Multiple Solutions for a Second-Order Impulsive Sturm-Liouville Equation

Abstract

We study the existence of solutions to a boundary value problem of a second-order impulsive Sturm-Liouville equation with a control parameter 𝜆. By employing some existing critical point theorems, we find the range of the control parameter in which the boundary value problem admits at least three solutions. It is also shown that, under certain conditions, there exists an interval of the control parameter in which the boundary value problem possesses infinitely many solutions. Some examples are given to demonstrate the main results in this paper.

1. Introduction

The aim of this paper is to investigate the existence of multiple solutions of the following Neumann boundary value problem with impulsive Sturm-Liouville type equation: −𝑝(𝑡)𝑢(𝑡)′′+𝑞(𝑡)𝑢(𝑡)=𝜆𝑓(𝑡,𝑢(𝑡)),𝑡≠𝑡_𝑗[],Δ𝑝𝑡,𝑡∈0,1_𝑗𝑢′𝑡_𝑗=𝐼_𝑗𝑢𝑡_𝑗(0)(1),𝑗=1,2,…,𝑚,𝑢′=𝑢′=0, (1)

where 𝑝∈𝐶¹([0,1], 𝑞∈𝐶([0,1]), 𝑓∈𝐶([0,1]×𝑅,𝑅), 𝐼_𝑗∈𝐶(𝑅,𝑅), 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑚<𝑡_𝑚+1=1, and 𝜆 is a positive real parameter. Δ𝑢′(𝑡_𝑗)=𝑢′(𝑡⁺_𝑗)−𝑢′(𝑡⁻_𝑗), where 𝑢′(𝑡⁺_𝑗) and 𝑢′(𝑡⁻_𝑗) denote the right and left limits, respectively, of 𝑢′(𝑡) at 𝑡=𝑡_𝑗, 𝑗=1,2,…,𝑚.

In the last few years, the existence of multiple solutions to Neumann problems has been widely investigated [1–5]. But little research has focused on the existence of multiple solutions for impulsive Sturm-Liouville equations whose right-hand side nonlinear term is depending on a parameter 𝜆. Processes subject to sudden changes in their states are modeled by the impulsive differential equations and have been investigated in various fields of science and technology. In the motion of spacecraft, one has to consider instantaneous impulses at a position with jump discontinuities in velocity, but no change in the position [6, 7]. This motivates us to consider problem (1).

In the literature, tools employed to establish the existence of solutions of impulsive differential equations include fixed point theorems, the upper and lower solutions method, the degree theory, critical point theory, and variational methods. See, for example, [8–20]. In this paper, our focus is on the existence of solutions of problem (1) with 𝜆 being the parameter. The problem is first transformed into the existence of critical points of some variational structure. Then with the help of critical point theory, results on the existence of at least three solutions and infinitely many solutions are established.

The rest of this paper is organized as follows. In Section 2 we present some preliminary results. Our main results and their proofs are given in Section 3.

2. Preliminaries

Throughout we assume that 𝑝(𝑡) and 𝑞(𝑡) satisfy 𝑝₀=min_[]𝑡∈0,1𝑝(𝑡)>0,𝑞₀=min_[]𝑡∈0,1𝑞(𝑡)>0. (2)

Take 𝑋=𝑊^1,2([0,1]) and define ‖𝑢‖_𝑋=¹₀𝑝(𝑡)𝑢′(𝑡)²+𝑞(𝑡)|𝑢(𝑡)|²𝑑𝑡^1/2,𝑢∈𝑋. (3)

For the norm in 𝐶([0,1]), we put ‖𝑢‖_∞=max_𝑡∈[0,1]|𝑢(𝑡)|. (4)

We have the following relation.

