛𝑥_𝑗−𝐼_𝑗𝑢𝑥_𝑗𝑢_𝑛𝑥_𝑗−𝑢𝑥_𝑗⟶0,_𝑇0𝑓𝑥,𝑢_𝑛−𝑓(𝑥,𝑢)𝑢_𝑛−𝑢d𝑥⟶0as𝑛⟶∞. (23) It follows from 𝑢_𝑛⇀𝑢 weakly in 𝑋 and (𝐽^′(𝑢_𝑛)−𝐽^′(𝑢),𝑢_𝑛−𝑢)→0 that ^𝑇₀𝑢^′_𝑛^𝑝−2𝑢^′_𝑛−𝑢^′𝑝−2𝑢^′𝑢^′_𝑛−𝑢^′d𝑥⟶0as𝑛⟶∞. (24) Noting (19), we have 𝑢_𝑛^𝑝−1−𝑢^𝑝−1𝑢_𝑛−𝑢≤^𝑇₀𝑢^′_𝑛^𝑝−2𝑢^′_𝑛−𝑢^′𝑝−2𝑢^′𝑢^′_𝑛−𝑢^′d𝑥, (25) and thus ‖𝑢_𝑛−𝑢‖→0 as 𝑛→∞. Hence, (i) of Definition 5 is satisfied. Next, we prove (ii) of Definition 5; if not, there exists a sequence {𝑢_𝑛}⊂𝑋 such that 𝐽𝑢_𝑛⟶𝑑,𝐽^′𝑢_𝑛⋅𝑢_𝑛⟶0,𝑛⟶∞,𝑢_𝑛⟶∞,𝑛⟶∞. (26)(27) By (26), there exists a constant 𝜀₁>0 such that 𝐽𝑢_𝑛−1𝑝𝐽^′𝑢_𝑛𝑢_𝑛≤𝜀₁. (28) On the other hand, (H4) implies that there is a 𝑀>0 such that −𝑝𝐹(𝑥,𝑢)+𝑓(𝑥,𝑢)𝑢≥𝑎|𝑢|, for all |𝑢|>𝑀 and 𝑥∈[0,𝑇]. Furthermore, −𝑝𝐹(𝑥,𝑢)+𝑓(𝑥,𝑢)𝑢 is bounded for |𝑢|≤𝑀 and 𝑥∈[0,𝑇]. Therefore, there exists 𝑐>0 such that −𝐹(𝑥,𝑢)+(1/𝑝)𝑓(𝑥,𝑢)𝑢≥(𝑎/𝑝)|𝑢|−𝑐, for all 𝑢∈ℝ, 𝑥∈[0,𝑇]. This, together with (H5), yields 𝐽𝑢_𝑛−1𝑝𝐽^′𝑢_𝑛𝑢_𝑛=^𝑚_𝑗=1^{𝑢𝑛(𝑥𝑗)}₀𝐼_𝑗(𝑥)d1𝑥−𝑝𝐼_𝑗𝑢_𝑛𝑥_𝑗𝑢_𝑛𝑥_𝑗+^𝑇₀−𝐹𝑥,𝑢_𝑛+1𝑝𝑓𝑥,𝑢_𝑛𝑢_𝑛d𝑥≥^𝑇₀−𝐹𝑥,𝑢_𝑛+1𝑝𝑓𝑥,𝑢_𝑛𝑢_𝑛d𝑥≥^𝑇₀𝑎𝑝𝑢_𝑛−𝑐d𝑥, (29) which implies ∫^𝑇₀|𝑢_𝑛|d𝑥≤𝑝𝑎⁻¹(𝑇𝑐+𝜀₁). Therefore, there is a 𝜀₂>0 such that ‖𝑢_𝑛‖_∞≤𝜀₂.
It follows from (H3) that there are 𝑐₃,𝑐₄>0 such that 𝐹(𝑥,𝑢)≤𝑐₃|𝑢|+𝑐₄|𝑢|^𝑝,∀𝑢∈ℝ,𝑥∈[0,𝑇]. (30) By this and (H5), we can find 𝐽𝑢_𝑛=1𝑝^𝑇₀𝑢^′_𝑛^𝑝d𝑥+^𝑚_𝑗=1^{𝑢𝑛(𝑥𝑗)}₀𝐼_𝑗(𝑥)d𝑥−^𝑇₀𝐹𝑥,𝑢_𝑛d𝑥≥1𝑝𝑢_𝑛^𝑝−^𝑇₀𝑐₃𝑢_𝑛+𝑐₄𝑢_𝑛^𝑝d𝑥≥1𝑝𝑢_𝑛^𝑝−𝑐₃𝑇𝑢_𝑛∞−𝑐₄𝑇𝑢_𝑛^𝑝_∞≥1𝑝𝑢_𝑛^𝑝−𝑐₃𝑇𝜀₂−𝑐₄𝑇𝜀^𝑝₂, (31) and thus 𝐽(𝑢_𝑛)→∞ if (27) holds, which contradicts 𝐽(𝑢_𝑛)→𝑑 in (26). This proves that 𝐽 satisfies condition (C).

