A NONLINEAR BOUNDARY PROBLEM INVOLVING THE p-BILAPLACIAN OPERATOR

A NONLINEAR BOUNDARY PROBLEM INVOLVING THE p-BILAPLACIAN OPERATOR

ABDELOUAHED EL KHALIL, SIHAM KELLATI, AND ABDELFATTAH TOUZANI Received 8 January 2005 and in revised form 3 May 2005

We show some new Sobolev’s trace embedding that we apply to prove that the fourth- order nonlinear boundary conditions∆²pu+|u|^p−2u=0 inΩand−(∂/∂n)(|∆u|^p−2∆u)= λρ|u|^p−2uon∂Ωpossess at least one nondecreasing sequence of positive eigenvalues.

1. Introduction and notations

LetΩbe a bounded domain of classC²inR,N≥2, 1< p <+∞, andρ∈L^r(∂Ω) a weight function which can change its sign, withr=r(N,p) satisfying

r > N−1

2p−1 forN p ^≥2, r=1 forN

p <2.

(1.1)

We assume that|(∂Ω)⁺| =0, where (∂Ω)⁺= {x∈∂Ω,ρ(x)>0}andλ∈R. We consider the following problem:

∆²pu+|u|^p−2u=0 inΩ,

− ∂

∂n

|∆u|^p−2∆u=λρ(x)|u|^p−2u on∂Ω, u∈W0^2,p(Ω).

(1.2)

∆²p:=∆(|∆u|^p−2∆u) is the operator of fourth order, so-called the p-biharmonic (or p-bilaplacian) operator. For p=2, the linear operator∆²2=∆²=∆·∆ is the iterated Laplacian that to a multiplicative positive constant appears often in the equations of Navier-Stokes as being a viscosity coefficient, and its reciprocal operator noted (∆²)⁻¹ is the celebrated Green’s operator (see [8]).

Existence results for nonlinear boundary problem have only been considered in recent years. For the second-orderp-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].

Copyright©2005 Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences 2005:10 (2005) 1525–1537 DOI:10.1155/IJMMS.2005.1525

In this paper, we study inTheorem 2.5the Sobolev’s trace embedding W^m,p(Ω) L^q(∂Ω), whereΩ⊂R^N is a bounded domain of classC^m,N≥2,q∈[1,p_m[ such that p_m=(N−1)p/(N−mp) ifmp < N and p_m=+∞ifmp≥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 byTheorem 3.3, is to show that problem (1.2) has at least one nondecreasing sequence of positive eigenvalues (λk)k≥1, by using some technical lemmas and the Ljusternick-Schnirelmann theory on C¹-manifolds, see [9]. So we give a direct characterization ofλ_k involving a minimax argument over sets of genus greater thank.

We set

λ1=inf

u_2,p^p ,u∈W^2,p(Ω);

∂Ωρ(x)|u|^pdx=1

, (1.3)

whereu2,p=(u^pp+∆u^pp)^{1/ p}is the norm ofW^2,p(Ω).

This paper is organized as follows. InSection 2, we establish the Sobolev’s trace em- bedding in the general case, that is, for anym∈N. InSection 3, we use a variational technique to prove the existence of a sequence of the positive eigenvalues of problem (1.2).

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 domainΩis of classC^kif∂Ωcan be covered by bounded open setsΘj

such that there is a mapping fj:Θj→B, whereBis the unit ball centered at the origin and

fj

Θj∩Ω=B∩R^N+, f_jΘj∩∂Ω=B∩∂R^N+, f_j∈C^kΩ, f_j⁻¹∈C^kB.

