Electronic Journal of Qualitative Theory of Differential Equations Spec. Ed. I, 2009 No. 10, 1–17; http://www.math.u-szeged.hu/ejqtde/

EXISTENCE OF THREE SOLUTIONS FOR SYSTEMS OF MULTI–POINT BOUNDARY VALUE PROBLEMS

Zengji Du ^{†}^{1} and Lingju Kong ^{‡}^{2}

† Department of Mathematics, Xuzhou Normal University Xuzhou, Jiangsu 221116, P. R. China e-mail: duzengji@163.com

‡ Department of Mathematics, University of Tennessee at Chattanooga Chattanooga, TN 37403, USA e-mail: Lingju-Kong@utc.edu

Honoring the Career of John Graef on the Occasion of His Sixty-Seventh Birthday

Abstract

By using a variational approach, we obtain some sufficient conditions for the existence of three classical solutions of a boundary value problem consisting of a system of differential equations and some multi–point boundary conditions. Applications of our results are discussed. Our results extend some related work in the literature.

Key words and phrases: Solutions, boundary value problems, critical points. AMS (MOS) Subject Classifications: 34B10, 34B15

1 Introduction

In recent years, there are many papers published on the existence of solutions of boundary value problems (BVPs) with various multi-point boundary conditions (BCs). For a small sample of the recent work on this topic, we refer the reader to [6, 10, 11, 15, 16, 19, 22, 23] for second order problems and to [2, 7, 8, 9, 12, 13, 17, 18, 24] for higher order ones. In this paper, we study the BVP consisting of the system of differential equations

(ϕ_{p_{i}}(u′_{i}))′ + λf_{i}(t, u_{1}, …, u_{n}) = 0, t ∈ (0, 1), i = 1, …, n,  (1)

1This author is supported by the Natural Science Foundation of Jiangsu Province (BK2008119), the NSF of the Education Department of Jiangsu Province (08KJB110011), and the Excellent Younger Teacher Program of Jiangsu Province in China (QL200613).

2Corresponding author. This author is supported in part by a Faculty Research Grant of the University of Tennessee at Chattanooga.

and the multi–point BCs

u_{i}(0) = ∑_{j=1}^{m} a_{j}u_{i}(t_{j}), u_{i}(1) = ∑_{j=1}^{m} b_{j}u_{i}(t_{j}), i = 1, …, n,  (2)

where λ is a real parameter, m, n ≥ 1 are integers, p_{i} > 1, ϕ_{p_{i}}(x) = |x|^{p_{i}−2}x, and f_{i} ∈ C([0, 1] × ℝ^{n}) for i = 1, …, n, a_{j}, b_{j} ∈ ℝ for j = 1, …, m, and 0 < t_{1} ≤ t_{2} ≤ … ≤ t_{m} < 1. We will obtain sufficient conditions for the existence of an open interval Λ ⊆ [0, ∞) such that, for each λ ∈ Λ, BVP (1), (2) has at least three classical solutions. Here, by a classical solution of BVP (1), (2), we mean a function u = (u_{1}, …, u_{n}) such that, for i = 1, …, n, u_{i} ∈ C^{1}[0, 1], ϕ_{p_{i}}(u′_{i}) ∈ C^{1}[0, 1], and u_{i}(t) satisfies (1), (2).

Our proof is based on a three critical point theorem of Ricceri [26]; see Lemma 1.2 below. For more applications of this theorem to various problems, we refer the reader to [3, 5, 6, 14, 21] for work on ordinary differential equations and [1, 4, 20] for work on partial differential equations. In particular, Bonannao [3] applied Ricceri’s theorem to the BVP

u′′ + λf(u) = 0, t ∈ (0, 1),  (3)
u(0) = u(1) = 0,  (4)

and obtained the following interesting result.

Proposition 1.1 ([3, Theorem 2]) Let f ∈ C(ℝ), F̄(x) = ∫_{0}^{x} f(ξ)dξ, and the norm

of the Sobolev space W_{0}^{1,2}([0, 1]) be defined by ||u|| = (∫_{0}^{1} |u′(s)|^{2}ds)^{1/2}. Assume that

there exist four positive constants c̄, d̄, ā, s̄ with c̄ < \sqrt{2}d̄ and γ̄ < 2 such that

(i) f(x) ≥ 0 for all x ∈ [−c̄, max{c̄, d̄}],

(ii) F̄(c̄) ≤ (c̄/2d̄)^{2}F̄(d̄),

(iii) F̄(x) ≤ ā(1 + |x|γ̄) for all x ∈ ℝ.

Then there exist an open interval Λ ⊆ [0, ∞) and a positive real number δ such that, for each λ ∈ Λ, BVP (3), (4) has at least three solutions belonging to C^{2}[0, 1] whose norms in W_{0}^{1,2}([0, 1]) are less than δ.

