Infinitely Many Weak Solutions for Fourth-order Equations Depending on Two Parameters

for every t1, t2 ∈ R, with p(0) = 0, and h : [0, 1]× R → [0,+∞) is a bounded and continuous function with m := inf(x,t)∈[0,1]×R h(x, t) > 0. There is an increasing interest in studying fourth-order boundary value problems, because the static form change of a beam can be described by a fourth-order equation, and also a model to study traveling waves in suspension bridges can be described by nonlinear fourth-order equations (for instance, see [13]). More general nonlinear fourth-order elliptic boundary value problems have been studied in recent

There is an increasing interest in studying fourth-order boundary value problems, because the static form change of a beam can be described by a fourth-order equation, and also a model to study traveling waves in suspension bridges can be described by nonlinear fourth-order equations (for instance, see [13]).More general nonlinear fourth-order elliptic boundary value problems have been studied in recent 132 S. Shokooh, G.A. Afrouzi, H. Zahmatkesh years.Several results are known concerning the existence of multiple solutions for fourth-order boundary value problems, and we refer the reader to [4,5,8,14,18] and the references cited therein.
The main features of this paper are the following: (I) the presence of a differential operator and the treatment in a suitable Sobolev function space; (II) the use of the Ricceri variational principle, which is a powerful analytic tool for multiplicity results in nonlinear problems with a variational structure.
Recently in [6], presenting a version of the infinitely many critical points theorem of Ricceri (see [17,Theorem 2.5]), the existence of an unbounded sequence of weak solutions for a Strum-Liouville problem, having discontinuous nonlinearities, has been established.In a such approach, an appropriate oscillating behavior of the nonlinear term either at infinity or at zero is required.This type of methodology has been used in several works in order to obtain existence results for different kinds of problems (see, for instance, [1,2,3,4,7,8,9,10,11,16] and references therein).We refer to [12] for several applications of the Ricceri variational principles.In [1], the existence of infinitely many classical solutions for the following Dirichlet quasilinear system has been obtained where for all M > 0 and all 1 ≤ i ≤ n.Now, starting from the results obtained in [1] and with the same method, we are interested in looking for a class of perturbations, namely µg + p, for which (1.1) still preserves multiple solutions.In particular, our goal in this paper is to obtain some sufficient conditions to guarantee that problem (1.1) has infinitely Infinitely Many Weak Solutions for Sourth-order Equations 133 many weak solutions.To this end, we require that the primitive F of f satisfies a suitable oscillatory behavior either at infinity (for obtaining unbounded solutions) or at zero (for finding arbitrarily small solutions), while G, the primitive of g, has an appropriate growth (see Theorems 3.1 and 3.7).Our approach is fully variational and the main tool is a general critical point theorem (see Lemma 2.1 below) contained in [6]; see also [17].Here, as an example, we state a special case of our results.
Then, the problem has a sequence of pairwise distinct weak solutions.

Preliminaries
The goal of this work is to establish some new criteria for problem (1.1) to have infinitely many weak solutions in X.Our analysis is mainly based on a recent critical point theorem of Bonanno and Molica Bisci [6] (see Lemma (2.1) below) which is a more precise version of Ricceri's variational principle [17,Theorem 2.5].
Lemma 2.1.Let X be a reflexive real Banach space, let Φ, Ψ : X → R be two Gâteaux differentiable functionals such that Φ is sequentially weakly lower semicontinuous, strongly continuous and coercive, and Ψ is sequentially weakly upper semicontinuous.For every r > inf X Φ, let ϕ(r) := inf

Then the following properties hold:
(a) For every r > inf X Φ and every λ ∈ (0, 1/ϕ(r)), the restriction of the functional (b) If γ < +∞, then for each λ ∈ (0, 1/γ), the following alternative holds: either Let us introduce some notation which will be used later.Define ) be the Sobolev space endowed with the usual norm defined as follows: We recall the following Poincaré type inequalities (see, for instance, [15, Lemma 2.3]): we have the following relation.
Proof: Taking (2.1) into account, the conclusion follows from the well-known inequality (c) for every ρ > 0 there exists a function Corresponding to f, g and p we introduce the functions F, G : We say that a function u ∈ X is a weak solution of problem (1.1) if holds for all v ∈ X.
In the following, let M := sup (x,t)∈[0,1]×R h(x, t) and suppose that the Lipschitz constant L > 0 of the function p satisfies the following condition: S. Shokooh, G.A. Afrouzi, H. Zahmatkesh