Lemma 1. Let 𝑀₁=min{𝑝₀,𝑞₀}. Then ‖𝑢‖_∞≤2𝑀₁‖𝑢‖_𝑋,∀𝑢∈𝑋. (5)

Proof. For any 𝑢∈𝑋, it follows from the mean value theorem that ¹₀𝑢(𝑠)𝑑𝑠=𝑢(𝜏) (6)
for some 𝜏∈[0,1]. Hence, for 𝑡∈[0,1], using Hölder inequality and (2), we have |𝑢(𝑡)|=𝑢(𝜏)+^𝑡_𝜏𝑢′(𝑠)𝑑𝑠≤|𝑢(𝜏)|+¹₀(𝑠≤𝑢′)𝑑𝑠₁₀|𝑢(𝑠)|𝑑𝑠+¹₀(𝑠≤1𝑢′)𝑑𝑠𝑞₀^1/21₀𝑞(𝑡)|𝑢(𝑡)|²𝑑𝑡^1/2+1𝑝₀^1/21₀𝑝(𝑡)𝑢′(𝑡)²𝑑𝑡^1/2≤2𝑀₁‖𝑢‖_𝑋. (7)

Define a functional 𝜑_𝜆 as 𝜑_𝜆(𝑢)=Φ(𝑢)−𝜆Ψ(𝑢),𝑢∈𝑋, (8)

where Φ(𝑢)=12‖𝑢‖²_𝑋+^𝑚_𝑗=1^{𝑢(𝑡𝑗)}₀𝐼_𝑗(𝑠)Ψ(𝑢)=𝑑𝑠,₁₀𝐹(𝑡,𝑢)𝑑𝑡, (9)(10)

with 𝐹(𝑡,𝑢)=₀^𝑢(𝑡)𝑓(𝑡,𝑠)𝑑𝑠. (11)

Note that 𝜑_𝜆 is Fréchet differentiable at any 𝑢∈𝑋, and for any 𝑣∈𝑋, we have 𝜑′_𝜆(𝑢)(𝑣)=lim_ℎ→0𝜑_𝜆(𝑢+ℎ𝑣)−𝜑_𝜆(𝑢)ℎ=¹₀𝑝(𝑡)(𝑡)(𝑡)(𝑡)𝑢(𝑡)𝑣(𝑡+𝑢′𝑣′+𝑞)𝑑𝑡_𝑚𝑗=1𝐼_𝑗𝑢𝑡_𝑗𝑣𝑡_𝑗−𝜆¹₀𝑓(𝑡,𝑢(𝑡))𝑣(𝑡)𝑑𝑡. (12)

Lemma 2. If 𝑢∈𝑋 is a weak solution of problem (1), then 𝑢 is a classical solution of problem (1).

Proof. For any function 𝑣∈𝑋, we get 0=¹₀𝑝(𝑡)(𝑡)(𝑡)(𝑡)𝑢(𝑡)𝑣(𝑡)+𝑢′𝑣′+𝑞𝑑𝑡_𝑚𝑗=1𝐼_𝑗𝑢𝑡_𝑗𝑣𝑡_𝑗. (13)
By the regularity theory, the weak solution 𝑢 is a classical solution of problem (1).

Next we show that a critical point of the functional 𝜑_𝜆 is a solution of problem (1).

Lemma 3. If 𝑢∈𝑋 is a critical point of 𝜑_𝜆, then 𝑢 is a classical solution of problem (1).

Proof. If 𝑢∈𝑋 is a critical point of 𝜑_𝜆, by (12) and Lemma 2, we have 𝑢 is a classical solution of problem (1)

For 𝑟∈ℝ, we define 𝜌(𝑟)=inf_{𝑣∈Φ−1((−∞,𝑟))}sup_𝑢∈Φ⁻¹_{((−∞,𝑟))}Ψ(𝑢)−Ψ(𝑣)(𝑣),𝛾𝑟−Φ₁∶=liminf_𝑟→+∞𝜌(𝑟),𝛿∶=liminf_{𝑟→inf𝑋Φ+}𝜌(𝑟). (14)(15)