Candito [5] extended Proposition 1.1 to the nonautonomous case. He and Ge [14] further extended the result in [5] to the BVP consisting of the equation

(ϕ_{p}(u′))′ + λf(t, u) = 0, t ∈ (0, 1),  (5)

and BC (4), where p > 1 and f ∈ C([0, 1] × ℝ). Recently, Du [6] extended the main results in [3, 5, 14] to the BVP consisting of Eq. (5) and the three–point BCs

u(0) = 0, u(1) = αu(η),  (6)

where α ∈ ℝ and η ∈ (0, 1).

Motivated by these works, in this paper, we establish some criteria for the existence of three classical solutions of BVP (1), (2) (see Theorem 2.1). As applications, we present some new existence results (see Corollaries 2.1 and 2.2) for the scalar BVP consisting of Eq. (5) and the multi–point BCs

u(0) = ∑_{j=1}^{m} a_{j}u(t_{j}), u(1) = ∑_{j=1}^{m} b_{j}u(t_{j}).  (7)

Observe that BCs (7) include BCs (4) and (6) as special cases. We also give an application to BVP (5), (4) (see Corollary 2.3). Our results extend the main results in [3, 5, 6, 14] to more general problems (see Remark 2.1).

For the reader’s convenience, we now recall the following two results that are fundamental tools in our discussion.

Lemma 1.1 ([25, Proposition 3.1]) Let X be a separable and reflexive real Banach space, and Φ, J two real functions on X. Assume that there exist r > 0 and u_{0}, u_{1} ∈ X such that

Φ(u_{0}) = J(u_{0}) = 0, Φ(u_{1}) > r,

sup_{u∈Φ^{−1}((−∞,r])} J(u) < r\frac{J(u_{1})}{Φ(u_{1})}.

Then, for each η satisfying

sup_{u∈Φ^{−1}((−∞,r])} J(u) < η < r\frac{J(u_{1})}{Φ(u_{1})},

one has

sup_{λ≥0} inf_{u∈X}(Φ(u) + λ(η − J(u))) < inf_{u∈X} sup_{λ≥0}(Φ(u) + λ(η − J(u))).

Lemma 1.2 ([26, Theorem 1]) Let X be a separable and reflexive real Banach space, Φ : X → ℝ a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X^{∗}, Ψ : X → ℝ a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Assume that

lim_{||u||→∞}(Φ(u) + λΨ(u)) = ∞

for all λ ∈ [0, ∞), and that there exists a continuous concave function h : [0, ∞) → ℝ such that

sup_{λ≥0} inf_{u∈X}(Φ(u) + λΨ(u) + h(λ)) < inf_{u∈X} sup_{λ≥0}(Φ(u) + λΨ(u) + h(λ)).

Then there exist an open interval Λ ⊆ [0, ∞) and a positive real number δ such that, for each λ ∈ Λ, the equation

Φ′(u) + λΨ′(u) = 0

has at least three solutions in X whose norms are less than δ.

The rest of this paper is organized as follows. After this introduction, in Section 2, we state the main results and give one simple example for illustrative purposes. The proofs of the main results, together with some technical lemmas, are presented in Section 3.

2 Main Results

In the sequel, for i = 1, …, n, let

X_{i} = {y ∈ W^{1,p_{i}}([0, 1]) : y(0) = ∑_{j=1}^{m} a_{j}y(t_{j}), y(1) = ∑_{j=1}^{m} b_{j}y(t_{j})}

and let X be defined by

X = X_{1} × X_{2} × … × X_{n}

with the norm

||u|| = ||(u_{1}, …, u_{n})|| = ∑_{i=1}^{n} ||u′_{i}||_{p_{i}},

where

||u′_{i}||_{p_{i}} = (∫_{0}^{1} |u′(s)|^{p_{i}}ds)^{1/p_{i}} .

Then, X is a separable and reflexive real Banach space.

We first make the following assumption.

(H1) ∑_{j=1}^{m} a_{j} ≠ 1 and ∑_{j=1}^{m} b_{j} ≠ 1.

We now introduce some notations. For any nonempty set S, let

S^{n} = \underbrace{S × S × … × S}_{n},

and for x > 0, define

S_{1,n}(x) = \{ \begin{matrix}[0, t_{1}/2] × [x ∑_{j=1}^{m} a_{j}, x]^{n} & if ∑_{j=1}^{m} a_{j} < 1, \\ [0, t_{1}/2] × [x, x ∑_{j=1}^{m} a_{j}]^{n} & if ∑_{j=1}^{m} a_{j} > 1, \end{matrix}  (8)

S_{2,n}(x) = \{ \begin{matrix}[(1 + t_{m})/2, 1] × [x ∑_{j=1}^{m} b_{j}, x]^{n} & if ∑_{j=1}^{m} b_{j} < 1, \\ [(1 + t_{m})/2, 1] × [x, x ∑_{j=1}^{m} b_{j}]^{n} & if ∑_{j=1}^{m} b_{j} > 1. \end{matrix}  (9)

