Infinitely Many Sign-Changing Solutions for Some Nonlinear Fourth-Order Beam Equations

Abstract

Several new existence theorems on positive, negative, and sign-changing solutions for the following fourth-order beam equation are obtained: 𝑢⁽⁴⁾=𝑓(𝑡,𝑢(𝑡)),   𝑡∈[0,1];  𝑢(0)=𝑢(1)=𝑢^′′(0)=𝑢^′′(1)=0, where 𝑓∈𝐶([0,1]×ℝ¹,ℝ¹). In particular, an infinitely many sign changing solution theorem is established. The method of the invariant set of decreasing flow is employed to discuss this problem.

1. Introduction and Main Results

It is well known that the following fourth-order two-point boundary value problem (BVP): 𝑢⁽⁴⁾((𝑡[],𝑢(0)(1)=𝑓𝑡,𝑢)),𝑡∈0,1=𝑢=𝑢^′′(0)=𝑢^′′(1)=0 (1) describes the deformation of an elastic beam both of whose ends are simply supported at 0 and 1. Owing to the importance of high-order differential equations both in theory and practice, much attention has been paid to such problems by a number of authors, see [1–15] and references therein.

Among these literatures, some of the authors used to deal with the existence of positive solutions by employing the cone expansion and compression fixed point theorem of norm type [1, 2, 5, 6], the five functionals fixed point theorem [10], and the abstract fixed point index theory [3, 4]. The main assumptions imposed on the nonlinear term 𝑓 included that it is superlinear or sublinear in 𝑢, and its growth on some intervals is restricted by suitable functions, or it is asymptotically linear in 0 and ∞.

Various methods have been applied to these problems in recent years. In [7], by using the strongly monotone operator principle and the critical point theory, Li et al. established some sufficient conditions for 𝑓 to guarantee that the problem has a unique solution, at least one nonzero solution, or infinitely many solutions. In [8], some new existence theorems on multiple positive, negative, and sign-changing solutions of BVP (1) were established by combining the critical point theory and the method of sub- and supsolution. In [11], the authors obtained a three-solution theorem and an infinitely-many-solution theorem by applying the Morse theory, in which they removed a condition in [7]. In [13, 15], Yang and Zhang got some infinitely many mountain pass solutions theorems for the problems with parameters by the mountain pass theorem in order interval, in which they imposed conditions on 𝑓 to guarantee that the equation has infinitely many pairs sub- and supsolutions.

Besides, a few papers are concerned with the existence and multiplicity of the sign-changing solutions for kinds of fourth-order boundary value problems [8, 9, 12] recently. One reason is from the following fact. Consider the linear eigenvalue problem 𝑢⁽⁴⁾[],𝑢(0)(1)=𝜆𝑢,𝑡∈0,1=𝑢=𝑢^′′(0)=𝑢^′′(1)=0. (2) It is known that (2) has an unbounded sequence of eigenvalues 𝜋⁴<(2𝜋)⁴<⋯<(𝑚𝜋)⁴<⋯ (3) with corresponding eigenfunctions sin𝜋𝑡,sin2𝜋𝑡,…,sin𝑚𝜋𝑡,…. (4)

Obviously, the first eigenfunction sin𝜋𝑡>0 for all 𝑡∈(0,1) and other eigenfunctions sin𝑚𝜋𝑡(𝑚=2,3,…) are all sign-changing functions on [0,1]. This fact suggests that BVP (1) regarded as a nonlinear perturbation of (2) should have more sign-changing solutions than positive and negative solutions. Another reason is that sign-changing solutions should have more complicated properties, such as the times of changing sign. Thus, sign-changing solutions are interesting challenges in mathematics. In a word, the study on sign-changing solutions is the natural extension and deepening of the previous research on positive solutions. In [12], using the fixed point index and the critical group, Li et al. obtained that the fourth-order Neumann problem has at least one positive solution and two sign-changing solutions under certain conditions. In [9], Li proved that the fourth-order problem in which 𝑓 contains the bending moment term 𝑢^′′ has multiple sign-changing solutions by the fixed point index theory in cones and an antisymmetrical extension method of solution.

In the present paper, motivated by the above results, we investigate the positive, negative, and sign-changing solutions for BVP (1) as well. By applying the method of the invariant set of decreasing flow, we establish several multiple solutions theorems. Furthermore, an infinitely many sign changing-solution theorem is obtained. The comparisons with the results in the literatures are stated in the remarks below our main theorems. And we present four simple examples to which our theorems can be applied, respectively.

For convenience, we list our conditions as follows:(H₁)𝑓∈𝐶([0,1]×ℝ¹,ℝ¹),(H₂)limsup_𝑢→0|𝑓(𝑡,𝑢)/𝑢|<𝜋⁴ for all 𝑡∈[0,1] uniformly,(H₃)there exist 𝜇∈(0,1/2) and 𝑀>0 such that 𝐹(𝑡,𝑢)≜^𝑢₀𝑓(𝑡,𝑣)d𝑣⩽𝜇𝑢𝑓(𝑡,𝑢)∀|𝑢|⩾𝑀,𝑡∈[0,1]; (5)(H₄)𝑓(𝑡,0)=0 for all 𝑡∈[0,1],(H₅) there exist two real numbers 𝑐>0 and 𝑞>2 such that 𝑓(𝑡,𝑢)⩽𝑐1+|𝑢|^𝑞−1∀(𝑡,𝑢)∈[0,1]×ℝ¹, (6)(H₆)𝑓(𝑡,𝑢) is odd in 𝑢, that is, 𝑓(𝑡,−𝑢)=−𝑓(𝑡,𝑢) for all (𝑡,𝑢)∈[0,1]×ℝ¹.

The following three conditions are a little weaker than (H₂) and (H₃), respectively: (H^′₂)limsup_𝑢→0+(𝑓(𝑡,𝑢)/𝑢)<𝜋⁴ for all 𝑡∈[0,1] uniformly, (H^′′₂)limsup_𝑢→0(𝑓(𝑡,𝑢)/𝑢)<𝜋⁴ for all 𝑡∈[0,1] uniformly, (H^′₃) there exist 𝜇∈(0,1/2) and 𝑀>0 such that 𝐹(𝑡,𝑢)≜^𝑢₀𝑓(𝑡,𝑣)d𝑣⩽𝜇𝑢𝑓(𝑡,𝑢)∀𝑢⩾𝑀,𝑡∈[0,1]. (7)

Now, the main results can be stated as follows.

Theorem 1. Assume that (H₁), (H₂′), and (H₃′) hold. In addition, 𝑓(𝑡,𝑢)⩾0 for all 𝑢∈[0,+∞). Then, BVP (1) has at least a positive solution in 𝐶⁴[0,1].

Remark 2. We will apply the method of the invariant set of decreasing flow to prove this theorem in Section 3. When dealing with BVP (1) by the cone expansion and compression theorems [16, 17], we usually assume that limsup_𝑢→0+𝑓(𝑡,𝑢)𝑢=0,limsup_𝑢→+∞𝑓(𝑡,𝑢)𝑢=+∞. (8) The first equation is stronger than (H^′₂), while the latter is weaker than (H₃). So, both of the two methods have their own characteristics.