Theorem 9. Suppose that (H2)–(H7) hold, and then (1) has infinitely many weak solutions.

Proof. By (H7), we know that 𝐽 is even. Denote 𝛽_𝑘=sup_{𝑢∈𝑍𝑘∩𝑆1}‖𝑢‖_𝑝, and by the compactness of the embedding 𝑋↪↪𝐿^𝑝([0,𝑇]), we see that 𝛽_𝑘→0 as 𝑘→∞ (see [26, Lemma 3.8]). Noting (30), we have by (H5) and Hölder inequality, for any 𝑢∈𝑍_𝑘 and ‖𝑢‖=𝑟_𝑘∶=𝛽_𝑘⁻¹, 𝐽(𝑢)≥1𝑝‖𝑢‖^𝑝−^𝑇₀𝑐₃|𝑢|+𝑐₄|𝑢|^𝑝d𝑥≥1𝑝‖𝑢‖^𝑝−𝑐₃𝑇^{(𝑝−1)/𝑝}‖𝑢‖_𝑝−𝑐₄‖𝑢‖^𝑝_𝑝≥1𝑝‖𝑢‖^𝑝−𝑐₃𝑇^{(𝑝−1)/𝑝}𝛽_𝑘‖𝑢‖−𝑐₄𝛽^𝑝_𝑘‖𝑢‖^𝑝≥𝛽_𝑘^−𝑝𝑝−𝑐₃𝑇^{(𝑝−1)/𝑝}−𝑐₄. (32) We easily have 𝑟_𝑘→∞ as 𝑘→∞ and 𝐽(𝑢)≥𝛽_𝑘^−𝑝/𝑝−𝑐₃𝑇^{(𝑝−1)/𝑝}−𝑐₄→∞ as 𝑘→∞. Hence, 𝑏_𝑘∶=inf_{𝑢∈𝑍𝑘,‖𝑢‖=𝑟𝑘}𝐽(𝑢)⟶∞as𝑘⟶∞. (33)
On the other hand, by (H6), we find that there are 𝑏,𝑐>0 such that 𝐹(𝑥,𝑢)≥𝑏|𝑢|^𝑝−𝑐,∀𝑢∈ℝ,𝑥∈[0,𝑇]. (34) Since all the norms of a finite dimensional normed space are equivalent, note that ‖⋅‖_𝑝 is a norm of 𝑌_𝑘, so there exists a 𝜉>0 such that ‖𝑢‖^𝑝_𝑝≥𝜉‖𝑢‖^𝑝,∀𝑢∈𝑌_𝑘. (35) Noting (9), we have 𝐽(𝑢)=1𝑝^𝑇₀𝑢^′𝑝d𝑥+^𝑚_𝑗=1^{𝑢(𝑥𝑗)}₀𝐼_𝑗(𝑥)d𝑥−^𝑇₀𝐹(𝑥,𝑢)d𝑥≤1𝑝‖𝑢‖^𝑝−^𝑇₀𝑏|𝑢|^𝑝−𝑐d𝑥+^𝑚_𝑗=1^{𝑢(𝑥𝑗)}₀𝑎_𝑗+𝑏_𝑗|𝑠|^𝛾𝑗−1d𝑥≤1𝑝‖𝑢‖^𝑝−𝑏‖𝑢‖^𝑝_𝑝++𝑐𝑇_𝑚𝑗=1𝑎_𝑗𝑢𝑥_𝑗+𝑏_𝑗𝛾_𝑗𝑢𝑥_𝑗^𝛾𝑗≤1𝑝‖𝑢‖^𝑝−𝑏𝜉‖𝑢‖^𝑝+^𝑚_𝑗=1𝑎_𝑗‖𝑢‖_∞+𝑏_𝑗𝛾_𝑗‖𝑢‖^𝛾𝑗_∞≤1+𝑐𝑇𝑝−𝑏𝜉‖𝑢‖^𝑝+^𝑚_𝑗=1𝑎_𝑗2𝑇^{(𝑝−1)/𝑝}‖𝑢‖+𝑏_𝑗𝛾_𝑗12𝑇^{(𝑝−1)/𝑝𝛾𝑗}‖𝑢‖^𝛾𝑗+𝑐𝑇. (36) Note that we can choose a large enough 𝑏 such that 1/𝑝−𝑏𝜉<0 by (H6) and 𝑝>𝛾_𝑗 by (H2), and then there exists positive constants 𝑑_𝑘 such that 𝐽(𝑢)≤0,foreach𝑢∈𝑌_𝑘,‖𝑢‖≥𝑑_𝑘. (37) By this and (33), we can take 𝜌_𝑘∶=max{𝑑_𝑘,𝑟_𝑘+1}, and thus 𝑎_𝑘∶=max_{𝑢∈𝑌𝑘,‖𝑢‖=𝜌𝑘}𝐽(𝑢)≤0. Up until now, we have proved that the functional 𝐽 satisfies all the conditions of Lemma 6, and then 𝐽 has infinitely many critical points. Equivalently, (1) has infinitely many weak solutions. This completes the proof.