Let the positive constants κ_{i}, i = 1, …, n, and ρ be defined by

κ_{i} = 2^{p_{i}−1} (t_{1}^{1−p_{i}} |1 − ∑_{j=1}^{m} a_{j}|^{p_{i}} + (1 − t_{m})^{1−p_{i}} |1 − ∑_{j=1}^{m} b_{j}|^{p_{i}})  (10)

and

ρ = \frac{1}{2} (1 + \frac{∑_{j=1}^{m} |a_{j}|}{|1 − ∑_{j=1}^{m} a_{j}|} + \frac{∑_{j=1}^{m} |b_{j}|}{|1 − ∑_{j=1}^{m} b_{j}|}) .  (11)

Throughout this paper, the following assumptions are also needed. (H2) there exists a function F : [0, 1]×ℝ^{n} → ℝ such that F (t, x_{1}, …, x_{n}) is continuous

in t and differentiable in x_{i}, i = 1, …, n. Moreover, ∂F/∂x_{i} = f_{i} for i = 1, …, n; (H3) there exist two positive constants c and d with c < d such that

F (t, x_{1}, …, x_{n}) ≥ 0 for (t, x_{1}, …, x_{n}) ∈ S_{1,n}(d) ∪ S_{2,n}(d)

and

∑_{i=1}^{n} \frac{κ_{i}d^{p_{i}}}{p_{i}} max_{(t,x_{1},...,x_{n})∈K} F (t, x_{1}, …, x_{n}) ≤ ∑_{i=1}^{n} \frac{κ_{i}c^{p_{i}}}{p_{i}} ∫_{t_{1}/2}^{(1+t_{m})/2} F (s, d, …, d)ds,

where

K = \{(t, x_{1}, …, x_{n}) : t ∈ [0, 1], ∑_{i=1}^{n} \frac{|x_{i}|^{p_{i}}}{p_{i}} < ρ^{p_{i}} ∑_{i=1}^{n} \frac{κ_{i}c^{p_{i}}}{p_{i}} \};

(H4) there exist θ ∈ L^{1}(0, 1) and n positive constants γ_{i} with γ_{i} < p_{i}, i = 1, …, n,

such that

F (t, x_{1}, …, x_{n}) ≤ θ(t) (1 + ∑_{i=1}^{n} |x_{i}|^{γ_{i}})

for t ∈ [0, 1] and x_{i} ∈ ℝ, i = 1, …, n; (H5) F (t, 0, …, 0) = 0 for t ∈ [0, 1].

We say that a function u = (u_{1}, …, u_{n}) ∈ X is a weak solution of BVP (1), (2) if

∫_{0}^{1} ∑_{i=1}^{n} ϕ_{p_{i}}(u′_{i}(s))v′_{i}(s)ds − λ ∫_{0}^{1} ∑_{i=1}^{n} f_{i}(s, u_{1}(s), …, u_{n}(s))v_{i}(s)ds = 0

EJQTDE Spec. Ed. I, 2009 No. 10

for any v = (v_{1}, …, v_{n}) ∈ W_{0}^{1,p_{1}}([0, 1]) × W_{0}^{1,p_{2}}([0, 1]) × … × W_{0}^{1,p_{n}}([0, 1]). Under the assumptions (H1)–(H5), Theorem 2.1 below shows that BVP (1), (2) has at least three classical solutions. To prove the theorem, we will first apply Lemmas 1.1 and 1.2 to obtain the existence of three weak solutions of BVP (1), (2), then we show that the three weak solutions are indeed the classical solutions. In the process of the proof, three functionals Φ, Ψ, and J are constructed in such a way that all the conditions of Lemmas 1.1 and 1.2 are satisfied.

We now make some brief comments about the assumptions (H1)–(H5). (H1) is needed to obtain some useful bounds for functions in X (see Lemma 3.1). The function F introduced in (H2) is used in the construction of the functional Ψ. (H3) is required in the proof of the existence of a function w ∈ X with some nice properties (see Lemma 3.2). Both Lemmas 3.1 and 3.2 are crucial to prove our existence results. (H4), together with Lemma 3.1, is used to show the functional Φ(u) + λΨ(u) is weakly coercive for λ ∈ [0, ∞), i.e.,

lim_{||u||→∞}(Φ(u) + λΨ(u)) = ∞, λ ∈ [0, ∞).

Finally, (H5) is needed to show that J(0, …, 0) = 0, which is necessary in order to apply Lemma 1.1.

Now, we state our main results. The first one is concerned with BVP (1), (2).

Theorem 2.1 Assume (H1)–(H5) hold. Then there exist an open interval Λ ⊆ [0, ∞) and a positive real number δ such that, for each λ ∈ Λ, BVP (1), (2) has at least three classical solutions whose norms in X are less than δ.

For n = 1, let

F̃(t, x) = ∫_{0}^{x} f(t, ξ)dξ.  (12)

The following corollaries are direct consequences of Theorem 2.1.

