Remarks on Some Fourth-order Boundary Value Problems with Non-monotone Increasing Eigenvalue Sequences

By Z 2-index theory, the existence and multiplicity of solutions for some fourth-order boundary value problems u (4) + au = µu + F u (t, u), 0 < t < L, u(0) = u(L) = u (0) = u (L) = 0 at resonance are studied, where a > 0 and µ ∈ R is an eigenvalue of the corresponding eigenvalue problem. The difficulty caused by the non-monotone eigenvalue sequence is handled concretely.


Introduction
In order to study physical, chemical and biological systems, one has considered some fourthorder semilinear differential equation models, such as the extended Fisher-Kolmogorov equation [4], the Swift-Hohenberg equation [10], suspended beam equations [2], etc.In the present paper, we are concerned with the existence and multiplicity of solutions for boundary value problems of fourth-order differential equations where a > 0, µ is a real parameter, F(t, u) ∈ C 1 ([0, L] × R, R), F u (t, u) denotes the gradient of F(t, u) with respect to the variable u.If F(t, u) satisfies lim |u|→∞ F(t, u)/u 2 = 0, we say F(t, u) is sublinear at infinity.
We first observe that the corresponding eigenvalue problem u (4) + au = λu, has the eigenvalues .3)and the eigenfunctions One says (1.1) is resonant at infinity if µ = λ k and F(t, u) is sublinear at infinity.On the other hand, for fixed a > 0, we can find that from (1.2) (iii) if L is sufficiently large , then there exists two integers k ≥ 2 and l satisfying We see easily that for the case (i) and (ii), the eigenvalue sequence {λ k } increases strictly.In the last two decades, much of the research in critical point theory has examined the existence and multiplicity of solutions of (1.1), so, we shall focus on the most complicated case (iii) with non-monotone increasing eigenvalue sequence.For the sake of simplicity, we assume that Up to now there have been a vast of literature about superlinear (1.1), namely, F(t, u) satisfies lim |u|→∞ F(t, u)/u 2 = ∞; we refer the reader to [1,5,6,11,12], and references therein.However, for the sublinear (1.1),only a few attempts have been done.In [9], Liu considered the existence of solutions with L = 1, µ = −a 2 /4.In [7], Han-Xu supposed L = 1, a = µ = 0, F(t, u) < γ|u| 2 + β (0 < γ < π 2 /2) and m 4 π 4 < f u (t, 0) < (m + 1) 4 π 4 with m ≥ 1 and proved the existence of three solutions.In [13] and discussed the existence of two solutions at resonance, basing on combining the minimax methods and the Morse theory.In [8], Li-Wang-Xiao used the Clark theorem to prove the following result.
Remark 1.4.In Theorem 1.2 and Corollary 1.3, if the integer k is between k + 1 and k + l , or k > k + l, then the similar results are still true.However, the details of their proofs must be adapted.Evidently, one can also handle case (i) and (ii) by the same arguments used by us in (iii), moreover, the process seems easier than that of (iii).In addition, it is also not hard to see that Theorem 1.1 is equivalent to the special situation of a = 0, µ < λ 1 in Corollary 1.3.This paper is organized as follows.In Section 2, we will prove some lemmas.In Section 3, the proof of Theorem 1.2 shall be given by the following Z 2 -type index theorem.Theorem 1.5.Let Y be a Banach space and the functional ϕ ∈ C 1 (Y, R) be even satisfying the Palais-Smale condition.Suppose that (i) there exist a subspace V of Y with V = r and δ > 0 such that sup w∈V, x =δ ϕ(w) < ϕ(0); (ii) there exists a closed subspace W of Y with W = s < r such that inf w∈W ϕ(w) > −∞.
Then f possesses at least r − s distinct pairs (u, −u) of critical points.
For the convenience of the reader, let us recall that the functional ϕ is said to satisfy the Palais-Smale condition if any sequence {u j } in Y is such that ϕ(u j ) is bounded, ϕ (u j ) → 0, possesses a convergent subsequence.

Preliminary
Let X = H 2 (0, L; R) ∩ H 1 0 (0, L; R) be a Hilbert space with the inner product and the corresponding norm we infer from the Poincaré inequality so • X is equivalent with • .It is well known that solutions of (1.1) are exactly the critical points of the corresponding functional given by on X. Direct computation shows, for ∀u, w ∈ X, Obviously, there is an orthogonal basis on (X, • ) as follows Ls j e j , we have v j = 1, and From (2.10) and (2.11), we obtain (2.12) Lemma 2.1.Under the assumptions of Theorem 1.2, there exists a norm • * equivalent with • on X; X has an orthogonal decomposition X = X + ⊕ X − ⊕ X 0 , and the functional I(u) in (2.6) is of the form ) Proof.Consider two cases.Case (i): µ = λ k > min{λ j } j≥1 .For this, there exists a unique integer k such that k and

.15)
We define and then we derive Moreover, one can estimate the terms in (2.15) as follows: with From (2.24) one gets (2.28) Case (ii): µ = λ k = min{λ j } j≥1 .Under this assumption, for ∀u ∈ X, u = ∑ ∞ j=1 α j v j , we also have and then we conclude and which implies a new norm • * as follows: (2.34) equivalent with • .Obviously, one obtains Consequently, we also have the same results as (2.27)-(2.28)with u − = 0 .
Lemma 2.2.Under the assumptions of Theorem 1.2, the functional I(u) defined in (2.6) satisfies the Palais-Smale condition.
Proof.Assume that {u m } ⊂ X satisfy Using ( f 2) and the Sobolev embedding inequality, we obtain with some constant C 1 > 0. Combining (2.37) with (2.38) derives thus we reduce that {u + m } is bounded since 1 ≤ α + 1 < 2. In the same way, {u − m } is also bounded.Therefore there is a C 2 > 0 such that (2.40) Next, with the aid of ( f 2) and the mean value theorem, we easily show for some constants C 3 , C 4 > 0. Since dim X 0 = 1, u 0 m ∈ X 0 , we know that u 0 m ∞ is equivalent with u 0 m , it is easy to conclude by (2.40) and (2.41) for some constants C 5 , C 6 > 0. By Lemma 2.1, I(u m ) can be written as From (2.42) and (2.43), we obtain

.45)
We note that for ∀m ≥ 1, u 0 m can be expressed by u Here, k is fixed, consequently, (2.45) contradicts ( f 4).Thus, we conclude that {u m } is bounded on X, and, using the standard method, {u m } has a convergent subsequence.Lemma 2.3.Under the assumptions of Theorem 1.2, the functional I(u) defined in (2.6) is bounded from below on X + .
Proof.According to ( f 2) , for any u ∈ X, we have with some constant C 8 > 0. In particular, if u ∈ X + , then for u * → ∞, we get

.47)
So I is bounded from below on X + .
3 Proof of the theorems

Proof of Theorem 1.2
Proof.In this section, we shall prove Theorem 1.2 by Theorem 1.5.Define then u = u + + u − , u + ∈ X + , u − ∈ X − , X = X + ⊕ X − , and We can continue to follow the same ideas as in Lemma 2.1, Lemma 2.2, and Lemma 2.3 to complete the proof of Corollary 1.3.In addition, it is not hard to see that, for the present case of λ k < µ < λ k+1 , condition ( f 4) appearing in Theorem 1.2 is not indispensable since X 0 = 0 in the orthogonal decomposition of X.The details should be left to the reader.Remark 3.1.For ∀β ∈ (0, 1  2 ), γ ∈ [0,