We show that this operator possesses at least one nondecreasing sequence of positive eigenvalues. A direct characterization of the principal eigenvalue (the first one) is given that we apply to study the spectrum of the -biharmonic operator with a critical Sobolev exponent and the nonlinear Steklov boundary conditions using variational arguments and trace critical Sobolev embedding.

1. Introduction

Let be a smooth bounded domain in . Consider the fourth-order nonlinear Steklov boundary eigenvalue problem where is the critical Sobolev exponent defined by Here is a real parameter which plays the role of an eigenvalue.

is the operator of fourth order called the -biharmonic operator. For , the linear operator . is the iterated Laplacian that multiplied with positive constant appears often in Navier-Stokes equations as being a viscosity coefficient. Its reciprocal operator, denoted by , is celebrated Green’s operator [1].

The nonlinear boundary condition describes a nonlinear flux through the boundary which depends on the solution itself and its normal derivation. Here denotes the outer normal derivative of on defined by .

Notice that the biharmonic equation , corresponding to , is a partial differential equation of fourth order which appears in quantum mechanics and in the theory of linear elasticity modeling Stokes’ flows. It is well known that elliptic problems with eigenvalues in the boundary conditions are usually called Steklov problems from their first appearance in [2]. For the fourth-order Steklov eigenvalue problems, the first eigenvalue plays a crucial role in the positivity preserving property for the biharmonic operator under conditions , on (see [3]). In [4], the authors investigated the bound for the first eigenvalue on the plane square and proved that the first eigenvalue is simple and its eigenfunction does not change sign. The authors of [5, 6] studied the spectrum of a fourth-order Steklov eigenvalue problem on a bounded domain in and gave the explicit form of the spectrum in the case where the domain is a ball. Let us mention that the spectrum of the fourth order Steklove has been completely determined by Ren and Yang [7] in the case , using the theory of completely continuous operators.

It is already evident from the well-studied second-order case that nonlinear equations with critical growth terms present highly interesting phenomena concerning the existence and nonexistence. For the fourth-order equations is more challenging, since the techniques depend strongly on the imposed boundary conditions.

For the case of , , with , the problem becomes linear on the right side and it is studied by Berchio et al. in [8].

It is well known that fourth-order elliptic problems arise in many applications, such as microelectromechanical system, in thin film theory, nonlinear surface diffusion on solids, interface dynamics, flow in Hele-Shaw cells, phase field models of multi-phase systems, and the deformation of a nonlinear elastic beam; see, for example, [9, 10], for more details.

In the nonlinear cases of , problems governed by -biharmonic operators attracted growing interest, and figure in variety of applications, where this operator is used to control the nonlinearity artificial viscosity of diffusion surface of non-Newtonian fluids [11].

Recently, El Khalil et al. in [12] proved that the following nonlinear boundary problem: has at least one nondecreasing sequence of positive eigenvalues .

In this paper, we use a variational technique to prove the existence of a sequence of positive eigenvalues of problem .

To present our result concerning , we consider the homogenous problem

Throughout this paper, we use the notation .

2. Definitions and Preliminaries

Definition 1. One says that a function is weak solution of if If is not identically zero, then one says that is an eigenvalue of corresponding to the eigenfunction .
The main objective of this work is to show that problem has at least one nondecreasing sequence positive eigenvalues , by using a variational technique based on mini-max theory on -manifolds [13]. In fact, we give a direct characterization of involving a mini-max argument over sets of genus greater than .

We set where denoted by is a norm on . Let us notice that equipped with this norm is a uniformly convex Banach space for . The norm is uniformly equivalent on to the usual norm of . Indeed, in [14] the scalar -polyharmonic operators , which coincide to the -biharmonic for , were recently introduced for all orders and independently in [15] only for even. The norms are proved to be equivalent; see also the vectorial case treated in [16]. The proof of the equivalence comes from the Poincar and Calderón-Zygmund type inequalities.

For reader’s convenience, we give below the proof of the equivalence between the standard Sobolev space norm and the norm . For that, consider the classical Dirichlet problem for the famous Poisson’s equation (see [17]): Let us denote by the norm in and by the norm in .

If is the inverse operator of , it is well known that (5) is uniquely solvable in for all and for any . Moreover, the operator solution satisfies the elliptic regularity estimation (continuity) This allows us to say the following.

If and is a solution of (5), then , on and . Moreover, That is, for a suitable positive constant independent of .

On the other hand, it is easy to remark that By the Closed Graph Theorem, we conclude that is equivalent to the norm induced by .

We see that the value defined in (4) can be written as

Finally, let us point out that the problem is naturally well defined taking in account the trace embedding introduced in [12].

Definition 2. Let be a real reflexive Banach space and let stand for its dual with respect to the pairing . We shall deal with mappings acting from into . The strong convergence in (and in is denoted by and the weak convergence by . is said to belong to the class if for any sequence in converging weakly to and it follows that converges strongly to in . We write .
Consider now the following two functionals defined on : and set .