Corollary 2.1 Assume (H1) and the following conditions hold:

(A1) there exist two positive constants c and d with c < d such that

F̃(t, x) ≥ 0 for (t, x) ∈ S_{1,1}(d) ∪ S_{2,1}(d)

and

max_{t∈[0,1],|x|<ρc(κ)^{1/p}} F̃(t, x) ≤ (\frac{c}{d})^{p} ∫_{t_{1}/2}^{(1+t_{m})/2} F̃(s, d)ds,

where S_{1,1} and S_{2,1} are defined by (8) and (9), respectively, with n = 1, ρ is defined by (11), and κ is defined by (10) with p_{i} = p, i.e.,

κ = 2^{p−1} (t_{1}^{1−p}|1 − ∑_{j=1}^{m} a_{j}|^{p} + (1 − t_{m})^{1−p}|1 − ∑_{j=1}^{m} b_{j}|^{p});  (13)

(A2) there exist θ ∈ L^{1}(0, 1) and γ > 0 with γ < p such that

F̃(t, x) ≤ θ(t) (1 + |x|^{γ})

for t ∈ [0, 1] and x ∈ ℝ.

Then there exist an open interval Λ ⊆ [0, ∞) and a positive real number δ such that, for each λ ∈ Λ, BVP (5), (7) has at least three classical solutions whose norms in X are less than δ.

Corollary 2.2 Assume (H1) and the following conditions hold:

(B1) f(t, x) = g(t)h(x) with g(t) and H(x) = ∫_{0}^{x} h(ξ)dξ being nonnegative;

(B2) there exist two positive constants c and d with c < d such that

max_{t∈[0,1]} g(t) max_{|x|<ρc(κ)^{1/p}} H(x) ≤ (\frac{c}{d})^{p} H(d)(G((1 + t_{m})/2) − G(t_{1}/2)),

where ρ is defined by (11) and κ is defined by (13), and G(t) = ∫_{0}^{t} g(s)ds.

(B3) there exists σ > 0 and γ > 0 with γ < p such that

H(x) ≤ σ(1 + |x|^{γ}) for x ∈ ℝ.

Then there exist an open interval Λ ⊆ [0, ∞) and a positive real number δ such that, for each λ ∈ Λ, BVP (5), (7) has at least three classical solutions whose norms in X are less than δ.

Corollary 2.3 Assume (A2) and the following condition hold:

(C1) there exist two positive constants c and d with c < d such that

F̃(t, x) ≥ 0 for (t, x) ∈ ([0, 1/4] ∪ [3/4, 1]) × [0, d]

and

max_{t∈[0,1],|x|<2^{1−1/p}c} F̃(t, x) ≤ (\frac{c}{d})^{p} ∫_{1/4}^{3/4} F̃(s, d)ds.

Then there exist an open interval Λ ⊆ [0, ∞) and a positive real number δ such that, for each λ ∈ Λ, BVP (5), (4) has at least three classical solutions whose norms in X are less than δ.

Remark 2.1 Corollaries 2.1–2.3 improve and extend the main results in [3, 5, 6, 14]. In particular, for the case when p = 2, by taking c̄ = \sqrt{2}c and d̄ = d, it is easy to see that Proposition 1.1 is a special case of Corollary 2.3.

We conclude this section with the following simple example.

Example Consider the BVP consisting of the equation

(|u′|u′)′ + λt(e^{u} + 2u) = 0, t ∈ (0, 1),  (14)

and the three–point BCs

u(0) = u(1) = \frac{1}{2}u (\frac{1}{2}) ,  (15)

where λ is a real parameter.

We claim that there exist an open interval Λ ⊆ [0, ∞) and a positive real number δ such that, for each λ ∈ Λ, BVP (14), (15) has at least three classical solutions whose norms in X are less than δ.

In fact, with p = 3, m = 1, a_{1} = b_{1} = t_{1} = 1/2, and f(t, x) = t(e^{x} +2x), it is easy to see that BVP (14), (15) is of the form BVP (5), (7). Let g(t) = t and h(x) = e^{x} + 2x. Then

max_{t∈[0,1]} g(t) = 1, G(t) = \frac{t^{2}}{2} , and H(x) = e^{x} + x^{2}.

Clearly, (B1) and (B3) with σ = 1 and γ = 2 hold.

Let c = 1 and d = 12. For ρ and κ defined by (11) and (13), by a simple calculation, we have that ρ = 3/2, κ = 4,

(\frac{c}{d})^{p} H(d)(G(1 + t_{1})/2 − G(t_{1}/2)) ≈ 23.57,

and

max_{t∈[0,1]} g(t) max_{|x|<ρc(κ)^{1/p}} H(x) ≈ 16.49.

Then

max_{t∈[0,1]} g(t) max_{|x|<ρc(κ)^{1/p}} H(x) ≤ (\frac{c}{d})^{p} H(d)(G(1 + t_{1})/2 − G(t_{1}/2)),

i.e., (B2) holds. The conclusion now readily follows from Corollary 2.2.

Remark 2.2 To the best of our knowledge, no known criteria in the literature can be applied to BVP (14), (15) to obtain the same conclusion as what we get here.