Two Examples. (1) Let 𝑝=4 and 𝑥₁=𝑇/2. Consider the problem −𝑢^′2𝑢^′′=𝑓(𝑥,𝑢),inΔ𝑢Ω,_′𝑥₁²𝑢^′𝑥₁=𝑢𝑥₁,𝑢(0)=−𝑢(𝑇),𝑢^′(0)=−𝑢^′(𝑇), (38) where 𝑓(𝑥,𝑢)=−𝜂₁−𝜂₂𝑢³, 𝜂₁≥0, 𝜂₂∈(0,(𝑇/2)⁻⁴). Clearly, (H1)–(H3) hold true. By Theorem 7, (38) has only a weak solution.

(2) (H6) can be weaken thatlim_|𝑢|→∞𝐹(𝑥,𝑢)|𝑢|^𝑝>𝑝𝜉⁻¹,uniformlyon𝑥∈[0,𝑇], (39) where 𝜉 is determined by (35). Indeed, by (39), we can obtain that there exist 𝑏>(𝑝𝜉)⁻¹ and 𝑐>0 such that (34) is satisfied. Furthermore, 1/𝑝−𝑏𝜉<0.

Let 𝑝=4 and 𝑥₁=𝑇/2. Consider the problem −𝑢^′2𝑢^′′=𝑓(𝑥,𝑢),inΔ𝑢Ω,_′𝑥₁²𝑢^′𝑥₁=³𝑢𝑥₁,𝑢(0)=−𝑢(𝑇),𝑢^′(0)=−𝑢^′(𝑇), (40) where 𝑓(𝑥,𝑢)=−𝜂+(𝜉⁻¹+4)𝑢³ and 𝐼_𝑗(𝑢(𝑥_𝑗))=³𝑢(𝑥₁), where 𝜂>0 and 𝜉 is defined by (35).

