A NONLINEAR BOUNDARY PROBLEM INVOLVING THE p-BILAPLACIAN OPERATOR

We show some new Sobolev's trace embedding that we apply to prove that the fourth-order nonlinear boundary conditions Δp2u

Existence results for nonlinear boundary problem have only been considered in recent years.For the second-order p-Laplacian with nonlinear boundary conditions of different type, see [5], see also [3].For a fourth-order elliptic equation with the ordinary boundary conditions, we cite [2] and with nonlinear boundary conditions, see [4].

Problem involving the p-Laplacian
In this paper, we study in Theorem 2.5 the Sobolev's trace embedding W m,p (Ω) L q (∂Ω), where Ω ⊂ R N is a bounded domain of class C m , N ≥ 2, q ∈ [1, p m [ such that p m = (N − 1)p/(N − mp) if mp < N and p m = +∞ if mp ≥ N.This embedding leads to a nonlinear eigenvalue problem (1.2), where the eigenvalue appears at the nonlinear boundary condition.Other main objective of this work, formulated by Theorem 3.3, is to show that problem (1.2) has at least one nondecreasing sequence of positive eigenvalues (λ k ) k≥1 , by using some technical lemmas and the Ljusternick-Schnirelmann theory on C 1 -manifolds, see [9].So we give a direct characterization of λ k involving a minimax argument over sets of genus greater than k.
We set where u 2,p = ( u p p + ∆u p p ) 1/p is the norm of W 2,p (Ω).This paper is organized as follows.In Section 2, we establish the Sobolev's trace embedding in the general case, that is, for any m ∈ N. In Section 3, we use a variational technique to prove the existence of a sequence of the positive eigenvalues of problem (1.2).

The Sobolev's trace embedding
We begin with the following definition and lemmas that will be helpful to prove the Sobolev's trace embedding.Definition 2.1.A domain Ω is of class C k if ∂Ω can be covered by bounded open sets Θ j such that there is a mapping f j : Θ j → B, where B is the unit ball centered at the origin and , where and there exists a positive constant depending only on p and N such that (2.7) Indeed, let q = p/(p − 1), (t − 1)q = N p/(N − p) and by using the Sobolev inequalities, see [6], By Hölder and (2.8), (2.9) On the other hand, ∂w/∂x j = ±t|u (t−1) |(∂u/∂x j ).By Hölder and (2.8), where c is a positive constant.Now, applying (2.9), (2.10), and Lemma 2.3, we find , with p m = (N − 1)p/(N − mp) and there exists a positive constant c depending only on p and N such that (2.12) Proof.By applying Sobolev inequality [6] to ∂u/∂x j , 1 ≤ j ≤ N, we obtain that u ∈ W 1,N p/(N−(m−1)p) (R N ).By Lemma 2.3, we deduce that v ∈ L p m (R N−1 ) with p m = (N − 1)p/(N − mp).
Theorem 2.5.Let Ω ⊂ R N , N ≥ 2, be a bounded domain of class C m .For all u ∈ W m,p (Ω), mp < N. The restriction of u to ∂Ω denoted also by u belongs to L q (∂Ω), for all q ∈ [1, p m ], and there exists a positive constant c depending only on p, m, and Ω such that

14)
Proof.There exists a continuous linear operator P that operates from W m,p (Ω) to W m,p (R N ), (cf.[1,6]), such that to every u element of W m,p (Ω) is associated an element P(u) ∈ W 2,p (R N ).By density, it is sufficient to study the properties of the trace on ∂Ω of the function C ∞ c (R N ).Let θ j and f j be as in the definition (2.2). ∂Ω is compact, therefore we can suppose that there exists a finite θ j , 1 ≤ j ≤ k, which covers ∂Ω.Let (P j , 1 ≤ j ≤ k) be a partition of unity of ∂Ω subordinate to this covering, see, for example, [1] . By Lemma 2.4, the trace w j of P j uo f −1 j on the hyperplane {(x 1 ,x 2 ,...,x N−1 ,0), x i ∈ R} satisfies the inequality where c j is a positive constant.We estimate the trace v j := w j o f j of the function P j u on where c j is a positive constant.We combine (2.15) and (2.16) as follows: (2.17) Abdelouahed El Khalil et al. 1529 On the other hand, u = j=k j=1 P j u = j=k j=1 v j , where v j = P j u, and suppv j ⊂ Γ j , ∂Ω ⊂ j=k j=1 Γ j .So On the other hand, ∂Ω is bounded, so u ∈ L q (∂Ω), for all q ∈ [1, p m ].
By using Theorem 2.5, the next corollary follows exactly as in the classical compact Sobolev embedding established in [1,6].
Corollary 2.6.Under the same hypotheses at the last theorem, The restriction of u to ∂Ω denoted also by u belongs to L q (∂Ω), for all q ∈ [1,+∞[.
Proof.By using the Sobolev embedding, W m,p (Ω) is compactly embedding in L ∞ (Ω) ∩ C(Ω).So the functions of W m,p (Ω) are continuous on Ω and bounded, therefore their traces are well defined, continuous, and bounded.So we have By ( * ) and ( * * ), we have the desired result.
1530 Problem involving the p-Laplacian