3 Proofs of the Main Results

Lemma 3.1 Assume (H1) holds. If u = (u_{1}, …, u_{n}) ∈ X, then

max_{t∈[0,1]} |u_{i}(t)| ≤ ρ||u′_{i}||_{p_{i}}, i = 1, …, n,

where ρ is defined by (11).

Existence of Three Solutions 9 Proof. For i = 1, …, n and t ∈ [0, 1], from

u_{i}(t) = ∫_{0}^{t} u′_{i}(s)ds + C_{1},

it follows that

u_{i}(0) = C_{1} and ∑_{j=1}^{m} a_{j}u_{i}(t_{j}) = ∑_{j=1}^{m} a_{j} ∫_{0}^{t_{j}} u′_{i}(s)ds + C_{1} ∑_{j=1}^{m} a_{j}.

Since u_{i}(0) = ∑_{j=1}^{m} a_{j}u_{i}(t_{j}), we have

C_{1} = ∑_{j=1}^{m} a_{j} ∫_{0}^{t_{j}} u′_{i}(s)ds + C_{1} ∑_{j=1}^{m} a_{j}.

Then,

C_{1} = \frac{1}{1 − ∑_{j=1}^{m} a_{j}} ∑_{j=1}^{m} a_{j} ∫_{0}^{t_{j}} u′_{i}(s)ds.

Thus,

u_{i}(t) = ∫_{0}^{t} u′_{i}(s)ds + \frac{1}{1 − ∑_{j=1}^{m} a_{j}} ∑_{j=1}^{m} a_{j} ∫_{0}^{t_{j}} u′_{i}(s)ds.  (16)

Similarly, from

u_{i}(t) = ∫_{1}^{t} u′_{i}(s)ds + C_{2} and u_{i}(1) = ∑_{j=1}^{m} b_{j}u_{i}(t_{j}),

we have that

u_{i}(t) = ∫_{1}^{t} u′_{i}(s)ds + \frac{1}{1 − ∑_{j=1}^{m} b_{j}} ∑_{j=1}^{m} b_{j} ∫_{1}^{t_{j}} u′_{i}(s)ds.  (17)

Now, (16) and (17) imply that

|u_{i}(t)| ≤ ∫_{0}^{t} |u′_{i}(s)|ds + \frac{1}{|1 − ∑_{j=1}^{m} a_{j}|} ∑_{j=1}^{m} |a_{j}| ∫_{0}^{t_{j}} |u′_{i}(s)|ds

and

|u_{i}(t)| ≤ ∫_{t}^{1} |u′_{i}(s)|ds + \frac{1}{|1 − ∑_{j=1}^{m} b_{j}|} ∑_{j=1}^{m} |b_{j}| ∫_{t_{j}}^{1} |u′_{i}(s)|ds.

EJQTDE Spec. Ed. I, 2009 No. 10

10 Z. Du & L. Kong Hence,

2|u_{i}(t)| ≤ ∫_{0}^{1} |u′_{i}(s)|ds + \frac{1}{|1 − ∑_{j=1}^{m} a_{j}|} ∑_{j=1}^{m} |a_{j}| ∫_{0}^{t_{j}} |u′_{i}(s)|ds

+\frac{1}{|1 − ∑_{j=1}^{m} b_{j}|} ∑_{j=1}^{m} |b_{j}| ∫_{t_{j}}^{1} |u′_{i}(s)|ds

≤ (1 + \frac{∑_{j=1}^{m} |a_{j}|}{|1 − ∑_{j=1}^{m} a_{j}|} + \frac{∑_{j=1}^{m} |b_{j}|}{|1 − ∑_{j=1}^{m} b_{j}|}) ∫_{0}^{1} |u′_{i}(s)|ds
= 2ρ ∫_{0}^{1} |u′_{i}(s)|ds.

Then, from Hölder’s inequality,

|u_{i}(t)| ≤ ρ ∫_{0}^{1} |u′_{i}(s)|ds ≤ ρ (∫_{0}^{1} |u′(s)|^{p_{i}})^{1/p_{i}} = ρ||u′_{i}||_{p_{i}} on [0, 1]. This completes the proof of the lemma. Lemma 3.2 Assume (H1)–(H3) hold. Then there exists w = (w_{1}, …, w_{n}) ∈ X such that

∑_{i=1}^{n} \frac{||w′_{i}||_{p_{i}}^{p_{i}}}{p_{i}} > r

and

∑_{i=1}^{n} \frac{||w′_{i}||_{p_{i}}^{p_{i}}}{p_{i}} max_{(t,x_{1},...,x_{n})∈K} F (t, x_{1}, …, x_{n}) < ∑_{i=1}^{n} \frac{κ_{i}c^{p_{i}}}{p_{i}} ∫_{0}^{1} F (s, w_{1}(s), …, w_{n}(s))ds,

where κ_{i} is defined by (10) and

r = ∑_{i=1}^{n} \frac{κ_{i}c^{p_{i}}}{p_{i}} .