Main results
In this section we establish the main abstract result of this paper.
Proof: Our aim is to apply Lemma 2.1(b) to problem (1.1).To this end, fix λ ∈ (λ 1 , λ 2 ) and g satisfying our assumptions.Since λ < λ 2 , we have Infinitely Many Weak Solutions for Sourth-order Equations 137 and put Note that the weak solutions of (1.1) are exactly the critical points of I λ .The functionals Φ, Ψ satisfy the regularity assumptions of Lemma 2.1.Indeed, by standard arguments, we have that Φ is Gâteaux differentiable and sequentially weakly lower semicontinuous and its Gâteaux derivative is the functional Φ ′ (u) ∈ X * , given by for any v ∈ X.Furthermore, the differential Φ ′ : X → X * is a Lipschitzian operator.Indeed, taking (2.1) and (2.2) into account, for any u, v ∈ X, there holds Recalling that p is Lipschitz continuous and h is bounded away from zero, the claim is true.In particular, we derive that Φ is continuously differentiable.Also, for any u, v ∈ X, we have By the assumption (A 0 ), it turns out that Φ ′ is a strongly monotone operator.So, by applying Minty-Browder theorem (Theorem 26.A of [19]), Φ ′ : X → X * admits a Lipschitz continuous inverse.On the other hand, the fact that X is compactly embedded into C 0 ([0, 1]) implies that the functional Ψ is well defined, continuously Gâteaux differentiable and with compact derivative, whose Gâteaux derivative is given by for any v ∈ X. Hence Ψ is sequentially weakly (upper) continuous (see [19, Corollary 41.9]).Since p is Lipschitz continuous and satisfies p(0) = 0, while h is bounded away from zero, we have from (2.2) that for all u ∈ X, and so Φ is coercive.First of all, we will show that λ < 1/γ.Hence, let {ξ n } be a sequence of positive numbers such that lim n→+∞ ξ n = +∞ and for all n ∈ N.Then, for all v ∈ X with Φ(v) < r n , taking (2.