Example 3. Let 𝑓()=(𝑡)()|𝑢|𝑡,𝑢𝑎𝑢+𝑏ln1+𝑢arctan𝑢+𝑐^𝛾()∈[]×[),()∈[]×(),𝑢,𝑡,𝑢0,10,+∞0,𝑡,𝑢0,1−∞,0 (9) where 𝑎∈[0,𝜋⁴), 𝑏∈𝐶([0,1],[0,+∞)) and 𝑐,𝛾>0. It is easy to check that all conditions of Theorem 1 are satisfied. So, BVP (1) with the nonlinear term (9) has at least a positive solution.

Theorem 4. Assume that (H₁), (H^′′₂), and (H₃) hold. In addition, 𝑓(𝑡,𝑢)𝑢⩾0 for all 𝑢∈ℝ¹. Then, BVP (1) has at least a positive solution and a negative solution in 𝐶⁴[0,1].

Remark 5. In [7], using the mountain pass lemma, Li et al. obtained that BVP (1) has at least one nonzero solution in 𝐶⁴[0,1] under the assumptions that (H₁), (H^′′₂), and (H₃) hold [7, Theorem 3.3]. By adding a condition 𝑓(𝑡,𝑢)𝑢⩾0, we get two solutions, one positive and the other negative. So, Theorem 4 can be seen as a complement of [7, Theorem 3.3].

Example 6. Let 𝑓(𝑡,𝑢)=𝑎arctan𝑢+𝑏(𝑡)𝑢𝑒^𝑢−1+𝑐|𝑢|^𝛾𝑢for(𝑡,𝑢)∈[0,1]×ℝ¹, (10) where 𝑎∈[0,𝜋⁴), 𝑏∈𝐶([0,1],[0,+∞)) and 𝑐,𝛾>0. It is easy to verify that all conditions of Theorem 4 are satisfied. So, BVP (1) with the nonlinear term (10) has at least two solutions, one positive and the other negative. Reference [7, Theorem 3.3] can only guarantee a nonzero solution for this example.

Theorem 7. Assume that (H₁)–(H₅) hold. Then, BVP (1) has at least a positive solution, a negative solution, and a sign-changing solution in 𝐶⁴[0,1].

Remark 8. Theorem 7 can be regarded as an improvement of [8, Corollary 18] though we add a growth condition (H₅). Firstly, the nonlinear term 𝑓 here only needs to be continuous while 𝑓 is locally Lipschitz continuous with respect to 𝑢 and strictly increasing in [8]. Secondly, as is known, when the method of the invariant set of decreasing flow is applied to differential equations, the main difficulty is that the cone has an empty interior in the function space we work in, such as the positive cone in 𝐿²[0,1]. Generally, one needs the 𝐸-regular operator or the bootstrap argument [8]. In our proof of this theorem, from the idea of [18, 19], we construct open sets in 𝐿²[0,1] directly instead of introducing 𝐶₀[0,1]↪𝐿²[0,1] as in [8].

Example 9. Let 𝑓(𝑡,𝑢)=𝑎𝑢+𝑏(𝑡)|sin𝑢|arctan𝑢+𝑐𝑢⁴for(𝑡,𝑢)∈[0,1]×ℝ¹, (11) where 𝑎∈[0,𝜋⁴), 𝑏∈𝐶[0,1] and 𝑐>0. It is easy to verify that all conditions of Theorem 7 are satisfied. So, Theorem 7 ensures that BVP (1) with the nonlinear term (11) has at least a positive solution, a negative solution, and a sign-changing solution. Since neither 𝑓(𝑡,𝑢) nor 𝑓(𝑡,𝑢)+𝑚𝑢 is strictly increasing, Corollary 18 in [8] cannot be applied to this example.

Theorem 10. Assume that (H₁)–(H₃), (H₅), and (H₆) hold. Then, BVP (1) has infinitely many sign-changing solutions in 𝐶⁴[0,1].

Remark 11. Using a symmetric mountain pass lemma [20, Theorem 9.12] due to Rabinowitz, Li et al. obtained an infinitely many solutions for BVP (1) [7, Theorem 3.4]. In [11], we obtained a similar conclusion [11, Theorem 1.3] after removing condition of [7, Theorem 3.4] and strengthening the differentiability of 𝑓. Yang and Zhang [13, 15] established some infinitely many mountain pass solutions theorems for the fourth-order boundary value problems with parameters by the mountain pass theorem in order interval, in which they supposed that 𝑓 is strictly increasing in 𝑢, and the problem has infinitely many pairs of sub- and supsolutions, such as the following [15, condition (H₃)].
There exist sequences {𝛼_𝑖},{𝛽_𝑖}⊂𝐶₀[0,1] satisfying 0<𝛼₁<𝛽₁<⋯<𝛼_𝑖<𝛽_𝑖<𝛼_𝑖+1<𝛽_𝑖+1<⋯<𝛼_𝑛<𝛽_𝑛<⋯, (12) and {𝛼_𝑖,𝛽_𝑖}(𝑖=1,2,…) is a pair of strict subsolution and supsolution of BVP….
This condition seems somewhat strong. Actually, it is not easy to impose conditions on the nonlinear term 𝑓 to guarantee that [15, condition (H₃)] holds. Besides, in [7, 11, 13, 15], though the authors have obtained the existence of infinitely many solutions, they have not given the signs of them. In fact, to our knowledge, none of the infinitely-many-sign-changing-solution theorem for BVP (1) has been found in the literatures so far. In contrast to [7, Theorem 3.4] and [11, Theorem 1.3], by adding a growth condition (H₅), Theorem 10 gets more information for those infinite solutions; that is, they all change their signs in the interval [0,1]. Compared with the theorems in [13, 15], our conditions are more natural and easier to verify.

Example 12. Let 𝑓(𝑡,𝑢)=𝑎tan𝑢+𝑏(𝑡)arctan𝑢ln1+𝑢²+𝑐|𝑢|^𝛾𝑢for(𝑡,𝑢)∈[0,1]×ℝ¹, (13) where 𝑎∈[0,𝜋⁴], 𝑏(𝑡)∈𝐶[0,1] and 𝑐,𝛾>0. It is easy to verify that all conditions of Theorem 10 are satisfied. So, Theorem 10 ensures that BVP (1) with the nonlinear term (13) has infinitely many sign-changing solutions. Theorem 3.4 in [7] and Theorem 1.3 in [11] can also guarantee that the problem has infinitely many solutions but cannot get their signs.
This paper is organized as follows. In Section 2, we recall some facts on the method of the invariant set of descending flow and prove two useful abstract theorems. The main results are proved in Section 3.

2. Preliminaries

In this section, we firstly outline some basic concepts on the method of the invariant set of descending flow. Secondly, four theorems which will be used in the proofs of our main results are listed. Among them, two are our new results, and the other two are due to [19]. Please refer to [21, 22] for more details about the method of the invariant set of descending flow.

Let 𝑋 be a real Banach space, 𝐽 a 𝐶¹ functional defined on 𝑋, 𝐽^′(𝑢) the gradient operator of 𝐽 at 𝑢∈𝑋, and 𝑊 a pseudogradient vector field for 𝐽. Let Cr(𝐽)=𝑢∈𝑋∶𝐽^′(𝑢)=𝜃,𝑋₀=𝑋⧵Cr(𝐽). (14)

For 𝑢₀∈𝑋₀, consider the following initial problem in 𝑋₀: d𝜑(𝑡)𝜑(𝑡)𝜑(0)d𝑡=−𝑊,𝑡⩾0,=𝑢₀. (15) By the theory of ordinary differential equations in Banach space, (15) has a unique solution in 𝑋₀, denoted by 𝜑(𝑡,𝑢₀), with the right maximal interval of existence [0,𝜂(𝑢₀)). Note that 𝜂(𝑢₀) may be either a positive number or +∞. It is easy to see that 𝐽(𝜑(𝑡,𝑢₀)) is monotonically decreasing on [0,𝜂(𝑢₀)), so 𝜑(𝑡,𝑢₀) is called a descending flow curve.