Proof. For i = 1, …, n, let

w_{i}(t) = \{ \begin{matrix}d (∑_{j=1}^{m} a_{j} + ^{\frac{2(1−∑_{j=1}^{m} a_{j})}{t_{1}}}t) , & t ∈ [0, t_{1}/2], \\ d, & t ∈ [t_{1}/2, (1 + t_{m})/2], \\ d (^{\frac{2−∑_{j=1}^{m} b_{j}−(∑_{j=1}^{m} b_{j})t_{m}}{1−t_{m}}} − ^{\frac{2(1−∑_{j=1}^{m} b_{j})}{1−t_{m}}}t) , & t ∈ [(1 + t_{m})/2, 1]. \end{matrix}

EJQTDE Spec. Ed. I, 2009 No. 10

Existence of Three Solutions 11 Let w(t) = (w_{1}(t), …, w_{n}(t)). Then, w ∈ X, and by a simple calculation, we obtain that

||w′_{i}||_{p_{i}}^{p_{i}} = ∫_{0}^{1} |w′_{i}(s)|^{p_{i}}ds
= (∫_{0}^{t_{1}/2} (\frac{2|1 − ∑_{j=1}^{m} a_{j}|}{t_{1}})^{p_{i}} ds + ∫_{(1+t_{m})/2}^{1} (\frac{2|1 − ∑_{j=1}^{m} b_{j}|}{1 − t_{m}})^{p_{i}} ds) d^{p_{i}}
= 2^{p_{i}−1} (t_{1}^{1−p_{i}} |1 − ∑_{j=1}^{m} a_{j}|^{p_{i}} + (1 − t_{m})^{1−p_{i}} |1 − ∑_{j=1}^{m} b_{j}|^{p_{i}}) d^{p_{i}} = κ_{i}d^{p_{i}}.  (18)

Thus, in view of the fact that c < d, we have

∑_{j=1}^{m} \frac{||w_{i}||_{p_{i}}^{p_{i}}}{p_{i}} = ∑_{j=1}^{m} \frac{κ_{i}d^{p_{i}}}{p_{i}} > ∑_{j=1}^{m} \frac{κ_{i}c^{p_{i}}}{p_{i}} = r.

Note that for i = 1, …, n,

d ∑_{j=1}^{m} a_{j} ≤ w_{i}(t) ≤ d on [0, t_{1}, /2] if ∑_{j=1}^{m} a_{j} < 1,

d ≤ w_{i}(t) ≤ d ∑_{j=1}^{m} a_{j} on [0, t_{1}, /2] if ∑_{j=1}^{m} a_{j} > 1,

d ∑_{j=1}^{m} b_{j} ≤ w_{i}(t) ≤ d on [(1 + t_{m})/2, 1] if ∑_{j=1}^{m} b_{j} < 1,

and

d ≤ w_{i}(t) ≤ d ∑_{j=1}^{m} b_{j} on [(1 + t_{m})/2, 1] if ∑_{j=1}^{m} b_{j} > 1.

Then, from (H3) and (18), it follows that

∑_{i=1}^{n} \frac{κ_{i}c^{p_{i}}}{p_{i}} ∫_{0}^{1} F (s, w_{1}(s), …, w_{n}(s))ds

≥ ∑_{i=1}^{n} \frac{κ_{i}c^{p_{i}}}{p_{i}} ∫_{t_{1}/2}^{(1+t_{m})/2} F (s, w_{1}(s), …, w_{n}(s))ds
≥ ∑_{i=1}^{n} \frac{κ_{i}d^{p_{i}}}{p_{i}} max_{(t,x_{1},...,x_{n})∈K} F (t, x_{1}, …, x_{n})
= ∑_{i=1}^{n} \frac{||w′_{i}||_{p_{i}}^{p_{i}}}{p_{i}} max_{(t,x_{1},...,x_{n})∈K} F (t, x_{1}, …, x_{n}).

EJQTDE Spec. Ed. I, 2009 No. 10

This completes the proof of the lemma.

In what follows, for i = 1, …, n, let ϕ_{p_{i}}^{−1} denote the inverse of ϕ_{p_{i}}. Then, ϕ_{p_{i}}^{−1}(x) = ϕ_{q_{i}}(x), where 1/p_{i} + 1/q_{i} = 1. Clearly, ϕ_{p_{i}} is increasing on ℝ, and

lim_{x→−∞} ϕ_{p_{i}}(x) = −∞ and lim_{x→∞} ϕ_{p_{i}}(x) = ∞.  (19)

Lemma 3.3 For any fixed λ ∈ ℝ, u = (u_{1}, …, u_{n}) ∈ (C[0, 1])^{n}, and i = 1, …, n, define α_{i}(·; u) : ℝ → ℝ by

α_{i}(x; u) = ∫_{0}^{1} ϕ_{p_{i}}^{−1} (x − λ ∫_{0}^{τ} f_{i}(s, u_{1}(s), …, u_{n}(s))ds) dτ

+ ∑_{j=1}^{m} a_{j}u_{i}(t_{j}) − ∑_{j=1}^{m} b_{j}u_{i}(t_{j}).