For 𝐼_𝑗(𝑢), we can easily have (H2) and (H7) hold. By computation, ∫^𝑢₀𝐼_𝑗(𝑠)d𝑠=(3/4)√³𝑢⁴≥0, 𝑝∫^𝑢₀𝐼_𝑗(𝑠)d𝑠−𝐼_𝑗(𝑢)𝑢=4×(3/4)√³𝑢⁴−√³𝑢⁴=2√³𝑢⁴≥0, for all𝑢∈ℝ, and hence we obtain that (H5) is satisfied.

For 𝑓(𝑥,𝑢) and 𝐹(𝑥,𝑢), we see 𝑓(𝑥,𝑢)=−𝜂+(𝜉⁻¹+4)𝑢³≤𝜂+(𝜉⁻¹+4)|𝑢|³, forall𝑢∈ℝ,𝑥∈[0,𝑇]. Therefore, (H3) holds.

It is obvious that 𝐹(𝑥,𝑢)=∫^𝑢₀𝑓(𝑥,𝑠)d𝑠=∫^𝑢₀(−𝜂+(𝜉⁻¹+4)𝑠³)d𝑠=−𝜂|𝑢|+((𝜉⁻¹+4)/4)|𝑢|⁴, so (H7) holds; furthermore, lim_|𝑢|→∞𝐹(𝑥,𝑢)|𝑢|^𝑝=lim_|𝑢|→∞−𝜂|𝑢|+𝜉⁻¹+4/4|𝑢|⁴|𝑢|⁴=𝜉⁻¹+44>𝑝𝜉⁻¹. (41) As a result of this, (39) (i.e., (H6)) is true.

By ()()𝑢|𝑢|+𝜉−𝑝𝐹𝑥,𝑢+𝑓𝑥,𝑢=−4−𝜂⁻¹+44|𝑢|⁴+−𝜂+𝜉⁻¹+4𝑢³𝑢|𝑢||𝑢|,=4𝜂−𝜂𝑢≥3𝜂 (42) we obtain that lim_|𝑢|→∞−𝑝𝐹(𝑥,𝑢)+𝑓(𝑥,𝑢)𝑢|𝑢|≥lim_|𝑢|→∞3𝜂|𝑢||𝑢|=3𝜂, (43) uniformly on 𝑥∈[0,𝑇]. Consequently, (H4) holds. Nowadays, we have proved that (H2)–(H7) hold, and then (40) has infinitely many weak solutions by Theorem 9.

In [14], the condition (𝑝₂) is ∃𝜇>𝑝∃𝛿>0∶∀|𝑡|≥𝛿,0≤𝜇𝐹(𝑥,𝑡)≤𝑡𝑓(𝑥,𝑡). (44) As known to all, this condition is originally introduced in [27] and is still present in many works which is used to guarantee the boundedness of (P.S.) sequences of the corresponding functional.

In the problem (40), note that there exists 𝛿>0 such that |𝑢|≥𝛿 and ()()|𝑢|+𝜉𝜇𝐹𝑥,𝑢≤𝑢𝑓𝑥,𝑢⟹−𝜂𝜇⁻¹+44𝜇|𝑢|⁴≤−𝜂𝑢+𝜉⁻¹+4|𝑢|⁴. (45) This is impossible if 𝜇>𝑝=4 and 𝛿 is an adequately large positive number. On the other hand, if (H4) holds, then there is a 𝛿 sufficiently large such that −𝑝𝐹(𝑥,𝑢)+𝑢𝑓(𝑥,𝑢)≥𝑎|𝑢|≥0,for𝑥∈[0,𝑇],|𝑢|≥𝛿. (46) Therefore, 𝑝𝐹(𝑥,𝑢)≤𝑢𝑓(𝑥,𝑢),for all 𝑥∈[0,𝑇]  and |𝑢|≥𝛿, which is also weaker than the condition (𝑝₂) if 𝐹(𝑥,𝑢)≥0. This means that our results generalize the corresponding results in [14].

Acknowledgments

This research is supported by the NNSF China (10971046), Shandong and Hebei Provincial Natural Science Foundation (ZR2012AQ007, A2012402036), GIIFSDU (yzc12063), IIFSDU (2012TS020), and the Project of Shandong Province Higher Educational Science and Technology Program (J09LA55).