Definition 13 (see [21]). A nonempty subset 𝑀 of 𝐸 is called an invariant set of descending flow of 𝐽 if 𝜑𝑡,𝑢₀∶0⩽𝑡<𝜂𝑢₀⊂𝑀 (16) for all 𝑢₀∈𝑀⧵Cr(𝐽).

Definition 14 (see [21]). Let 𝑀 and 𝐷 be invariant sets of descending flow of 𝐽, 𝐷⊂𝑀. Denote 𝐶_𝑀(𝐷)=𝑢₀∶𝑢₀∈𝐷or𝑢₀∈𝑀⧵𝐷andthereexists0<𝑡^′<𝜂𝑢₀suchthat𝜑𝑡^′,𝑢₀∈𝐷. (17) If 𝐷=𝐶_𝑀(𝐷), then 𝐷 is called a complete invariant set of descending flow relative to 𝑀.

For a subset 𝑀 of 𝑋, 𝐽 is called satisfying PS condition on 𝑀 if any sequence {𝑢_𝑛}⊂𝑀 such that {𝐽(𝑢_𝑛)} is bounded and 𝐽^′(𝑢_𝑛)→0 as 𝑛→+∞ possesses a convergent subsequence.

Following this, we use 𝜕_𝐴𝐵, and int_𝐴𝐵  and clos_𝐴𝐵 denote the boundary, the interior, and the closure of the set 𝐵 in set 𝐴, respectively.

Theorem 15 (see [21]). Assume that 𝑀 is closed and connected and is an invariant set of descending flow for 𝐽, and 𝐷 is an open subset of 𝑀 and an invariant set of descending flow for 𝐽 as well. If 𝐶_𝑀(𝐷)≠𝑀, inf_{𝑢∈𝜕𝑀𝐷}𝐽(𝑢)>−∞, and 𝐽 satisfies PS condition on 𝑀⧵𝐷, then inf_{𝑢∈𝜕𝑀𝐶𝑀(𝐷)}𝐽(𝑢)⩾inf_{𝑢∈𝜕𝑀𝐷}𝐽(𝑢)>−∞, (18)inf_{𝑢∈𝜕𝑀𝐶𝑀(𝐷)}𝐽(𝑢) is a critical value of 𝐽, and there exists at least one point on 𝜕_𝑀𝐶_𝑀(𝐷) corresponding to this value.

Next, we list four theorems, of which two are our new results, and two are due to [19].

Assume that 𝐻 is a real Hilbert space, 𝑃 is a positive cone in 𝐻, and the partial order on 𝐻 is given by 𝑃. 𝐽 is a 𝐶¹ functional on 𝐻, and 𝐽^′(𝑢) can be expressed in the form of 𝐽^′(𝑢)=𝑢−𝐴𝑢.

Theorem 16. Suppose that 𝐽 satisfies PS condition on 𝑃 and 𝐴∶𝑃→𝑃. 𝐷 is an open convex subset of 𝑃 and 𝐴(𝜕_𝑃𝐷)⊂𝐷. If there exists 𝑢₀∈𝑃⧵𝐷 such that inf_{𝑢∈𝜕𝑃𝐷}𝐽(𝑢)>𝐽(𝑢₀), then 𝐽 has at least a positive critical point.

Proof. According to [21, Lemma 2.5], since 𝐴∶𝑃→𝑃 and 𝐴(𝜕_𝑃𝐷)⊂𝐷, one can construct a pseudogradient vector field 𝑊 for 𝐽 such that 𝑃 and 𝐷 both are invariant sets of descending flow for 𝐽 determined by 𝑊. We only need to show 𝐶_𝑃(𝐷)≠𝑃. In fact, 𝑢₀∉𝐶_𝑃(𝐷). Otherwise, if 𝑢₀∈𝐶_𝑃(𝐷), then there exists 0<𝑡^′⩽𝜂(𝑢₀) such that 𝜑(𝑡^′,𝑢₀)∈𝐷. Then, we can find 0<𝑡₁<𝑡^′ with 𝜑(𝑡₁,𝑢₀)∈𝜕_𝑃𝐷. Thus, 𝐽(𝑢₀)⩾𝐽(𝜑(𝑡₁,𝑢₀)), which contradicts the fact that 𝐽(𝑢₀)<inf_{𝑢∈𝜕𝑃𝐷}𝐽(𝑢). Theorem 15 implies the conclusion. The proof is completed.

Remark 17. In contrast to the cone expansion and compression theorems, the operator 𝐴 needs not to be completely continuous, and 𝐷 needs not to be bounded as well. But 𝐴 is a gradient operator, and 𝐽 satisfies PS condition. These facts indicate that both methods have their own characteristics.

By symmetry, we can easily obtain the following theorem, and we omit its proof.

Theorem 18. In addition to all the conditions of Theorem 16, suppose that 𝐽 satisfies PS condition on −𝑃, and 𝐴∶−𝑃→−𝑃, 𝐷₁ is an open convex subset of −𝑃 and 𝐴(𝜕_−𝑃𝐷₁)⊂𝐷₁. If there exists 𝑢₁∈−𝑃⧵𝐷₁ such that inf_{𝑢∈𝜕−𝑃𝐷}𝐽(𝑢)>𝐽(𝑢₁), then 𝐽 has at least two critical points, one is positive, and the other is negative.

The following two theorems are due to [19]. For ease of use in Section 3, here we write their special cases. See [19] for more general results.

Let 𝒟^± be two closed convex subsets of 𝐻. We need the following assumptions:(A1)𝒪=int_𝐻𝒟⁺∩int_𝐻𝒟⁻≠∅,(A2)𝐴(𝒟^±)⊂int_𝐻𝒟^±,(A3) there exists a path ℎ∶[0,1]→𝐻 such that ℎ(0)∈(int_𝐻𝒟⁺)⧵𝒟⁻, ℎ(1)∈(int_𝐻𝒟⁻)⧵𝒟⁺ and max_[]𝑠∈0,1𝐽(ℎ(𝑠))<𝛼₀=inf_{𝑢∈𝒟+∩𝒟−}𝐽(𝑢), (19)(A4) there exist a number 𝛼₁, a sequence {𝐻_𝑛} of subspaces of 𝐻, and a sequence {𝑅_𝑛} of positive numbers satisfying dim𝐻_𝑛⩾𝑛for𝑛∈ℕ,sup_{𝑢∈𝐻𝑛⧵𝐵𝑛}𝐽(𝑢)⩽𝛼₁<𝛼₀, (20) where 𝐵_𝑛={𝑢∈𝐻_𝑛∶‖𝑢‖⩽𝑅_𝑛}.

Remark 19. According to [21, Lemma 2.5], we deduce from (A₁) and (A₂) that 𝒪 and 𝒟^± are the invariant sets of decreasing flow.

Theorem 20. Assume that (A₁)–(A₃) hold, and 𝐽 satisfies PS condition on 𝐻. Then, 𝐽 has a critical point in each of the four mutually disjoint sets: 𝜕_𝐻𝐶_𝐻(𝒪)⧵(𝒟⁺∪𝒟⁻), 𝜕_𝐻𝐶_𝐻(𝒪)∩int_𝐻𝒟⁺, 𝜕_𝐻𝐶_𝐻(𝒪)∩int_𝐻𝒟⁻, and 𝒪.