Then the equation

α_{i}(x; u) = 0  (20)

has a unique solution x_{u,i}.

Proof. From (19), we see that

lim_{x→−∞} α(x; u) = −∞ and lim_{x→∞} α(x; u) = ∞.

Hence, the existence and uniqueness of a solution of Eq. (20) follow from the fact that α(·; u) is continuous and increasing on ℝ.

Lemma 3.4 The function u(t) = (u_{1}(t), …, u_{n}(t)) is a solution of BVP (1), (2) if and only if u_{i}(t), i = 1, …, n, is a solution of the system of the integral equations

u_{i}(t) = ∑_{j=1}^{m} a_{j}u_{i}(t_{j}) + ∫_{0}^{t} ϕ_{p_{i}}^{−1} (x_{u,i} − λ ∫_{0}^{τ} f_{i}(s, u_{1}(s), …, u_{n}(s))ds) dτ, i = 1, …, n,

where x_{u,i} is the unique solution of Eq. (20).

Proof. This can be verified by direct computations.

Proof of Theorem 2.1. For u = (u_{1}, …, u_{n}) ∈ X, let

Φ(u) = ∑_{i=1}^{n} \frac{||u′_{i}||_{p_{i}}^{p_{i}}}{p_{i}}

and

Ψ(u) = − ∫_{0}^{1} F (s, u_{1}(s), …, u_{n}(s))ds.

Then, Φ is a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X^{∗}, and Ψ is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. In particular, for u = (u_{1}, …, u_{n}) ∈ X and v = (v_{1}, …, v_{n}) ∈ X, we have

Φ′(u)(v) = ∫_{0}^{1} ∑_{i=1}^{n} ϕ_{p_{i}}(u′_{i}(s))v′_{i}(s)ds

and

Ψ′(u)(v) = − ∫_{0}^{1} ∑_{i=1}^{n} f_{i}(s, u_{1}(s), …, u_{n}(s))v_{i}(s)ds.

Hence, the weak solutions of BVP (1), (2) are exactly the solutions of the equation

Φ′(u) + λΨ′(u) = 0.  (21)

Let λ ∈ [0, ∞). From (H4) and Lemma 3.1, we see that

Φ(u) + λΨ(u) ≥ ∑_{i=1}^{n} \frac{||u′_{i}||_{p_{i}}^{p_{i}}}{p_{i}} − λ ∫_{0}^{1} θ(s) (1 + ∑_{i=1}^{n} |u_{i}(s)|^{γ_{i}}) ds

≥ ∑_{i=1}^{n} \frac{||u′_{i}||_{p_{i}}^{p_{i}}}{p_{i}} − λ (1 + ρ ∑_{i=1}^{n} ||u′_{i}||_{p_{i}}^{γ_{i}}) ∫_{0}^{1} θ(s)ds.

Since γ_{i} < p_{i} for i = 1, …, n, we have that

lim_{||u||→∞}(Φ(u) + λΨ(u)) = ∞.

Lemma 3.1 implies that

sup_{t∈[0,1]} ∑_{i=1}^{n} \frac{|u_{i}(t)|^{p_{i}}}{p_{i}} ≤ ρ^{p_{i}} ∑_{i=1}^{n} \frac{||u′_{i}||_{p_{i}}^{p_{i}}}{p_{i}} .  (22)

Let w and r be as introduced in Lemma 3.2. Then,

Φ(w) = ∑_{i=1}^{n} \frac{||w′_{i}||_{p_{i}}^{p_{i}}}{p_{i}} > r = ∑_{i=1}^{n} \frac{κ_{i}c^{p_{i}}}{p_{i}} .

Let u_{0} = (0, …, 0), u_{1} = w, and J = −Ψ. Then, in view of (H5), it is clear that

Φ(u_{0}) = J(u_{0}) = 0, Φ(u_{1}) > r.

Moreover, from Lemma 3.2 and (22), it follows that

sup_{u∈Φ((−∞,r])}(J(u)) = sup_{u∈Φ((−∞,r])}(−Ψ(u))
= ∑_{∑_{i=1}^{n} \frac{||u′_{i}||{p_{i} \atop p_{i}}}{p_{i}} ≤r} ∫_{0}^{1} F (s, u_{1}(s), …, u_{n}(s))ds

≤ max_{(t,x_{1},...,x_{n})∈K} F (t, x_{1}, …, x_{n})
< \begin{matrix}∑ & {n \atop i=1} & \frac{κ_{i}c^{p_{i}}}{p_{i}} \\ ∑ & {n \atop i=1} & \frac{||w′_{i}||_{p_{i}}^{p_{i}}}{p_{i}} \end{matrix} ∫_{0}^{1} F (s, w_{1}(s), …, w_{n}(s))ds

< r\frac{−Ψ(w)}{Φ(w)} = r\frac{J(u_{1})}{Φ(u_{1})}.