3) into account, one
Infinitely Many Weak Solutions for Sourth-order Equations 139 has v ∞ < ξ n .Note that Φ(0) = Ψ(0) = 0.Then, for all n ∈ N, Moreover, from the assumption (A 2 ) and the condition (3.1), we have A < +∞ and Therefore, The assumption µ ∈ (0, µ G,λ ) immediately yields Hence, Let λ be fixed.We claim that the functional I λ is unbounded from below.Since there exist a sequence {η n } of positive numbers and τ > 0 such that lim n→+∞ η n = +∞ and S. Shokooh, G.A. Afrouzi, H. Zahmatkesh for each n ∈ N large enough.For all n ∈ N define For any fixed n ∈ N, it is easy to see that w n ∈ X and, in particular, one has Taking (2.1), (2.2) and (3.6) into account, we have On the other hand, bearing (A 1 ) in mind and since G is non-negative, from the definition of Ψ, we infer By (3.4), (3.7) and (3.8), we observe that for every n ∈ N large enough.Since λτ > 1 and lim n→+∞ η n = +∞, we have Then, the functional I λ is unbounded from below, and it follows that I λ has no global minimum.Therefore, by Lemma 2.1(b), there exists a sequence {u n } of critical points of I λ such that lim n→+∞ u n = +∞, and the conclusion is achieved.✷ Remark 3.2.Under the conditions A = 0 and B = +∞, from Theorem 3.1 we see that for every λ > 0 and for each µ ∈ 0, 2k2π 2 g∞ , problem (1.1) admits a sequence of weak solutions which is unbounded in X.Moreover, if g ∞ = 0, the result holds for every λ > 0 and µ ≥ 0.
The following result is a special case of Theorem 3.1 with µ = 0. Theorem 3.3.Assume that all the assumptions in the Theorem 3.1 hold.Then, for each the problem has an unbounded sequence of weak solutions in X.
Here, we point out the following consequence of Theorem 3.3.
Corollary 3.5.Assume that the assumption (A 1 ) in the Theorem 3.1 holds.Suppose that Then, the problem has an unbounded sequence of weak solutions in X.
Corollary 3.6.Let g 1 : [0, 1] → R be a non-negative continuous function.Put dt for all ξ ∈ R and assume that Then, for every and with α 1 = 0, and for every non-negative continuous where the problem has an unbounded sequence of weak solutions in X.
Proof: Set f (x, t) = n i=1 α i (x)g i (t) for all (x, t) ∈ [0, 1]×R.From the assumption (A 4 ) and the condition min lim inf Using Lemma 2.1(c) and arguing as in the proof of Theorem 3.1, we can obtain the following result.
Theorem 3.7.Assume that the assumption (A 1 ) in the Theorem 3.1 holds and Then, setting for every λ ∈ (λ 3 , λ 4 ) and for every arbitrary non-negative function satisfying the condition if we put where µ ′ g,λ = +∞ when g 0 = 0, for every µ ∈ [0, µ ′ g,λ ) problem (1.1) has a sequence of weak solutions, which strongly converges to zero in X.
Proof: Fix λ ∈ (λ 3 , λ 4 ) and let g be a function that satisfies the condition (3.10).Since λ < λ 4 , we obtain for all (x, t) ∈ [0, 1] × R. We take Φ, Ψ and I λ as in the proof of Theorem 3.1.Now, as it has been pointed out before, the functionals Φ and Ψ satisfy the regularity assumptions required in Lemma 2.1.As first step, we will prove that λ < 1/δ.Then, let {ξ n } be a sequence of positive numbers such that lim n→+∞ ξ n = 0 and By the fact that inf X Φ = 0 and the definition of δ, we have δ = lim inf r→0 + ϕ(r).
Then, as in showing (3.3) in the proof of Theorem 3.1, we can prove that δ < +∞.From µ ∈ (0, µ ′ g,λ ), the following inequalities hold Let λ be fixed.We claim that the functional I λ has not a local minimum at zero.Since there exist a sequence {η n } of positive numbers and τ > 0 such that lim n→+∞ η n = 0 + and for each n ∈ N large enough.For all n ∈ N, let w n (x) defined by (3.5) with the above η n .Note that λτ > 1.Then, as in showing (3.9), we can obtain that we see that zero is not a local minimum of I λ .This, together with the fact that zero is the only global minimum of Φ, we deduce that the energy functional I λ has not a local minimum at the unique global minimum of Φ.Therefore, by Lemma 2.1(c), there exists a sequence {u n } of critical points of I λ which converges weakly to zero.In view of the fact that the embedding X ֒→ C 0 ([0, 1]) is compact, we know that the critical points converge strongly to zero, and the proof is complete.✷ Remark 3.8.Under the conditions A ′ = 0 and B ′ = +∞, Theorem 3.7 ensures that for every λ > 0 and for each µ ∈ 0, 2k2π 2 g0 , problem (1.1) admits a sequence of weak solutions which strongly converges to 0 in X.Moreover, if g 0 = 0, the result holds for every λ > 0 and µ ≥ 0.
Infinitely Many Weak Solutions for Sourth-order Equations 145 Remark 3.9.Applying Theorem 3.7, results similar to Theorems 1.1 and 3.3 Corollaries 3.5 and 3.6 can be obtained.We omit the discussions here.
We conclude this paper with the following example to illustrate our results.has a sequence of weak solutions which is unbounded in X.

(b 1 )
I λ possesses a global minimum, or (b 2 ) there is a sequence {u n } of critical points (local minima) of I λ such that lim n→+∞ Φ(u n ) = +∞.(c) If δ < +∞, then for each λ ∈ (0, 1/δ), the following alternative holds: either (c 1 ) there is a global minimum of Φ which is a local minimum of I λ , or (c 2 ) there is a sequence {u n } of pairwise distinct critical points (local minima) of I λ that converges weakly to a global minimum of Φ.