3. Main Results

3.1. Existence of At Least Three Solutions

In this section we derive conditions under which problem (1) admits at least three solutions. For this purpose, we introduce the following assumptions. (H1) Assume that there exists a positive constant 𝑘₁ such that for each 𝑢∈𝑋0≤^𝑚_𝑗=1^{𝑢(𝑡𝑗)}₀𝐼_𝑗(𝑠)𝑑𝑠≤𝑘₁max_{𝑗∈{1,2,…,𝑚}}𝑢𝑡_𝑗². (16)(H2) Assume that there exist positive constants 𝑎, 𝑏, and 𝛾∈[0,1), such that 0<𝑓(𝑡,𝑢)≤𝑎+𝑏|𝑢|^𝛾,for(𝑡,𝑢)∈[0,1]×ℝ. (17)

Let 𝑘₀=∫¹₀𝑞(𝑡)𝑑𝑡 and let 𝑘₂=(1/2)𝑘₀+𝑘₁ with 𝑘₁ given in (16). Clearly, 𝑀₁≤𝑘₀. For constants 𝑐₁, 𝑐, we define 𝜆₁=4𝑀₁∫¹₀max_{|𝑢|≤𝑐1}𝐹(𝑡,𝑢)𝑑𝑡𝑐²₁,𝜆₂=23∫¹₀𝐹(𝑡,𝑐)𝑑𝑡.𝑘₂𝑐². (18)

Theorem 4. Assume that (H1), (H2) are satisfied. If there exist two positive constants 𝑐₁, 𝑐 satisfying 𝑐₁<𝑐, and 𝜆₁<𝜆₂, (19)
then, for each 𝜆∈(1/𝜆₂,1/𝜆₁), problem (1) admits at least three solutions.

Proof. By Lemma 3, it suffices to show that the functional 𝜑_𝜆 defined in (8) has at least three critical points. We prove this by verifying the conditions given in [21, Theorem  3.2]. Note that Φ defined in (9) is a nonnegative Gâteaux differentiable, coercive, and sequentially weakly lower semicontinuous functional and its Gâteaux derivative admits a continuous inverse on 𝑋^∗. Moreover, Ψ defined in (10) is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Set 𝑀𝑟=¹𝑐²₁4. (20)
Note that 𝑢₁(𝑡)=𝑐∈𝑋. It then follows from (H1) that Φ𝑢₁=12𝑢₁²_𝑋+^𝑚_𝑗=1^{𝑢1(𝑡𝑗)}₀𝐼_𝑗(𝑠)=1𝑑𝑠2𝑐²¹₀𝑞(𝑠)𝑑𝑠+^𝑚_𝑗=1^𝑐₀𝐼_𝑗(𝑠)≤1𝑑𝑠2𝑘₀𝑐²+𝑘₁𝑐²≤𝑘₂𝑐²,Φ𝑢₁≥12𝑢₁²_𝑋=12𝑘₀𝑐². (21)
Then, we have 𝑀2𝑟=¹𝑐²₁2<12𝑘₀𝑐²≤Φ𝑢₁. (22)
For 𝑢∈𝑋 satisfying Φ(𝑢)<𝑟, by Lemma 1, one has |𝑢|²≤‖𝑢‖²_∞≤2𝑀₁‖𝑢‖²_𝑋≤4𝑀₁Φ(𝑢)<4𝑀₁𝑟=𝑐²₁,𝑡∈[0,1], (23)
which implies that Ψ(𝑢)=¹₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡≤¹₀max_{|𝑢|≤𝑐1}𝐹(𝑡,𝑢)𝑑𝑡. (24)
Hence sup_{𝑢∈Φ−1((−∞,𝑟))}Ψ(𝑢)≤¹₀max_{|𝑢|≤𝑐1}𝐹(𝑡,𝑢)𝑑𝑡. (25)
So, one has 0≤sup_𝑢∈Φ⁻¹_{((−∞,𝑟))}Ψ(𝑢)𝑟≤4𝑀₁∫¹₀max_{|𝑢|≤𝑐1}𝐹(𝑡,𝑢)𝑑𝑡𝑐²₁,23Ψ𝑢₁Φ𝑢₁≥23∫¹₀𝐹(𝑡,𝑐)𝑑𝑡.𝑘₂𝑐². (26)
Making use of (19), we obtain sup_𝑢∈Φ⁻¹_{((−∞,𝑟))}Ψ(𝑢)𝑟<23Ψ𝑢₁Φ𝑢₁. (27)
By (H1) and (H2), when 𝜆∈(1/𝜆₂,1/𝜆₁), we easily obtain that the functional 𝜑_𝜆 is coercive.
Thus all conditions in [21, Theorem  3.2] are verified, and hence for each 𝜆∈(1/𝜆₂,1/𝜆₁), the functional 𝜑_𝜆=Φ−𝜆Ψ admits at least three critical points. Consequently, problem (1) admits at least three solutions.