Fix η such that

sup_{u∈Φ((−∞,r])}(J(u)) < η < r\frac{J(u_{1})}{Φ(u_{1})}

and define h(λ) = λη for λ ≥ 0. Then, from Lemma 1.1, we have

sup_{λ≥0} inf_{u∈X}(Φ(u) + λΨ(u) + h(λ)) < inf_{u∈X} sup_{λ≥0}(Φ(x) + λΨ(x) + h(λ)).

Therefore, by Lemma 2, there exist an open interval Λ ⊆ [0, ∞) and a positive real number δ such that, for each λ ∈ Λ, Eq. (21) has at least three solutions

u^{k} = (u_{k1}, …, u_{kn}), k = 1, 2, 3,

in X that are three weak solutions of BVP (1), (2) and whose norms are less than δ. Then, for k = 1, 2, 3, and i = 1, …, n, we have

∫_{0}^{1} ∑_{i=1}^{n} ϕ_{p_{i}}(u′_{ki}(s))v′_{i}(s)ds − λ ∫_{0}^{1} ∑_{i=1}^{n} f_{i}(s, u_{k1}(s), …, u_{kn}(s))v_{i}(s)ds = 0  (23)

for any v = (v_{1}, …, v_{n}) ∈ W_{0}^{1,p_{1}}([0, 1])×W_{0}^{1,p_{2}}([0, 1])×…×W_{0}^{1,p_{n}}([0, 1]). Recall that, in one dimension, any weakly differentiable function is absolutely continuous, so that its classical derivative exists almost everywhere, and that the classical derivative coincides with the weak derivative. Now, applying integration by parts to (23) yields that

∑_{i=1}^{n} ∫_{0}^{1} ((ϕ_{p_{i}}(u′_{ki}(s)))′ + λf_{i}(s, u_{k1}(s), …, u_{kn})(s))v_{i}(s)ds = 0,

from which it follows that

(ϕ_{p_{i}}(u′_{ki}))′ + λf_{i}(t, u_{k1}, …, u_{kn}) = 0 for a.e. t ∈ (0, 1).  (24)

Then, by Lemmas 3.3 and 3.4, it is easy to see that

u_{ki}(t) = ∑_{j=1}^{m} a_{j}u_{ki}(t_{j}) + ∫_{0}^{t} ϕ_{p_{i}}^{−1} (x_{u^{k},i} − λ ∫_{0}^{τ} f_{i}(s, u_{k1}(s), …, u_{kn}(s))ds) dτ,

where x_{u^{k},i} is the unique solution of Eq. (20) with u = u^{k}. Consequently, u_{ki} ∈ C^{1}[0, 1] and ϕ(u′_{ki}) ∈ C^{1}[0, 1], i.e., u^{k}, k = 1, 2, 3, are classical solutions of BVP (1), (2). This completes the proof of the theorem.

Proof of Corollary 2.1. The conclusion follows directly from Theorem 2.1.

Proof of Corollary 2.2. Let F̃ be defined by (12). Then, from (B1) and (B2), we see that

F̃(t, x) = ∫_{0}^{x} g(t)h(ξ)dξ = g(t)H(x) ≥ 0 for (t, x) ∈ [0, 1] × ℝ

and

max_{t∈[0,1],|x|<(ρκ)^{1/p}c} F̃(t, x) = max_{t∈[0,1],|x|<ρc(κ)^{1/p}} ∫_{0}^{x} g(t)h(ξ)dξ
= max_{t∈[0,1]} g(t) max_{|x|<ρc(κ)^{1/p}} H(x)
≤ (\frac{c}{d})^{p} H(d)(G((1 + t_{m})/2) − G(t_{1}/2))
= (\frac{c}{d})^{p} ∫_{0}^{d} h(ξ)dξ ∫_{t_{1}/2}^{(1+t_{m})/2} g(s)ds
= (\frac{c}{d})^{p} ∫_{t_{1}/2}^{(1+t_{m})/2} ∫_{0}^{d} g(s)h(ξ)dξds
= (\frac{c}{d})^{p} ∫_{t_{1}/2}^{(1+t_{m})/2} ∫_{0}^{d} f(s, ξ)dξds
= (\frac{c}{d})^{p} ∫_{t_{1}/2}^{(1+t_{m})/2} F̃(s, d)ds.

Thus, (A1) holds. By (B1) and (B3), we have

F̃(t, x) = ∫_{0}^{x} g(t)h(ξ)dξ = g(t)H(x) ≤ σg(t)(1 + |x|^{γ})

for (t, x) ∈ [0, 1] × ℝ. Then (A2) holds with θ(t) = σg(t). The conclusion then readily follows from Corollary 2.1.

Proof of Corollary 2.3. By taking m = 1, a_{1} = b_{1} = 0, and t_{1} = 1/2, it is easy to see that the assumption (A1) reduces to (C1). The conclusion then readily follows from Corollary 2.1.

Acknowledgments

The authors thank Dr. Boris Belinkskiy for his useful suggestions and comments during the preparation of this work.

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