Theorem 21. Assume that (A₁), (A₂), and (A₄) hold, and 𝐽 is an even functional and satisfies PS condition on 𝐻. Then, 𝐽 has a sequence of solutions {±𝑢_𝑛} in ℳ=𝜕_𝐻𝐶_𝐻(𝒪)⧵(𝐶_𝐻(int_𝐻𝒟⁺)∪𝐶_𝐻(int_𝐻𝒟⁻)) such that 𝐽𝑢_𝑛⟶+∞𝑎𝑠𝑛⟶∞. (21)

3. Proof of the Main Results

In this section, we will employ the abstract theorems in Section 2 to prove Theorems 1–10.

Let 𝐸=𝐶[0,1] denote the usual real Banach space with the norm ‖𝑢‖_𝐶=max_𝑡∈[0,1]|𝑢(𝑡)| for all 𝑢∈𝐶[0,1]. By 𝐻=𝐿²[0,1] we denote the usual real Hilbert space with the norm ‖𝑢‖=(∫¹₀|𝑢(𝑡)|d𝑡)^1/2 for 𝑢∈𝐻. Let 𝑃={𝑢∈𝐻∶𝑢(𝑡)⩾0a.e.𝑡∈[0,1]}, (22) and then 𝑃 is a cone in 𝐻 and has an empty interior in 𝐻.

Define a functional 𝐽∶𝐻→ℝ¹ by 𝐽(𝑢)=12‖𝑢‖²−¹₀𝐹(𝑡,𝐾𝑢(𝑡))d𝑡,𝑢∈𝐻, (23) where 𝐹(𝑡,𝑢)=∫^𝑢₀𝑓(𝑡,𝑣)d𝑣, 𝐾𝑢(𝑡)=∫¹₀𝐺(𝑡,𝑠)𝑢(𝑠)d𝑠 and 𝐺()=𝑠()𝑡()𝑡,𝑠1−𝑡,0⩽𝑠⩽𝑡⩽1,1−𝑠,0⩽𝑡⩽𝑠⩽1. (24) Then, it is easy to see that 𝐽∈𝐶¹(𝐻,ℝ¹) with derivatives given by 𝐽^′(𝑢)=𝑢−𝐾𝐟𝐾𝑢≜𝑢−𝐴𝑢,∀𝑢∈𝐻, (25) where (𝐟𝑢)(𝑡)=𝑓(𝑡,𝑢(𝑡)) is a Nemytskii operator.

Remark 22. The following results are known. See [7] for details.(i)𝐺∶[0,1]×[0,1]→[0,1] is nonnegative continuous and max_{(𝑡,𝑠)∈[0,1]×[0,1]}𝐺(𝑡,𝑠)=1/4.(ii)𝐟∶𝐸→𝐸 is bounded and continuous.(iii)𝐾 is linear completely continuous as an operator both from 𝐸→𝐸 and 𝐻 to 𝐻. Moreover, ‖𝐾‖_{ℒ(𝐻,𝐻)}=1/𝜋², where ℒ(𝐻,𝐻) denotes the Banach space of all bounded linear operators from 𝐻 to 𝐻.(iv)𝐾²=𝐾∘𝐾∶𝐻→𝐻 is linear, compact, and symmetric, and the norm ‖𝐾²‖_{ℒ(𝐻,𝐻)}=1/𝜋⁴. We omit the subscript ℒ(𝐻,𝐻) in the following.(v)𝐴=𝐾𝐟𝐾∶𝐻→𝐻 is a completely continuous operator.(vi)According to [7, Lemma 3.1], BVP (1) has a nontrivial solution in 𝐶⁴[0,1] if and only if the functional 𝐽 has a nontrivial critical point in 𝐻 (i.e., 𝐴 has a nontrivial fixed point in 𝐻). More precisely, if 𝑢∈𝐶⁴[0,1] is a solution of (1), then 𝑣=𝐾𝐟𝑢 is a critical point of 𝐽; on the other hand, if 𝑣∈𝐻 is a critical point of 𝐽, then 𝐾𝑣 is a solution of (1) in 𝐶⁴[0,1]. Moreover, since 𝐾 is a positive operator, 𝐾𝑣 has the same sign as 𝑣; that is, if 𝑣∈𝐻 is a positive/negative/sign-changing critical point of 𝐽, then 𝐾𝑣 is a positive/negative/sign-changing solution of (1) in 𝐶⁴[0,1], respectively.
Before proving main results, we first give several lemmas.

Lemma 23. Let 𝜈=1/𝜇.(i)Assume that (H₁) and (𝐻^′₃) hold. Then, 𝐽 satisfies PS condition on 𝑃, and there exist three positive numbers 𝐶₁, 𝐶₂, and 𝐶₃ such that 𝐹(𝑡,𝑢)⩽𝜇𝑢𝑓(𝑡,𝑢)+𝐶₁∀()∈[]×[),𝐹()𝑡,𝑢0,10,+∞𝑡,𝑢⩾𝐶₂|𝑢|^𝜈−𝐶₃∀(𝑡,𝑢)∈[0,1]×[0,+∞). (26)(27)(ii)Assume that (H₁) and (H₃) hold. Then, 𝐽 satisfies PS condition on 𝐻, and there exist three positive numbers 𝐶₁, 𝐶₂, and 𝐶₃ such that 𝐹(𝑡,𝑢)⩽𝜇𝑢𝑓(𝑡,𝑢)+𝐶₁∀(𝑡,𝑢)∈×[0,1]×ℝ¹,𝐹(𝑡,𝑢)⩾𝐶₂|𝑢|^𝜈−𝐶₃∀(𝑡,𝑢)∈[0,1]×ℝ¹. (28)(29)

Proof. Since the proof of (i) is analogous to that of (ii), we only need to prove (ii).
𝐽 is the special case of 𝐽₁ defined in (53) as 𝑚=0 and 𝐾₁=𝐾. See the proof of Lemma 24 for the fact that 𝐽 satisfies PS condition.
Since 𝜈𝐹(𝑡,𝑢)−𝑢𝑓(𝑡,𝑢) is continuous on [0,1]×[−𝑀,𝑀], there exists 𝐶₁>0 such that 𝐹(𝑡,𝑢)⩽𝜇𝑢𝑓(𝑡,𝑢)+𝐶₁∀𝑡∈[0,1],𝑢∈[−𝑀,𝑀]. (30) So, by (H₃), we obtain 𝐹(𝑡,𝑢)⩽𝜇𝑢𝑓(𝑡,𝑢)+𝐶₁∀𝑡∈[0,1],𝑢∈ℝ¹. (31) According to (H₃), for all 𝑡∈[0,1] and 𝑢⩾𝑀, we have 𝐹(𝑡,𝑢)𝑢^𝜈′_𝑢=𝑢^𝜈𝑓(𝑡,𝑢)−𝜈𝑢^𝜈−1𝐹(𝑡,𝑢)𝑢^2𝜈=𝑢𝑓(𝑡,𝑢)−𝜈𝐹(𝑡,𝑢)𝑢^𝜈+1⩾0. (32) Hence, 𝐹(𝑡,𝑢)𝑢^𝜈⩾𝐹(𝑡,𝑀)𝑀^𝜈⩾𝑀^−𝜈min_[]𝑡∈0,1𝐹(𝑡,𝑀)=𝐶^′>0 (33) for all 𝑡∈[0,1] and 𝑢⩾𝑀. This implies that 𝐹(𝑡,𝑢)⩾𝐶^′|𝑢|^𝜈 for all 𝑡∈[0,1] and 𝑢⩾𝑀. Similarly, we can prove that there is a constant 𝐶^′′>0 such that 𝐹(𝑥,𝑢)⩾𝐶^′′|𝑢|^𝜈 for all 𝑡∈[0,1] and 𝑢⩽−𝑀. Since 𝐹(𝑡,𝑢)−𝐶₂|𝑢|^𝜈 is continuous on [0,1]×[−𝑀,𝑀], there exists 𝐶₃>0 such that 𝐹(𝑡,𝑢)−𝐶₂|𝑢|^𝜈>−𝐶₃ on [0,1]×[−𝑀,𝑀]. Thus, we have 𝐹(𝑡,𝑢)⩾𝐶₂|𝑢|^𝜈−𝐶₃∀(𝑡,𝑢)∈[0,1]×ℝ¹, (34) where 𝐶₂=min{𝐶^′,𝐶^′′}.