(2.1)

Lemma2.2. Letu∈W^1,1(R^N),N >1. For ally∈R,v(x) :=u(x,y)∈L¹(R^N−1)and vL¹(R^N−1)≤ uL¹(R^N)+ ∂u

∂x_N

L¹(_R^N)

. (2.2)

Proof. W^1,1(R^N)=C_c^∞(R^N). So, it suffices to prove the lemma foru∈C_c^∞(R^N). Thus,

R^N−1u(x,y)dx≤

R^N−1

+_∞ y

∂u

∂x_N(x,t)dt dx

≤

R^N−1

R

∂u

∂xN(x,t)dt dx,

(2.3)

that is,

vL¹(R^N−1)≤ ∂u

∂xN

L¹(R^N)≤ ∂u

∂xN

L¹(R^N)

+uL¹(R^N). (2.4) Lemma2.3. Letu∈W^1,p(R^N),p < N. For ally∈R,v(x) :=u(x,y)∈L^t(R^N−1), where

t=(N₋1)p

N₋p ⁼1 +(p₋1)N

N₋p , (2.5)

and there exists a positive constant depending only onpandNsuch that

vL^t(R^N−1)≤cu1,p,R^N. (2.6) Proof. W^1,p(R^N)=C^∞_c (R^N). So it suffices to prove the lemma foru∈C^∞_c (R^N). If we set w= |u|^t, thenw∈W^1,1(R^N) and

wL¹(_R^N)≤c∇u^(t_L^p−_(R¹⁾N)uL^p(_R^N), ∂w

∂xj

L¹(R^N)≤c∇u^t_Lp(R^N). (2.7) Indeed, letq=p/(p−1), (t−1)q=N p/(N−p) and by using the Sobolev inequalities, see [6],

u|^t−1

L^q(R^N)≤c∇u^{N p/(N}_Lp(R^N)^−p). (2.8) By H¨older and (2.8),

w1=

R^N|u|^(t−1)|u|dx≤ up|u|^(t−1)

q≤cup∇u^(t_Lp⁻(_R^1)N). (2.9) On the other hand,∂w/∂x_j= ±t|u^(t−1)|(∂u/∂x_j). By H¨older and (2.8),

∂w

∂xj

L¹(R^N)≤tu^(t−1)L^q(_R^N)

∂u

∂xj

L^p(R^N)≤c∇u^t_L^p_(RN), (2.10) wherecis a positive constant.

Now, applying (2.9), (2.10), andLemma 2.3, we find uL^t(R^N−1)≤cuL^p(R^N)+∇uL^p(R^N)

≤cu1,p,R^N. (2.11)

Lemma2.4. Letu∈W^m,p(R^N),N >1,m∈R, andmp < N. For ally∈R,v(x) :=u(x,y)

∈L^pm(R^N−1), withp_m=(N−1)p/(N−mp)and there exists a positive constantcdepend- ing only onpandNsuch that

v_Lpm(R^N−1)≤cum,p,R^N. (2.12) Proof. By applying Sobolev inequality [6] to ∂u/∂xj, 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^pm(R^N−1) with p_m=(N−

1)p/(N−mp).

Theorem2.5. LetΩ⊂R^N,N≥2, be a bounded domain of classC^m. For allu∈W^m,p(Ω), mp < N. The restriction ofuto∂Ωdenoted also byubelongs toL^q(∂Ω), for allq_∈[1,p_m],

p_m=(N−1)p

N−mp (2.13)

and there exists a positive constantcdepending only on p,m, andΩsuch that

up_m(∂Ω)≤cum,p,Ω. (2.14) 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 everyu element ofW^m,p(Ω) is associated an ele- mentP(u)∈W^2,p(R^N). By density, it is sufficient to study the properties of the trace on

∂Ωof the functionC_c^∞(R^N).

Letθj and fj be as in the definition (2.2).∂Ωis compact, therefore we can suppose that there exists a finiteθ_j, 1≤j≤k, which covers∂Ω. Let (P_j, 1≤j≤k) be a parti- tion of unity of∂Ωsubordinate to this covering, see, for example, [1]. Ifu∈C^∞_c (R^N), thenPjuo f_j⁻¹∈C0^m(B). We extendPjuo f_j⁻¹toC^m0(R^N). ByLemma 2.4, the tracewj of Pjuo f_j⁻¹on the hyperplane{(x1,x2,. . .,xN−1, 0),xi∈R}satisfies the inequality

w_jLpm(_R^N−1)≤cP_juo f_j⁻¹_m,p,B≤c_jum,p,R^N, (2.15)

wherecjis a positive constant. We estimate the tracevj:=wjo fjof the functionPjuon Γj:=θj∩∂Ω. Then

v_jΓ^pm

j ≤c_j

R^N−1∩B

w_j^pmdx, (2.16)

wherec_jis a positive constant. We combine (2.15) and (2.16) as follows:

vj

L^pm(Γj)≤Mjum,p,R^N. (2.17)

