On the p-Biharmonic Operator with Critical Sobolev Exponent and Nonlinear Steklov Boundary Condition

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.


Introduction
Let Ω be a smooth bounded domain in R  .Consider the fourth-order nonlinear Steklov boundary eigenvalue problem Here  is a real parameter which plays the role of an eigenvalue.Δ 2   := Δ(|Δ| −2 Δ) is the operator of fourth order called the -biharmonic operator.For  = 2, the linear operator Δ 2 2 = Δ 2 = Δ.Δ 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 (Δ 2 ) −1 , is celebrated Green's operator [1].The nonlinear boundary condition describes a nonlinear flux through the boundary Γ which depends on the solution itself and its normal derivation.Here / denotes the outer normal derivative of  on Γ defined by / = ∇⋅ → .Notice that the biharmonic equation Δ 2  = 0, corresponding to  = 2, 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  = 0, Δ − (/) = 0 on Γ (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 fourthorder Steklov eigenvalue problem on a bounded domain in 2 International Journal of Analysis R  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  = 2, 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.
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  ̸ = 2, 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 (  ) ≥1 .
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

Definitions and Preliminaries
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 (  ) ≥1 , by using a variational technique based on mini-max theory on  1 -manifolds [13].In fact, we give a direct characterization of   involving a mini-max argument over sets of genus greater than .
We set where Let us notice that W  (Ω) equipped with this norm is a uniformly convex Banach space for 1 <  < +∞.The norm ‖Δ ⋅ ‖  is uniformly equivalent on  2, 0 (Ω) to the usual norm of  2, 0 (Ω).Indeed, in [14] the scalar -polyharmonic operators Δ   , which coincide to the -biharmonic for  = 2, 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 ‖Δ ⋅ ‖  .For that, consider the classical Dirichlet problem for the famous Poisson's equation (see [17]): Let us denote by and by ‖‖ 2, = (‖Δ‖   + ‖‖   ) 1/ the norm in W  (Ω).
This allows us to say the following.
On the other hand, it is easy to remark that By the Closed Graph Theorem, we conclude that  → ‖Δ‖  = (∫ Ω |Δ|  ) 1/ is equivalent to the norm induced by  2, (Ω).
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 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 lim sup  → +∞ ⟨Δ 2    ,   − ⟩ ≤ 0 it follows that   converges strongly to  in .We write  ∈ ( + ).
Lemma 3. One has the following statements.
(i) Φ and  are even and of class  1 on W  (Ω).
Proof.It is clear that  and Φ are even and of class  1 on W  (Ω) and M =  −1 {1/}.Therefore M is closed.The derivative operator   satisfies   () ̸ = 0 for all  ∈ M (i.e.,   () is onto for all  ∈ M), so  is a submersion; then M is a  1 -manifold.
is the duality mapping of W  (Ω) associated with the norm ‖Δ ⋅ ‖  .
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 Φ satisfies the Palais-Smale condition on M, that is, for {  } ⊂ M if {Φ(  )}  is bounded and where Proof.Let , V ∈ W  (Ω).We have where  is the constant given by the embedding of W  (Ω) in   (Γ).Here ‖ ⋅ ‖ * is the dual norm associated with ‖Δ ⋅ ‖  .Now, by the definition of Φ we have that ‖Δ(  )‖  is bounded in R.
Thus, without loss of generality, we can assume that   converges weakly in W  (Ω) for some function  ∈ W  (Ω) and ‖Δ(  )‖  → .For the rest we distinguish two cases.
If  ̸ = 0, then let us prove that lim sup Indeed, notice that Applying   of ( 14) to , we deduce that This implies that Combining with the above equalities, we obtain lim sup We deduce lim sup We can write According to (31), we conclude that lim sup where  is the duality mapping defined in Remark 4. Thus it satisfies the condition  + given in [18].Therefore,   →  strongly in W  (Ω).This achieves the proof of the lemma.

Main Results
Set where () =  is the genus of , that is, the smallest integer  such that there exists an odd continuous map from  to R  \ {0}.
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  ∈ N * , is a critical value of Φ restricted on M.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  ∈ N * , Γ  ̸ = 0 and the limit (37).Indeed, since W  (Ω) is separable, there exists (  ) ≥1 linearly dense in   (Ω) such that supp   ∩ supp   = 0 if  ̸ = .We may assume that   ∈ M (if not, we take    =   /(  )).
Let now  ∈ N * and denote that Clearly,   is a vector subspace with dim   = .
It follows that the map defines a norm on   .Consequently, there is a constant  > 0 such that This implies that the set is bounded, since  ⊂ (0, 1/), where Thus,  is a symmetric bounded neighborhood of 0 ∈   .Moreover,   ∩ M is a compact set.By Proposition 2.3 in [13], we conclude that (  ∩M) =  and then we obtain finally that Γ  ̸ = 0.This completes the proof of the theorem.Now, we claim that Let (  ,  *  ) , be a biorthogonal system such that   ∈ W  (Ω) and  *  ∈ (W  (Ω))  , the   are linearly dense in W  (Ω), and the  *  are total for the dual (W  (Ω))  .

Corollary 7. One has the following statements:
Proof.(i) For  ∈ M, set  1 = {, −}.It is clear that ( 1 ) = 1, Φ is even, and On the other hand, for all  ∈ Γ 1 , for all  ∈ , we have sup It follows that inf Thus (ii) For all  ≥ , we have Γ  ⊂ Γ  and in view of definition of   ,  ∈ N * , we get   ≥   .Regarding   → ∞, it is proved before in Theorem 6.
We now turn to the fourth-order nonlinear Steklov boundary eigenvalue problem (  )  .( If  ∈ W  (Ω) \ {0}, then it is called an eigenfunction of problem (  )  .
We formulate our second main result of this paper as follows.

International Journal of Analysis
Proof of Theorem 10.Consider the following minimization problem: The existence of least energy solution follows from the following proposition.
Proof.Let {  } ≥0 be a minimizing sequence for Λ  such that Then, Moreover, from (11), we have which implies that Exploiting the compactness of the embedding W  (Ω) →   (Γ) and  1, (Ω) →   (Ω), we deduce that there exists  ∈ W  (Ω) such that up to a subsequence.That is, if we set V  :=   − , then On the other hand, in view of (56), we have ‖Δ  ‖   ≥ , so that, from (57), we obtain which remains bounded away from 0 since Λ  < .From this, we deduce that  ̸ = 0.
which proves (54) and then the proof of Theorem 10 is achieved.