Proof of Theorem 1. From (H₁), we have 𝐴∶𝑃→𝑃.
By (H^′₂), there exists a sufficiently small number 𝑟>0 such that 𝑓(𝑡,𝑢)⩽𝜋⁴()∈[]×[],𝑓()𝑢,∀𝑡,𝑢0,10,𝑟𝑡,𝑢<𝜋⁴𝑢,∀(𝑡,𝑢)∈[0,1]×(0,𝑟]. (35)(36) Define 𝐷={𝑢∈𝑃∶‖𝑢‖<𝑟} as an open convex subset of 𝑃 then 𝜕_𝑃𝐷={𝑢∈𝑃∶‖𝑢‖=𝑟}. (37) For given 𝑢∈𝐻, it follows from (i) of Remark 22 that |𝐾𝑢(𝑡)|=¹₀𝐺()𝑢(𝑠)⩽1𝑡,𝑠d𝑠4¹₀|𝑢(𝑠⩽1)|d𝑠4¹₀|𝑢(𝑠)|²d𝑠^1/2=14‖𝑢‖,𝑡∈[0,1]. (38) This implies that ‖𝐾𝑢‖_𝐶⩽14‖𝑢‖,𝑢∈𝐻. (39) Thus, ‖𝐾𝑢‖_𝐶⩽14‖𝑢‖=14𝑟,∀𝑢∈𝜕_𝑃𝐷. (40) Thereafter, for 𝑢∈𝜕_𝑃𝐷, we have from (35) that (𝐴𝑢)(𝑡)=¹₀𝐺()𝑓((𝑠⩽𝑡,𝑠𝑡,𝐾𝑢)())d𝑠₁₀𝐺(𝑡,𝑠)𝜋⁴(𝐾𝑢)(𝑠)d𝑠=𝜋⁴𝐾²𝑢(𝑡),𝑡∈[0,1]. (41) Then, by (41), (36), and (iii) of Remark 22, we have ‖𝐴𝑢‖<𝜋⁴𝐾²𝑢⩽‖𝑢‖,∀𝑢∈𝜕_𝑃𝐷, (42) namely, 𝐴(𝜕_𝑃𝐷)⊂𝐷.
Since 𝐸↪𝐻, 𝐟∶𝐸→𝐸 is a bounded continuous operator and 𝐾 is bounded linear operator, there exists 𝑀^′>0 such that ‖𝐟𝐾𝑢‖⩽𝑀^′‖𝑢‖=𝑀^′𝑟,∀𝑢∈𝜕_𝑃𝐷. (43) Thus, by (26), Hölder's inequality, and (43), we have 𝐽(𝑢)=12‖𝑢‖²−¹₀𝐹((𝑡⩾1𝑡,𝐾𝑢)())d𝑡2𝑟²−𝜇¹₀𝑓(𝑡,(𝐾𝑢)(𝑡))(𝐾𝑢)(𝑡)d𝑡−𝐶₁⩾12𝑟²−𝜇‖𝐟(𝐾𝑢)‖‖𝐾𝑢‖−𝐶₁⩾12−𝜇𝑀^′𝜋²𝑟²−𝐶₁=constant,∀𝑢∈𝜕_𝑃𝐷. (44)
Choose 𝑅>0. Let 𝑢_𝑅(𝑡)=𝑅sin𝜋𝑡,𝑡∈[0,1]. Obviously, 𝑢_𝑅∈𝑃. Since 𝜈>2, 𝐿^𝜈[0,1]↪𝐻, that is, there exists 𝐶₄>0 such that ‖⋅‖⩽𝐶₄‖⋅‖_𝐿^𝜈_[0,1]. (45) Consequently, we have from (27) that 𝐽𝑢_𝑅=12𝑢_𝑅²−¹₀𝐹𝑡,𝐾𝑢_𝑅(𝑡)⩽1d𝑡4𝑅²−¹₀𝐶₂𝐾𝑢_𝑅(𝑡)^𝜈−𝐶₃=1d𝑡4𝑅²−𝐶₂𝐾𝑢_𝑅^𝜈_{𝐿𝜈[]0,1}+𝐶₃⩽14𝑅²−𝐶₂𝐶₄^−𝜈𝐾𝑢_𝑅^𝜈+𝐶₃=14𝑅²−𝐶₁𝐶₄^−𝜈𝜋^−2𝜈12^𝜈/2𝑅^𝜈+𝐶₂. (46) Since 𝜈>2, we have lim_𝑅→+∞𝐽𝑢_𝑅=−∞. (47) Combining (47) and (44), we obtain that there exists 𝑢₁∈𝑃⧵𝐷 such that inf_{𝑢∈𝜕𝑃𝐷}𝐽(𝑢)>𝐽(𝑢). (48) Now, all the conditions of Theorem 16 are satisfied. Therefore, Theorem 16 ensures that BVP (1) has at least a positive solution. The proof is completed.

Proof of Theorem 4. By the symmetry of 𝑃 and −𝑃, it is easy to verify that all the conditions of Theorem 18 are satisfied. Theorem 18 ensures that BVP (1) has at least a positive solution and a negative solution. This completes the proof.

In order to prove Theorem 7 and Theorem 10, we need to construct convex subset 𝒟^± of 𝐻 and an operator 𝐴 satisfying the assumptions (A₁) and (A₂). We begin by transforming BVP (1) into the following equivalent boundary value problem: 𝑢⁽⁴⁾+𝑚𝑢=𝑓₁((𝑡[],𝑢(0)(1)𝑡,𝑢)),𝑡∈0,1=𝑢=𝑢^′′(0)=𝑢^′′(1)=0, (49) where 𝑚>0 and 𝑓₁(𝑡,𝑢)=𝑓(𝑡,𝑢)+𝑚𝑢 for all (𝑡,𝑢)∈[0,1]×ℝ¹. Let 𝐺₁(𝑡,𝑠) be Green's function for the linear boundary value problem −𝑢^′′+𝑚𝑢=0,𝑢(0)=𝑢(1)=0, (50) which is explicitly given by 𝐺₁()=()𝑡,𝑠𝜔sinh𝜔⁻¹()()⋅sinh𝜔𝑠⋅sinh𝜔1−𝑡,0⩽𝑠⩽𝑡⩽1,𝜔sinh𝜔⁻¹⋅sinh𝜔𝑡⋅sinh𝜔(1−𝑠),0⩽𝑡⩽𝑠⩽1, (51) where 𝜔=√𝑚, sinh𝑥=(𝑒^𝑥−𝑒^−𝑥)/2 is the hyperbolic sine function. It is easy to verify that 𝐺₁(𝑡,𝑠)>0 for all 𝑡,𝑠∈[0,1]. Define operators 𝐾₁,𝐟₁∶𝐶[0,1]→𝐶[0,1], respectively, by 𝐾₁𝑢(𝑡)=¹₀𝐺₁()𝑢(𝑠)[],[]𝑡,𝑠d𝑠,𝑡∈0,1∀𝑢∈𝐶0,1,𝐾²₁=𝐾₁∘𝐾₁,𝐟₁𝑢(𝑡)=𝑓₁(𝑡,𝑢(𝑡)),𝑡∈[0,1],∀𝑢∈𝐶[0,1]. (52) Obviously, 𝐾₁ and 𝐟₁ have the same properties of 𝐾 and 𝐟 as in Remark 22, respectively. Besides, ‖𝐾₁‖=1/(𝜋²+𝑚) and ‖𝐾²₁‖=1/(𝜋⁴+𝑚).