Example 5. Consider the boundary value problem −𝑒^𝑡𝑢′(𝑡)′+𝑢(𝑡)=𝜆𝑓(𝑡,𝑢),𝑡≠𝑡₁[],Δ𝑒,𝑡∈0,1_𝑡1𝑢′𝑡₁=𝑢𝑡₁,𝑡₁=12,𝑢′(0)=𝑢′(1)=0, (28)
where 𝑓()=𝑒𝑡,𝑢^𝑢[],𝑢,𝑢≤16,𝑡∈0,1_1/2+𝑒¹⁶−4,𝑢>16,𝑡∈[0,1]. (29)
Here, 𝐼₁(𝑠)=𝑠, 𝑝(𝑡)=𝑒^𝑡, 𝑞(𝑡)=1, and 𝑚=1. Note that (H1), (H2) are satisfied. Moreover, we have 𝑘₀=1, 𝑘₁=1/2, 𝑘₂=1, 𝑀₁=1, and 𝐹()=𝑒𝑡,𝑢^𝑢[],2−1,𝑢≤16,𝑡∈0,13𝑢^3/2+𝑒¹⁶𝑢+−4613−15𝑒¹⁶,𝑢>16,𝑡∈[0,1]. (30)
Choose 𝑐₁=1, 𝑐=16. Direct calculations give 𝜆₁=4𝑀₁∫¹₀max_{|𝑢|≤𝑐1}𝐹(𝑡,𝑢)𝑑𝑡𝑐²₁()𝜆=4𝑒−1<8,₂=23∫¹₀𝐹(𝑡,𝑐)𝑑𝑡.𝑘₂𝑐²=1𝑒384¹⁶−1>100. (31)

Therefore it follows from Theorem 4 that (28) admits at least three solutions in 𝑋 provided that 𝜆∈(0.01,0.125).

3.2. Existence of Infinitely Many Solutions

In this section, we derive some conditions under which problem (1) admits infinitely many distinct solutions. To this end, we need the following assumption. (H3) Assume that 𝐹(𝑡,𝑢)≥0,(𝑡,𝑢)∈[0,1]×ℝ. (32)

Set 𝜇₁=4𝑀₁liminf_𝜉→+∞∫¹₀max_{|𝑢|≤𝜉}𝐹(𝑡,𝑢)𝑑𝑡𝜉²,𝜇₂=1𝑘₂limsup_𝜉→+∞∫¹₀𝐹(𝑡,𝜉)𝑑𝑡𝜉². (33)

Theorem 6. Assume that (H3) is satisfied. If 𝜇₁<𝜇₂ (34)
holds, then for each 𝜆∈(1/𝜇₂,1/𝜇₁), problem (1) has an unbounded sequence of solutions in 𝑋.