On the other hand, u=j=k

j=1Pju=j=k

j=1vj, where vj=Pju, and suppvj⊂Γj,∂Ω⊂

j=k j=1Γj. So

u_Lpm(∂Ω)≤

j=k

j=1

vj

L^pm(∂Ω)=

j=k

j=1

vj

L^pm(Γj). (2.18) From (2.18),

u_Lpm(∂Ω)≤



^j=k

j=1

Mj



um,p,R^N. (2.19)

On the other hand,∂Ωis bounded, sou∈L^q(∂Ω), for allq∈[1,p_m].

By usingTheorem 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, W^m,p(Ω)is compactly embedding inL^q(∂Ω)for allq∈[1,p_m[.

Theorem2.7. LetΩ⊂R^N,N≥2, be a bounded domain of classC^m. For allu∈W^m,p(Ω), mp≥N. The restriction ofuto∂Ωdenoted also byubelongs toL^q(∂Ω), for allq∈[1, +∞[.

Proof. Let an arbitraryq∈[1,∞[. We can findqsuch that (a)mq < N;

(b)q=Nq/(N−1 +mq).

From (b),q=q_m=(N−1)q/(N−mq). Sincemq < N, thenq < p(becausemp≥N).

So

(1)W^m,p(Ω) is continuously embedding inW^m,q(Ω).

Sinceq=q_mandmq < N, thus fromTheorem 2.5,

(2)W^m,q(Ω) is continuously embedding inL^t(∂Ω) for allt∈[1,q], and fromCorollary 2.6,

(3)W^m,q(Ω) is compactly embedding inL^t(∂Ω) for allt∈[1,q[.

By combining (1), (2), and (3), we conclude that

(i)W^m,p(Ω) is continuously embedding inL^t(∂Ω) for allt∈[1,q], (ii)W^m,p(Ω) is compactly embedding inL^t(∂Ω) for allt∈[1,q[.

qbeing arbitrary, then we have the desired result.

Theorem2.8. LetΩ⊂R^N,N≥2, be a bounded domain of classC^m,mp > N.W^m,p(Ω)is compactly embeddingL^∞(∂Ω)∩C(∂Ω).

Proof. By using the Sobolev embedding,W^m,p(Ω) is compactly embedding inL^∞(Ω)∩ C(Ω). So the functions ofW^m,p(Ω) are continuous onΩand bounded, therefore their traces are well defined, continuous, and bounded. So we have

(∗)W^m,p(Ω) is compactly embedding inL^∞(Ω)∩C(Ω),

(∗∗)L^∞(Ω)∩C(Ω) is continuously embedding inL^∞(∂Ω)∩C(∂Ω).

By (∗) and (∗∗), we have the desired result.

3. Main results

Through this paper, all solutions are weak, that is,u_∈W^2,p(Ω) is a solution of (1.2), if for allv∈W^2,p(Ω), we have

(S1)∆²_pu,v+_Ω|u|^p−2uv=0;

(S2)−

∂Ω(∂/∂n)(|∆u|^p−2∆u)v=λ_∂Ωρ(x)|u|^p−2uv.

If we replace S2in (S1), then we deduce that

Ω|∆u|^p−2∆u∆v+

Ω|u|^p−2uv=λ

∂Ωρ|u|^p−2uv dσ. (3.1) Ifu∈W^2,p(Ω)− {0}, thenuis called the eigenfunction of (1.2) associated to the eigen- valueλ.

We will use the Ljusternick-Schnirelmann theory onC¹-manifolds [9].

Consider the following two functionals defined onW^2,p(Ω):

A(u)= 1

puW^2,p(Ω), B(u)= 1 p

∂Ωρ(x)|u|^pdσ, (3.2) whereuW^2,p(Ω)=(up+∆up)^{1/ p}. We set

ᏹ=

u∈W^2,p(Ω); pB(u)=1. (3.3) Lemma3.1. (i)AandBare even and of classC¹onW^2,p(Ω).

(ii)ᏹis a closedC¹-manifold.