Define a functional 𝐽₁∶𝐻→ℝ¹ by 𝐽₁(𝑢)=12‖𝑢‖²−¹₀𝐹₁𝑡,𝐾₁𝑢(𝑡)d𝑡,𝑢∈𝐻, (53) where 𝐹₁(𝑡,𝑢)=∫^𝑢₀𝑓₁(𝑡,𝑣)d𝑣. Then, it is easy to see that 𝐽₁∈𝐶¹(𝐻,ℝ¹) with derivatives given by 𝐽^′₁(𝑢)=𝑢−𝐾₁𝐟₁𝐾₁𝑢 for all 𝑢∈𝐻. Set 𝐴₁=𝐾₁𝐟₁𝐾₁. (54) Then, according to [7, Lemma 3.1], BVP (49) (i.e. BVP (1)) has a nontrivial solution in 𝐶⁴[0,1] if and only if the functional 𝐽₁ has a nontrivial critical point in 𝐻 (i.e., 𝐴₁ has a nontrivial fixed point in 𝐻). Similarly, since 𝐾₁ is a positive operator, if 𝑣∈𝐻 is a positive/negative/sign-changing critical point of 𝐽₁, then 𝐾₁𝑣 is a positive/negative/sign-changing solution of BVP (1) in 𝐶⁴[0,1], respectively.

Lemma 24. Assume that (H₁) and (H₃) hold. Then, the functional 𝐽₁ satisfies PS condition on 𝐻.

Proof. Suppose that {𝑢_𝑛}⊂𝐻, and there exists 𝑀₁>0 such that |𝐽₁(𝑢_𝑛)|⩽𝑀₁ and 𝐽^′₁𝑢_𝑛=𝑢_𝑛−𝐴₁𝑢_𝑛⟶0in𝐻as𝑛⟶∞. (55) Notice that 𝐽^′₁𝑢_𝑛,𝑢_𝑛=𝑢_𝑛−𝐾₁𝐟₁𝐾₁𝑢_𝑛,𝑢_𝑛=𝑢_𝑛²−¹₀𝑓₁𝑡,𝐾₁𝑢_𝑛(𝑡)𝐾₁𝑢_𝑛(𝑡)d𝑡 (56) and ‖𝐾₁‖²=1/(𝜋²+𝑚)². It follows from (28) and the definition of 𝐽₁ that 𝑀₁⩾𝐽𝑢_𝑛=12𝑢_𝑛²−¹₀𝐹₁𝑡,𝐾₁𝑢_𝑛(𝑡)=1d𝑡2𝑢_𝑛²−¹₀𝐹𝑡,𝐾₁𝑢_𝑛(𝑡+𝑚)2𝐾₁𝑢_𝑛(𝑡)²⩾1d𝑡2𝑢_𝑛²−𝜇¹₀𝑓𝑡,𝐾₁𝑢_𝑛(𝑡)𝐾₁𝑢_𝑛(𝑡)−𝑚d𝑡2¹₀𝐾₁𝑢_𝑛(𝑡)²d𝑡−𝐶₁=12𝑢_𝑛²−𝜇¹₀𝑓₁𝑡,𝐾₁𝑢_𝑛(𝑡)𝐾₁𝑢_𝑛(𝑡)−1d𝑡2−𝜇𝑚𝐾₁𝑢_𝑛²−𝐶₁⩾12𝑢_𝑛²−𝜇¹₀𝑓₁𝑡,𝐾₁𝑢_𝑛(𝑡)𝐾₁𝑢_𝑛(𝑡)−1d𝑡2−𝜇𝑚𝐾₁²𝑢_𝑛²−𝐶₁=12−𝜇𝑢_𝑛²+𝜇𝐽^′₁𝑢_𝑛,𝑢_𝑛−12𝑚−𝜇𝜋²+𝑚²𝑢_𝑛²−𝐶₁⩾12𝑚−𝜇1−𝜋²+𝑚²𝑢_𝑛²−𝜇𝐽^′₁𝑢_𝑛𝑢_𝑛−𝐶₁,𝑛=1,2,…. (57) Since 𝐽^′₁(𝑢_𝑛)→0 as 𝑛→∞, there exists 𝑁₀∈ℕ such that 𝑀₁⩾12𝑚−𝜇1−𝜋²+𝑚²𝑢_𝑛²−𝑢_𝑛−𝐶₁,𝑛>𝑁₀. (58) This implies that {𝑢_𝑛}⊂𝐻 is bounded. Since 𝐴₁∶𝐻→𝐻 is completely continuous, we have from (55) that {𝑢_𝑛} has a convergent subsequence in 𝐻. Thus, 𝐽₁ satisfies PS condition on 𝐻.

Lemma 25. Assume that (H₁)--(H₅) hold. Then, there exist 𝑚>0 and 𝜀₀>0 such that 𝐴₁𝒟^±_𝜀⊂int𝒟^±_𝜀𝑓𝑜𝑟𝜀∈0,𝜀₀, (59) where 𝐴₁ is as defined in (54) and 𝒟^±_𝜀={𝑢∈𝐻∶dist(𝑢,±𝑃)⩽𝜀}. (60)