Proof. We apply [22, Theorem  2.1] to show that the functional 𝜑_𝜆 defined in (8) has an unbounded sequence of critical points.
We first show that 𝛾₁<+∞. Let {𝜉_𝑛} be a sequence of positive numbers such that 𝜉_𝑛→+∞ as 𝑛→∞ and lim_𝑛→+∞∫¹₀max_{|𝑢|≤𝜉𝑛}𝐹(𝑡,𝑢)𝑑𝑡𝜉²_𝑛=liminf_𝜉→+∞∫¹₀max_{|𝑢|≤𝜉}𝐹(𝑡,𝜉)𝑑𝑡𝜉². (35)
For any positive integer 𝑛, we let 𝑟_𝑛=𝑀₁𝜉²_𝑛/4. For 𝑢∈𝑋 satisfying Φ(𝑢)<𝑟_𝑛, similar to the proof of Theorem 4, one can show that ‖𝑢‖²_∞≤4𝑀₁Φ(𝑢)<𝜉²_𝑛,𝑡∈[0,1], (36)
which implies that Ψ(𝑢)=¹₀𝐹(𝑡,𝑢)𝑑𝑡≤¹₀max_{|𝑢|≤𝜉𝑛}𝐹(𝑡,𝑢)𝑑𝑡. (37)
Note that Ψ(0)=Φ(0)=0; thus we have 𝜌𝑟_𝑛=inf_{𝑣∈Φ−1−∞,𝑟𝑛}sup_𝑢∈Φ⁻¹_{((−∞,𝑟𝑛))}Ψ(𝑢)−Ψ(𝑣)𝑟_𝑛(𝑣)≤−Φsup_𝑢∈Φ⁻¹_{((−∞,𝑟𝑛))}Ψ(𝑢)−Ψ(0)𝑟_𝑛(0)=−Φsup_𝑢∈Φ⁻¹_{((−∞,𝑟𝑛))}Ψ(𝑢)𝑟_𝑛≤4∫¹₀max_{|𝑢|≤𝜉𝑛}𝐹(𝑡,𝑢)𝑑𝑡𝑀₁𝜉²_𝑛, (38)
which, together with (15), gives us 𝛾₁≤4𝑀₁liminf_𝜉→+∞∫¹₀max_{|𝑢|≤𝜉}𝐹(𝑡,𝑢)𝑑𝑡𝜉²=𝜇₁<+∞. (39)
This shows that (1/𝜇₂,1/𝜇₁)⊆(0,1/𝛾₁). For any fixed 𝜆∈(1/𝜇₂,1/𝜇₁), it follows from [22, Theorem  2.1] that either 𝜑_𝜆=Φ−𝜆Ψ has a global minimum or there is a sequence {𝑢_𝑛} of critical points (local minima) of 𝜑_𝜆 such that lim_𝑛→+∞‖𝑢_𝑛‖_𝑋=+∞.
Next we show that the functional 𝜑_𝜆 has no global minimum for 𝜆∈(1/𝜇₂,1/𝜇₁). Since 𝜆>1/𝜇₂=𝑘₂/limsup_𝜉→+∞(∫¹₀𝐹(𝑡,𝜉)𝑑𝑡/𝜉²), we can choose a constant 𝑀₂ such that, for each 𝑛∈𝑁={1,2,…}, sup_𝜉≥𝑛∫¹₀𝐹(𝑡,𝜉)𝑑𝑡𝜉²>𝑀₂>𝑘₂𝜆. (40)
Thus, there exists 𝜉_𝑛≥𝑛 such that ∫¹₀𝐹𝑡,𝜉_𝑛𝑑𝑡𝜉²_𝑛>𝑀₂. (41)
Define 𝑢_𝑛(𝑡) as follows: 𝑢_𝑛(𝑡)=𝜉_𝑛,𝑡∈[0,1]. (42)
This yields Φ𝑢_𝑛=12𝑘₀𝜉²_𝑛+^𝑚_𝑗=1^𝜉𝑛₀𝐼_𝑗(𝑠)≤1𝑑𝑠2𝑘₀𝜉²_𝑛+𝑘₁𝜉²_𝑛=𝑘₂𝜉²_𝑛. (43)
Then 𝜑_𝜆𝑢_𝑛=Φ𝑢_𝑛−𝜆Ψ𝑢_𝑛≤𝑘₂𝜉²_𝑛−𝜆¹₀𝐹𝑡,𝜉_𝑛𝑑𝑡≤𝜉²_𝑛𝑘₂−𝜆𝑀₂. (44)
Note that 𝑘₂−𝜆𝑀₂<0. Thus the functional 𝜑_𝜆 has no lower bound and hence it has no global minimum and the proof is complete.