Main results
Through this paper, all solutions are weak, that is, u ∈ W 2,p (Ω) is a solution of (1.2), if for all v ∈ W 2,p (Ω), we have (S 1 ) If we replace S 2 in (S 1 ), then we deduce that 2) associated to the eigenvalue λ.
The following lemma is the key to show the existence.
(ii) The functional A satisfies the Palais-Smale condition on ᏹ, that is, for ) . By Hölder's inequality, we have Abdelouahed El Khalil et al. 1531 where p 2 = (N − 1)p/(N − 2p), and s is given by Therefore, Then it suffices that max(1, and B is well defined.Second case.If N/ p = 2, r > (N − 1)/(2p − 1).In this case, from Theorem 2.7, for any q ∈ [1,+∞[.There is q ≥ 1 such that (3.10) We obtain that By Hölder's inequality, we arrive at for any u,v ∈ W 2,p (Ω).Then in this case, B is well defined.Third case.If N/ p < 2, r = 1.In this case, from Theorem 2.8, 1532 Problem involving the p-Laplacian Therefore for any u,v ∈ W 2,p (Ω), we have with ρ ∈ L 1 (Ω), and B is well defined also in this case.
where q is given by (3.11).By Sobolev's trace embedding, there exists c > 0 such that where c is the constant given by embedding of Hence B is completely continuous also in this case.
(ii) {u n } is bounded in W 2,p (Ω).Hence without loss of generality, we can assume that u n converges weakly in W 2,p (Ω) to some function u ∈ W 2,p (Ω) and u n 2,p → c.For the rest, we distinguish two cases.If c = 0, then u n converges strongly to 0 in W 2,p (Ω).
If c = 0, the claim is to prove that u n is of Cauchy in W 2,p (Ω).Set (3.23) We remark that On the other hand, with n defined as in (3.4), and where • * is the dual norm associated to • 2,p .This implies that h n converges, for a subsequence if necessary, in W 2,p (Ω).Indeed, from (i) of Lemma 3.2 B : W 2,p (Ω) → (W 2,p (Ω)) is completely continuous.On the other hand, for a subsequence if necessary, u n 2,p → c ≥ 0. It follows that (h n ) n≥0 is convergent in (W 2,p (Ω)) .Then, G u n ,u m −→ 0, as n −→ +∞.(3.27) 1534 Problem involving the p-Laplacian From [7], we have the inequality By applying Hölder's inequality, we deduce that where c is a positive constant independent of n and m, γ = p if 1 < p < 2, and From [7], we have where By integrating these two relations over Ω, we find On the other hand, G 1 ≤ G and G 2 ≤ G. Then from (3.27) and (3.32), Then from (3.29) and (3.30), Therefore (u n ) n is a Cauchy's sequence in W 2,p (Ω).This achieves the proof of the lemma. Set where γ(K) = k is the genus of K, that is, the smallest integer k such that there exists an odd continuous map from K to R k − {0}.Now, by the Ljusternick-Schnirelmann theory, see, for example, [9], we have our main result.
Abdelouahed El Khalil et al. 1535 is a critical value of A restricted on ᏹ.More precisely, there exists 2) associated to the positive eigenvalue λ k .Moreover, Proof.We need only to prove that for any k ∈ N * , Γ k = ∅ and the least assertion.Indeed, since W 2,p (Ω) is separable, there exists (e i ) i≥1 linearly dense in W 2,p (Ω) such that suppe i ∩ suppe j = ∅ if i =j.We can assume that e i ∈ ᏹ.Let k ∈ N * , denote F k = span{e 1 , e 2 ,...,e k }.F k is a vectorial subspace and dim This implies that the set Now we claim that λ k → +∞, as k → +∞.Let (e n ,e * j ) n, j be a biorthogonal system such that e n ∈ W 2,p (Ω), e * j ∈ (W 2,p (Ω)) , the e n are linearly dense in W 2,p (Ω); and the e * j are total for (W 2,p (Ω)) , see, for example, [9].Set now, for k ∈ N * , for some M > 0 independent of k.Therefore, u k 2,p ≤ M. (3.45)This implies that (u k ) k is bounded in W 2,p (Ω).For a subsequence of {u k } if necessary, we can assume that {u k } converges weakly in W 2,p (Ω) and strongly in L p (Ω).By our choice of F ⊥ k−1 , we have u k 0 weakly in W 2,p (Ω) because e * n ,e k = 0, for all k ≥ n.This contradicts the fact that u k p = 1 for all k.Since λ k ≥ t k , the claim is proved, which completes the proof.
On the other hand, for all K ∈ Γ 1 , for all u ∈ K, Then, From the definition of λ i , i ∈ N * , we have λ i ≥ λ j .λ n → +∞ is already proved in Theorem 3.3.The proof is achieved.
Proof.First of all, we remark that (∂Ω

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
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. 3 )
Lemma 3.1.(i) A and B are even and of class C 1 on W 2,p (Ω).(ii) ᏹ is a closed C 1 -manifold.Proof.(i) It is clear that A and B are even and of class C 1 on W 2,p (Ω), A (u) = ∆ 2 p u + |u| p−2 u, and B (u) = ρ|u| p−2 u.

Mathematical Problems in Engineering Special Issue on Modeling Experimental Nonlinear Dynamics and Chaotic Scenarios
.51) λ n (−ρ) n≥0 is a nondecreasing positive sequence, so (λ −n )(ρ) n≥0 is a negative decreasing sequence.