Proof. As a consequence of (H₁)--(H₃) and (H₅), there exists 𝑚>0 such that 𝑢𝑓(𝑡,𝑢)+𝑚𝑢²>0∀𝑡∈[0,1],𝑢∈ℝ¹with𝑢≠0. (61) By (H₂) and (H₅), there exist 𝛿>0 and 𝑐₁>0 such that 𝑓(𝑡,𝑢)+𝑚𝑢⩽𝜋⁴+𝑚−𝛿|𝑢|+𝑐₁|𝑢|^𝑞−1∀(𝑡,𝑢)∈[0,1]×ℝ¹. (62) Next, we show that there exists 𝜀₀>0 such that dist𝐴₁𝑢,𝑃<dist(𝑢,𝑃)as0<dist(𝑢,𝑃)⩽𝜀₀,dist𝐴₁𝑢,−𝑃<dist(𝑢,−𝑃)as0<dist(𝑢,−𝑃)⩽𝜀₀. (63)(64) For any 𝑢∈𝐻, define its positive part and negative part as follows: 𝑢⁺(𝑡)=max{𝑢(𝑡),0},𝑢⁻(𝑡)=min{𝑢(𝑡),0}. (65) Obviously, 𝑢=𝑢⁺+𝑢⁻ for all 𝑢∈𝐻. From (61), we have 𝑓₁()()𝑓𝑡,𝑢=𝑓𝑡,𝑢+𝑚𝑢>0∀𝑢>0,₁(𝑡,𝑢)=𝑓(𝑡,𝑢)+𝑚𝑢<0∀𝑢<0. (66) This implies that 𝑓₁(𝑡,𝑢)⁻=𝑓(𝑡,𝑢)+𝑚𝑢⁻=𝑓(𝑡,𝑢⁻)+𝑚𝑢⁻ (67) for all 𝑢∈𝐻. Since 𝐾₁ is a positive operator, (𝐾₁𝑢)⁻=𝐾₁𝑢⁻. So, we have from (67) that 𝐴₁𝑢⁻=𝐾₁𝐟₁𝐾₁𝑢⁻=𝐾₁𝐟₁𝐾₁𝑢⁻=𝐾₁𝐟₁𝐾₁𝑢⁻=𝐾₁𝐟₁𝐾₁𝑢⁻ (68) for all 𝑢∈𝐻. For any 𝑢∈𝐻, it follows that 𝐾₁𝑢⁻(𝑡)=¹₀𝐺₁(𝑡,𝑠)𝑢⁻(𝑠)d𝑠⩽max_{()∈[]×[]𝑡,𝑠0,10,1}𝐺₁(𝑡,𝑠)¹₀|𝑢⁻(𝑠)|d𝑠⩽max_{()∈[]×[]𝑡,𝑠0,10,1}𝐺₁(𝑡,𝑠)¹₀|𝑢⁻(𝑠)|²d𝑠^1/2=max_{()∈[]×[]𝑡,𝑠0,10,1}𝐺₁(𝑡,𝑠)‖𝑢⁻‖,𝑡∈[0,1]. (69) Thereafter, for 𝑡∈[0,1], we have 𝐾₁𝐾₁𝑢⁻(𝑡)^𝑞−1⩽𝐾₁max_{()∈[]×[]𝑡,𝑠0,10,1}𝐺₁(𝑡,𝑠)‖𝑢⁻‖^𝑞−1=𝑐₂‖𝑢⁻‖^𝑞−1, (70) where 𝑐₂=(max_{(𝑡,𝑠)∈[0,1]×[0,1]}∫¹₀𝐺(𝑡,𝑠)d𝑠)^𝑞>0.
Consider 𝑢∈𝐻 and set 𝑣=𝐴₁𝑢. Then, by (68), (62), and (70), we obtain ‖𝑣⁻‖²=¹₀𝐾₁𝐟₁𝐾₁𝑢⁻²=d𝑡₁₀𝐾₁𝑓𝑡,𝐾₁𝑢⁻+𝑚𝐾₁𝑢⁻²⩽d𝑡₁₀𝐾₁𝜋⁴+𝑚−𝛿𝐾₁𝑢⁻+𝑐₁𝐾₁𝑢^−𝑞−12=d𝑡₁₀𝜋⁴+𝑚−𝛿𝐾²₁𝑢⁻+𝑐₁𝐾₁𝐾₁𝑢^−𝑞−12⩽d𝑡₁₀𝜋⁴+𝑚−𝛿𝐾²₁𝑢⁻+𝑐₁𝑐₂‖𝑢‖^𝑞−12=𝜋d𝑡₄+𝑚−𝛿𝐾²₁𝑢⁻+𝑐₁𝑐₂‖𝑢‖^𝑞−12. (71) Therefore, we obtain that ‖𝑣⁻‖⩽𝜋⁴+𝑚−𝛿𝐾²₁𝑢⁻+𝑐₁𝑐₂‖𝑢‖^𝑞−1⩽𝜋⁴+𝑚−𝛿𝐾²₁𝑢⁻+𝑐₁𝑐₂‖𝑢⁻‖^𝑞−1⩽1−𝛿𝜋⁴+𝑚⁻¹‖𝑢⁻‖+𝑐₁𝑐₂‖𝑢⁻‖^𝑞−1. (72) Since 𝑣=𝑣⁺+𝑣⁻ and 𝑣⁺∈𝑃, dist(𝑣,𝑃)⩽𝑣−𝑣⁺=‖𝑣⁻‖. (73) Thus, it follows from (72) that dist(𝑣,𝑃)⩽1−𝛿𝜋⁴+𝑚⁻¹‖𝑢⁻‖+𝑐₁𝑐₂‖𝑢⁻‖^𝑞−1. (74) For any 𝑤∈𝑃, we have ‖𝑢⁻‖⩽‖𝑢−𝑤‖. (75) In fact, let 𝐼₁={𝑡∈[0,1]∶𝑢(𝑡)⩾0}={𝑡∈[0,1]∶𝑢⁻(𝑡)=0}, 𝐼₂=[0,1]⧵𝐼₁. Then, 𝑢⁻(𝑡)=𝑢(𝑡)<0 for all 𝑡∈𝐼₂. So, 𝑢(𝑡)−𝑤(𝑡)⩽𝑢(𝑡)=𝑢⁻(𝑡)<0 (76) for all 𝑡∈𝐼₂, and then |𝑢(𝑡)−𝑤(𝑡)|⩾|𝑢⁻(𝑡)| (77) for all 𝑤∈𝑃 and 𝑡∈𝐼₂. Thus ‖𝑢−𝑤‖²⩾_𝐼2|𝑢−𝑤|²d𝑡⩾_𝐼2|𝑢⁻|²d𝑡=‖𝑢⁻‖². (78) Combining (74) and (75), we deduce that dist(𝑣,𝑃)⩽1−𝛿𝜋⁴+𝑚⁻¹‖𝑢−𝑤‖+𝑐₁𝑐₂‖𝑢−𝑤‖^𝑞−1 (79) for all 𝑤∈𝑃. Consequently, dist(𝑣,𝑃)⩽1−𝛿𝜋⁴+𝑚⁻¹dist(𝑢,𝑃)+𝑐₁𝑐₂(dist(𝑢,𝑃))^𝑞−1<dist(𝑢,𝑃)+𝑐₁𝑐₂(dist(𝑢,𝑃))^𝑞−1. (80) Since 𝑞>2, there exists 𝜀₀>0 such that dist(𝑣,𝑃)<dist(𝑢,𝑃)as0<dist(𝑢,𝑃)⩽𝜀₀. (81) Similarly, we can find 𝜀₀>0 small enough such that (64) holds. Up to now, these constants 𝑚>0 and 𝜀₀>0 are as required. The proof is completed.