Let 𝜇₃=4𝑀₁liminf_𝜔→0+∫¹₀max_{|𝑢|≤𝜔}𝐹(𝑡,𝑢)𝑑𝑡𝜔²,𝜇₄=1𝑘₂limsup_𝜔→0+∫¹₀𝐹(𝑡,𝜔)𝑑𝑡𝜔². (45)

Theorem 7. Assume that (H3) is satisfied. If 𝜇₃<𝜇₄ (46)
holds, then, for each 𝜆∈(1/𝜇₄,1/𝜇₃), problem (1) has a sequence of nonzero solutions in X, which weakly converges to 0.

Proof. The proof is similar to that of Theorem 6 by showing that 𝛿<+∞ and 0 is not a local minimum of the functional 𝜑_𝜆=Φ−𝜆Ψ.

Example 8. Consider the boundary value problem −𝑒^𝑡𝑢′(𝑡)′+𝑢(𝑡)=𝜆𝑓(𝑡,𝑢),𝑡≠𝑡₁[],Δ𝑒,𝑡∈0,1_𝑡1𝑢′𝑡₁=𝑢𝑡₁,𝑡₁=12,𝑢′(0)=𝑢′(1)=0, (47)
where 𝑓(𝑡,𝑢)=8𝑡𝑢(1−sin𝑢)−4𝑡𝑢²cos𝑢.
Here, 𝐼₁(𝑠)=𝑠, 𝑝(𝑡)=𝑒^𝑡, 𝑞(𝑡)=1, and 𝑚=1. Hence we have 𝑘₀=1, 𝑘₁=1/2, 𝑘₂=1, 𝑀₁=1, and 𝐹(𝑡,𝑢)=4𝑡(1−sin𝑢)𝑢², (48)
so (H3) is satisfied. Moreover, we have lim_𝜉→+∞∫inf¹⁰max_{|𝑢|≤𝜉}𝐹(𝑡,𝑢)𝑑𝑡𝜉²=0,lim_𝜉→+∞∫sup¹⁰𝐹(𝑡,𝜉)𝑑𝑡𝜉²=2. (49)
Therefore, condition (34) holds and Theorem 6 applies: for 𝜆∈(0.5,+∞), problem (47) admits an unbounded sequence of solutions in 𝑋.

Example 9. Consider the boundary value problem −𝑒^𝑡𝑢^′(𝑡)′+𝑢(𝑡)=𝜆𝑓(𝑡,𝑢),𝑡≠𝑡₁[],Δ𝑒,𝑡∈0,1_𝑡1𝑢′𝑡₁=𝑢𝑡₁,𝑡₁=12,𝑢′(0)=𝑢′(1)=0, (50)
where 𝑓((𝑡=1𝑡,𝑢))𝑡𝑢0.35+4(|𝑢+1sinln|)8(|𝑢{0}[]cosln|)𝑢∈𝑅⧵,𝑡∈0,10,𝑢=0. (51)
In this example, 𝐼₁(𝑠)=𝑠, 𝑝(𝑡)=𝑒^𝑡, 𝑞(𝑡)=1, and 𝑚=1. Hence we have 𝑘₀=1, 𝑘₁=1/2, 𝑘₂=1, 𝑀₁=1, and 𝐹((𝑡=1𝑡,𝑢))2𝑡𝑢²10.35+4(|𝑢,{0}[],sinln|)𝑢∈𝑅⧵,𝑡∈0,10,𝑢=0. (52)
Hence, one has 4𝑀₁lim_𝑤→0+∫inf¹⁰max_{|𝑢|≤𝑤}𝐹(𝑡,𝑢)𝑑𝑡𝑤²1=0.1<𝑘₂lim_𝑤→0+∫sup¹⁰𝐹(𝑡,𝑤)𝑑𝑡𝑤²=0.15. (53)
Therefore (46) holds. Owing to Theorem 7, when 𝜆∈(6.67,10) problem (50) admits a sequence of pairwise distinct classical solutions strongly converging at 0 in 𝑋.

Acknowledgments

The authors are very grateful to the referees for their valuable comments and suggestions, which greatly improved the presentation of this paper. This work is supported by Hunan Provincial Natural Science Foundation of China (no. 11JJ3012).