Existence of Multiple Solutions for a Quasilinear Biharmonic Equation

Using three critical points theorems, we prove the existence of at least three solutions for a quasilinear biharmonic equation.


Introduction
In this paper, we show the existence of at least three weak solutions for the Navier boundary value problem Δ 2 − div (∇ ) = ( , ) + ( ) in Ω, where Ω ⊂ R (4 > ≥ 1) is a nonempty bounded open set with a sufficient smooth boundary Ω, > 0, : Ω × R → R is an 1 -Carathéodory function, and : R → R is a Lipschitz continuous function with Lipschitz constant > 0; that is, for every 1 , 2 ∈ R and (0) = 0. Motivated by the fact that such problems are used to describe a large class of physical phenomena, many authors looked for existence and multiplicity of solutions for forthorder nonlinear equations. For an overview on this subject, we cite the papers . For instance, when = 1, in [22], Liu and Li, using Ricceri's three critical points theorem [24], established the existence of at least three weak solutions for the following problem: where , are real constants and is a positive parameter and : [ , ] × R → R is a 2 -Carathéodory function. Later, some authors generalized this type of equation (see [1-5, 10, 12, 15, 19]). When > 1, Liu and Su [21] also used Ricceri's three critical points theorem [24] to established the existence of at least three weak solutions for the following problem: where Ω ⊂ R ( ≥ 1) is a nonempty bounded open set with a sufficient smooth boundary Ω, > max{1, /2}, > 0, and : Ω × R → R is an 1 -Carathéodory function. After that some authors used different critical point theorems to get one nontrivial, at least three, and infinitely many solutions (see [6-8, 16, 18]). Elliptic systems were also considered by [9,11,13,14,17,20,23]. The goal of the present paper is to establish some new criteria for (1) to have at least three weak solutions (Theorems 4 and 5). Our analysis is mainly based on recent critical point theorems that are contained in Theorems 2 and 3. In fact, employing rather different three critical points theorems, under different assumptions on the nonlinear term , we obtain the exact collections of for which (1) admits at least three weak solutions in the space 2,2 (Ω) ∩ 1,2 0 (Ω).

International Scholarly Research Notices
A special case of our main results is the following theorem. Theorem 1. Let : R → R be a Lipschitz continuous function with the Lipschitz constant > 0 and (0) = 0 such that < 1/ 2 (Ω), where is defined by (17). Let : R → R be a continuous function and put ( ) = ∫ 0 ( ) for each ∈ R. Assume that ( ) > 0 for some > 0 and ( ) ≥ 0 in [0, ] and Then, there is * > 0 such that for each > * the problem admits at least three weak solutions.

Preliminaries
First we here recall for the reader's convenience our main tools to prove the results. The first result has been obtained in [25] and the second one in [26].
Theorem 2 (see [25,Theorem 3.1]). Let be a separable and reflexive real Banach space, Φ : → R a nonnegative continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on * , and Ψ : → R a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Assume that there exists 0 ∈ such that Φ( 0 ) = Ψ( 0 ) = 0 and that Further, assume that there are > 0, 1 ∈ , such that < Φ( 1 ) and the equation has at least three solutions in and, moreover, for each ℎ > 1,

there exists an open interval
and a positive real number such that, for each ∈ Λ 2 , (10) has at least three solutions in whose norms are less than .
Theorem 3 (see [26,Theorem 3.6]). Let be a reflexive real Banach space; let Φ : → R be a sequentially weakly lower semicontinuous, coercive, and continuously Gâteaux differentiable whose Gâteaux derivative admits a continuous inverse on * , and let Ψ : → R be a sequentially weakly upper semicontinuous and continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Assume that there exist ∈ R and 1 ∈ with 0 < < Φ( 1 ), such that Then, for each ∈ Λ , the functional Φ − Ψ has at least three distinct critical points in .
Let : Ω × R → R be an 1 -Carathéodory function and let : R → R be a Lipschitz continuous function with the Lipschitz constant > 0; that is, for every 1 , 2 ∈ R, and (0) = 0. Put for all ∈ Ω and ∈ R. Denote the usual norm in is defined by Note that is a separable and reflexive real Banach space. We say that a function ∈ is a weak solution of problem for all ∈ . Since < 4, one has < +∞. Suppose that the Lipschitz constant > 0 of the function satisfies < 1/ 2 (Ω). For other basic notations and definitions, we refer the reader to [27][28][29].
International Scholarly Research Notices 3

Main Results
Our main results are the following theorems.

Theorem 4. Assume that there exist a function
∈ , a positive function ∈ 1 , and two positive constants and with < such that Then, for each in problem (1) admits at least three weak solutions in and, moreover, for each ℎ > 1, there exists an open interval and a positive real number such that, for each ∈ Λ 2 , the problem (1) admits at least three weak solutions in whose norms are less than .

Theorem 5. Assume that there exists a function ∈ and a positive constant such that
Then, for each , problem (1) admits at least three weak solutions.
Let us give particular consequences of Theorems 4 and 5 for a fixed test function . Now, fix 0 ∈ Ω and pick with > 0 such that ( 0 , ) ⊂ Ω where ( 0 , ) denotes the ball with center at 0 and radius of . Put denotes the volume of Ω. (Ω))( ) 2 )).

International Scholarly Research Notices
Then, for each in

problem (1) admits at least three weak solutions in and, moreover, for each ℎ > 1, there exist an open interval
and a positive real number such that, for each ∈ Λ 2 , problem (1) admits at least three weak solutions in whose norms are less than .

Corollary 7.
Assume that there exist two positive constants and with < such that the assumption (A4) in Corollary 6 holds. Furthermore, suppose that

Then, for each
problem (1) admits at least three weak solutions.
Proof. All the assumptions of Theorem 5 are fulfilled by choosing as given in (25) and := ((1− 2 (Ω))/2)( / ) 2 and bearing in mind that and recalling Hence, by applying Theorem 5, we have the conclusion.

Proofs
Proof of Theorem 4. Our aim is to apply Theorem 2 to our problem. To this end, for each ∈ , we let the functionals Φ, Ψ : → R be defined by and put 6 International Scholarly Research Notices The functionals Φ and Ψ satisfy the regularity assumptions of Theorem 2. Indeed, by standard arguments, we have that Φ is Gâteaux differentiable and sequentially weakly lower semicontinuous and its Gâteaux derivative at the point ∈ is the functional Φ ( ) ∈ * , given by for every V ∈ . Furthermore, the differential Φ : → * is a Lipschitzian operator. Indeed, for any , V ∈ , there holds Recalling that is Lipschitz continuous and the embedding → 2 (Ω) is compact, the claim is true. In particular, we derive that Φ is continuously differentiable. The inequality (17) yields for any , V ∈ the estimate By the assumption < 1/ 2 (Ω), it turns out that Φ is a strongly monotone operator. So, by applying Minty-Browder theorem [29,Theorem 26.A], Φ : → * admits a Lipschitz continuous inverse. On the other hand, the fact that is compactly embedded into 0 (Ω) implies that the functional Ψ is well defined, continuously Gâteaux differentiable, and with compact derivative, whose Gateaux derivative at the point ∈ is given by for every V ∈ . Note that the weak solutions of (1) are exactly the critical points of . Also, since is Lipschitz continuous and satisfies (0) = 0, we have from (17) that for all ∈ , and so Φ is coercive. Furthermore from (A3) for any fixed ∈ [0, +∞[, using (46), taking (17) into account, we have