Proof of Theorem 7. We only need to verify that all the conditions of Theorem 20 hold. From Lemmas 25 and 24, 𝐽₁ satisfies PS condition on 𝐻, and it is easy to see that 𝐴₁ and 𝒟^±_𝜀 satisfy (A₁) and (A₂). Now, we show that (A₃) holds.
It follows from (59) that 𝐴₁ has no fixed point on 𝜕_𝐻𝒟^±_𝜀 for 𝜀∈(0,𝜀₀]. This implies that Cr𝐽₁∩𝒟⁺_𝜀∩𝒟⁻_𝜀={𝜃}, (82) where 𝜃 denotes the function 𝑢(𝑡)≡0 for 𝑡∈[0,1]. Then, by (59) and [19, Lemma 3.2], inf_{𝑢∈𝒟+𝜀∩𝒟−𝜀}𝐽₁(𝑢)=𝐽₁(𝜃)=0. (83) Define a path ℎ_𝑅∶[0,1]→𝐻 as ℎ_𝑅(𝑠)=𝑅cos𝜋𝑠sin𝜋𝑡+𝑅sin𝜋𝑠sin2𝜋𝑡,𝑅>0. (84) Then, ℎ_𝑅(0)=𝑅sin𝜋𝑡 and ℎ_𝑅(1)=−𝑅sin𝜋𝑡. Since ()=‖‖=𝑅dist±𝑅sin𝜋𝑡,∓𝑃𝑅sin𝜋𝑡√2, (85) it is easy to see that ℎ_𝑅(0)∈(int_𝐻𝒟⁺_𝜀)⧵𝒟⁻_𝜀 and ℎ_𝑅(1)∈(int_𝐻𝒟⁻_𝜀)⧵𝒟⁺_𝜀 as 𝑅 is large enough.
It follows from (29), (45), and a direct computation that 𝐽₁ℎ_𝑅(𝑠)=12ℎ_𝑅(𝑠)²−¹₀𝐹₁𝑡,𝐾₁ℎ_𝑅(𝑠)⩽1d𝑡4𝑅²−¹₀𝐶₂𝐾₁ℎ_𝑅(𝑠)^𝜈−𝐶₃+12𝑚𝐾₁ℎ_𝑅(𝑠)²=1d𝑡4𝑅²−𝐶₂𝐾₁ℎ_𝑅(𝑠)^𝜈_{𝐿𝜈[]0,1}−12𝑚𝐾₁ℎ_𝑅(𝑠)²+𝐶₃⩽14𝑅²−𝐶₂𝐶₄^−𝜈𝐾₁ℎ_𝑅(𝑠)^𝜈−12𝑚𝐾₁ℎ_𝑅(𝑠)²+𝐶₃=14𝑅²−𝐶₂𝐶₄^−𝜈𝑔(𝑠)^𝜈𝑅^𝜈−12𝑚𝑔(𝑠)²𝑅²+𝐶₃⩽𝐶₅𝑅²−𝐶₆𝑅^𝜈+𝐶₃, (86) where 𝑔(𝑠)=𝑚²+2𝜋²𝑚+𝜋⁴sin²𝜋𝑠+𝑚²+8𝜋²𝑚+16𝜋⁴cos²𝜋𝑠2𝑚⁴+20𝜋²𝑚³+66𝜋⁴𝑚²+80𝜋⁶𝑚+32𝜋^81/2,𝐶₅=14−12𝑚min_𝑠∈[0,1]𝑔(𝑠)²,&𝐶₆=𝐶₂𝐶₄^−𝜈min_𝑠∈[0,1]𝑔(𝑠)^𝜈>0. (87) Since 𝜈>2, we have lim_𝑅→+∞max_[]𝑠∈0,1𝐽₁ℎ_𝑅(𝑠)=−∞. (88) Therefore, max_[]𝑠∈0,1𝐽₁ℎ_𝑅(𝑠)<0=inf_{𝑢∈𝒟+𝜀∩𝒟−𝜀}𝐽₁(𝑢) (89) as 𝑅 is large enough.
Now, all the conditions of Theorem 20 are satisfied, and Theorem 20 ensures that BVP (1) has at least four solutions. According to the construction of 𝒟^±_𝜀 and the locations of these solutions, we can easily see that one is zero, one is positive, one is negative, and one is sign changing. This completes the proof.

Proof of Theorem 10. From Lemmas 25 and 24, 𝐽₁ satisfies PS condition on 𝐻, and it is easy to see that 𝐴₁ and 𝒟^±_𝜀 satisfy (A₁) and (A₂). From (H₆), 𝐽₁ is an even functional. Next, we show that (A₄) holds.
Denote that 𝐻_𝑛=span{𝑒₁,𝑒₂,…,𝑒_𝑛}, where 𝑒_𝑘=√2sin𝑘𝜋𝑡,𝑘=1,2,…,𝑛. For 𝑢∈𝐻_𝑛, there exist 𝑎₁,𝑎₂,…,𝑎_𝑛∈ℝ¹ such that 𝑢=𝑎₁𝑒₁+𝑎₂𝑒₂+⋯+𝑎_𝑛𝑒_𝑛,𝑟=‖𝑢‖=^𝑛_𝑖=1𝑎²_𝑖^1/2. (90) So, we have from (29) that 𝐽₁(𝑢)=12‖𝑢‖²−¹₀𝐹₁𝑡,𝐾₁𝑢(𝑡)⩽1d𝑡2𝑟²−¹₀𝐶₂𝐾₁𝑢(𝑡)^𝜈−𝐶₃+12𝑚𝐾₁𝑢(𝑡)²=1d𝑡2𝑟²−𝐶₂𝐾₁𝑢^𝜈_{𝐿𝜈[]0,1}−12𝑚𝐾₁𝑢²+𝐶₃⩽12𝑟²−𝐶₂𝐶₄^−𝜈𝐾₁𝑢^𝜈−12𝑚𝐾₁𝑢²+𝐶₃=12𝑟²−𝐶₂𝐶₄^−𝜈𝑛_𝑖=1𝑎_𝑖𝜆_𝑖𝑒_𝑖^𝜈−12𝑚^𝑛_𝑖=1𝑎_𝑖𝜆_𝑖𝑒_𝑖²+𝐶₃=12𝑟²−𝐶₂𝐶₄^−𝜈𝑛_𝑖=1𝑎²_𝑖𝜆²_𝑖^𝜈/2−12𝑚^𝑛_𝑖=1𝑎²_𝑖𝜆²_𝑖+𝐶₃⩽12𝑟²−𝐶₂𝐶₄^−𝜈𝜆²_𝑛^𝑛_𝑖=1𝑎²_𝑖^𝜈/2−12𝑚𝜆²_𝑛^𝑛_𝑖=1𝑎²_𝑖+𝐶₃=121−𝑚𝜆²_𝑛𝑟²−𝐶₂𝐶₄^−𝜈𝜆_𝑛𝑟^𝜈+𝐶₃, (91) where 𝜆_𝑖 denotes the 𝑖th eigenvalue of 𝐾₁. Consequently, lim_{‖𝑢‖→+∞}𝐽(𝑢)=−∞,𝑢∈𝐻_𝑛. (92) This implies that sup_{𝑢∈𝐻𝑛⧵𝐵𝑛}𝐽₁(𝑢)⩽𝛼₁<𝛼₀=0=inf_{𝑢∈𝒟+𝜀∩𝒟−𝜀}𝐽₁(𝑢), (93) where 𝐵_𝑛={𝑢∈𝐻_𝑛∶‖𝑢‖_𝐻⩽𝑅_𝑛}. Up to now, all the conditions of Theorem 21 are satisfied. So, BVP (1) has infinitely many solutions in ℳ=𝜕_𝐻𝐶_𝐻(𝒪)⧵𝐶_𝐻int_𝐻𝒟⁺_𝜀∪𝐶_𝐻int_𝐻𝒟⁻_𝜀. (94) Obviously, all of them are sign changing. This completes the proof.

Acknowledgments

The authors would like to thank the anonymous reviewers for their helpful comments. This paper was supported by National Natural Science Foundation of China (11101253 and 10826081), the Fundamental Research Funds for the Central Universities (Program no. GK200902046), and the Scientific Research Foundation of Xi'an University of Science and Technology (no. 200843).