Lemma 3. One has the following statements.(i) and are even and of class on .(ii) is a closed -manifold.

Proof. It is clear that and are even and of class on and . Therefore is closed. The derivative operator satisfies for all (i.e., is onto for all , so is a submersion; then is a -manifold.

Remark 4. Observe that defined by is the duality mapping of associated with the norm .

The following lemma is the key of our result related to the existence.

Lemma 5. One has the following statements.(i) is completely continuous.(ii)The functional satisfies the Palais-Smale condition on , that is, for if is bounded and where . Then has a convergent subsequence in .

Proof. Let . We have Then By applying Hölder’s inequality, we obtain where and are conjugate by the equality . Therefore Hence where is the constant given by the embedding of in . Here is the dual norm associated with .
Now, by the definition of we have that is bounded in .
Thus, without loss of generality, we can assume that converges weakly in for some function and . For the rest we distinguish two cases.If , then converges strongly to in .If , then let us prove that Indeed, notice that Applying of (14) to , we deduce that Thus Therefore That is, On the other hand, from Lemma 5,    is completely continuous. Thus Then It follows that This implies that Combining with the above equalities, we obtain We deduce We can write , since for large enough. Therefore According to (31), we conclude that where is the duality mapping defined in Remark 4. Thus it satisfies the condition given in [18]. Therefore, strongly in .
This achieves the proof of the lemma.

3. Main Results

Set where is the genus of , that is, the smallest integer such that there exists an odd continuous map from to .

Let us now state our first main result of this paper using mini-max theory; we have our main result formulated as follows.

Theorem 6. For any integer , is a critical value of restricted on . More precisely, there exist , such that and is a solution of   associated with positive eigenvalue . Moreover,

Proof. We only need to prove that, for any , and the limit (37). Indeed, since is separable, there exists linearly dense in such that if . We may assume that (if not, we take ).
Let now and denote that Clearly, is a vector subspace with .
If , then there exist in such that . Thus It follows that the map defines a norm on . Consequently, there is a constant such that This implies that the set is bounded, since , where Thus, is a symmetric bounded neighborhood of . Moreover, is a compact set.
By Proposition  2.3 in [13], we conclude that and then we obtain finally that .
This completes the proof of the theorem.
Now, we claim that Let be a biorthogonal system such that and , the are linearly dense in , and the are total for the dual .
For , set We have, for any . Thus Indeed, if not, for is large, there exists with such that for some independent of . Thus . This implies that is bounded in . For a subsequence of , if necessary, we can assume that converges weakly in and strongly in . By our choice of , we have weakly in , because , for any . This contradicts the fact that for all . Since the claim is proved. This achieved the proof of the theorem.

Corollary 7. One has the following statements:(i), ; ;(ii).

Proof. (i) For , set . It is clear that , is even, and Hence On the other hand, for all , for all , we have It follows that Thus
(ii) For all , we have and in view of definition of , we get . Regarding , it is proved before in Theorem 6.

We now turn to the fourth-order nonlinear Steklov boundary eigenvalue problem .

Definition 8. One says that a function is weak solution of if If , then it is called an eigenfunction of problem .

Lemma 9 (see [19]). Assume that is a bounded domain of in .
Then for all .

We formulate our second main result of this paper as follows.

Theorem 10. There exists such that if then admits at least energy solutions satisfying where is the first eigenfunction of such that , associated with the principal eigenvalue defined by (11).

Proof of Theorem 10. Consider the following minimization problem: The existence of least energy solution follows from the following proposition.

Proposition 11. Assume that . If , then the minimum in (55) is achieved, where is the Sobolev constant for the embedding .

Proof. Let be a minimizing sequence for such that Then, Moreover, from (11), we have which implies that is bounded in . Hence is bounded in .
Exploiting the compactness of the embedding and , we deduce that there exists such that up to a subsequence. That is, if we set , then On the other hand, in view of (56), we have , so that, from (57), we obtain which remains bounded away from since . From this, we deduce that .
Now, In view of (59) and (60), we may rewrite (57) as Moreover, by (56) and Brezis-Lieb Lemma [20], we have where we use also the fact that both and do not exceed . Since for every , the last inequality gives By combining this inequality with (62), we obtain which shows that is minimizer for (53).
Let be a positive eigenfunction of and Thus and, for , we have We now prove the first part of (54). Indeed, in view of the characterization of in (53), we have Since is a least energy solution of , we have Moreover, by taking in (53), we get Identities (69)-(70) readily imply that . In turn, this and (62) show that Moreover, by (69) and (68) we obtain Thus Then in view of (11), we get Hence Consequently, using (76), the last inequality implies that Finally, we conclude that From [21], the inequality holds true for any , with if and if .
By applying Hölder’s inequality, we have where Then using again Hölder’s inequality we obtain Hence
Finally, we conclude that which proves (54) and then the proof of Theorem 10 is achieved.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


The authors would like to thank the anonymous reviewer for the helpful and constructive comments that greatly contributed to improving the final version of the paper.