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

and Applied Analysis 3 where a ∈ [0, π), b ∈ C([0, 1], [0, +∞)) and c, γ > 0. It is easy to verify that all conditions ofTheorem 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 signchanging solution in C[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 termf 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 L[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 L[0, 1] directly instead of introducing 0 1] 󳨅→ L 2 [0, 1] as in [8]. Example 9. Let f (t, u) = au + b (t) |sin u| arctan u + cu for (t, u) ∈ [0, 1] ×R, (11) where a ∈ [0, π), 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 signchanging 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 C 4 [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 themountain pass theorem in order interval, inwhich they supposed that f is strictly increasing in u, and the problem has infinitely many pairs of suband supsolutions, such as the following [15, condition (H3)]. There exist sequences i i ⊂ 0 1] satisfying 0 < 1 < 1 < ⋅ ⋅ ⋅ < i < i < i+1 < i+1 < ⋅ ⋅ ⋅ < n < n < ⋅ ⋅ ⋅ , (12) and i 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-manysign-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 getsmore 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) = a tan u + b (t) arctan u ln (1 + u) + c|u|u for (t, u) ∈ [0, 1] ×R, (13) where a ∈ [0, π], 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 infinitelymany 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.Themain 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 C 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} , 0 = X \ Cr (J) . (14) For 0 ∈ X0, consider the following initial problem inX0: d dt φ (t) = −W(φ (t)) , t ⩾ 0,