Proof. (i) It is clear thatAandBare even and of classC¹ onW^2,p(Ω),A(u)=∆²pu+

|u|^p−2u, andB(u)=ρ|u|^p−2u.

(ii)ᏹ=B⁻¹{1/ p}, soBis closed. Its derivative operatorBsatisfiesB(u)=0, for all u∈ᏹ(i.e.,B(u) is onto for allu∈ᏹ), soBis a submersion, thenᏹis aC¹-manifolds.

The following lemma is the key to show the existence.

Lemma3.2. (i)B:W^2,p(Ω)→(W^2,p(Ω))is completely continuous.

(ii)The functionalAsatisfies the Palais-Smale condition onᏹ, that is, for{u_n} ⊂ᏹ, if A(un)is bounded and

n:=Aun

−gnBun

−→0 asn−→+∞, (3.4)

where gn = A(un),un/B(un),un. Then {un}n≥1 has a convergent subsequence in W^2,p(Ω).

Proof. (i)Step1 (definition ofB).

First case. IfN/ p >2,r >(N−1)/(2p−1). Letu,v∈W^2,p(Ω). By H¨older’s inequality, we have

∂Ωρ(x)u(x)^p−2u(x)v(x)dσ≤ ρrus^p−1vp2, (3.5)

wherep₂ =(N−1)p/(N−2p), andsis given by p₋1

s + 1 p₂ +1

r ⁼1. (3.6)

Therefore,

p−1

s ⁼1−1 r⁻

1

p₂ >1−2p−1 N−1 ⁻

N−2p (N−1)p⁼

p−1

p₂ . (3.7)

Then it suffices that

max(1,p−1)< s < p2 (3.8)

andBis well defined.

Second case. IfN/ p=2,r >(N−1)/(2p−1). In this case, fromTheorem 2.7,

W^2,p(Ω)L^q(∂Ω) (3.9)

for anyq∈[1, +∞[. There isq≥1 such that 1

q+1 r+ p−1

p ⁼ 1 q+1

r+ 1

p⁼1. (3.10)

We obtain that

1 q ⁼1−

1 r+ 1

p

≤1. (3.11)

By H¨older’s inequality, we arrive at

∂Ωρ(x)u(x)^p−2u(x)v(x)dx≤ ρru^pp⁻¹vq (3.12) for anyu,v∈W^2,p(Ω). Then in this case,Bis well defined.

Third case. IfN/ p <2,r=1. In this case, fromTheorem 2.8,

W^2,p(Ω)C(∂Ω)∩L^∞(∂Ω). (3.13)

Therefore for anyu,v∈W^2,p(Ω), we have

∂Ωρ(x)u(x)^p−2u(x)v(x)dx<∞, (3.14) withρ∈L¹(Ω), andBis well defined also in this case.

Step 2.Bis completely continuous. Let (u_n)⊂W^2,p(Ω) be a sequence such thatu_n→u weakly inW^2,p(Ω). We must show thatB(un)→B(u) strongly in (W^2,p(Ω)), that is,

sup

v∈W2,p(Ω) v2,p≤1

∂Ωρu_n^p−2u_n− |u|^p−2uv dx−→0 asn−→+∞. (3.15) For this end, we distinguish three cases as in Step 1 above forN/ p >2, andr >(N− 1)/(2p−1). Letsbe as in (3.8). Then,

sup

v∈W2,p(Ω) v2,p≤1

∂Ωρun^p−2un− |u|^p−2uvdx

≤ sup

v∈W2,p(Ω) v2,p≤1

ρ_ru_n^p−2u_{n− |}u_|^p−2u

s/(p−1)v_p2

≤cρru_n^p−2u_n− |u|^p−2u_s/(p−1),

(3.16)

wherecis the constant of Sobolev’s embedding [1].

On other hand, the Nemytskii’s operatoru→ |u|^p−2uis continuous fromL^s(∂Ω) into L^s/(p−1)(∂Ω), andu_n→uweakly inW^2,p(Ω). So, we deduce thatu_n→ustrongly inL^s(∂Ω) becauses < p₂. Hence,

u_n^p−2u_{n− |}u_|^p−2u

s/(p−1)−→0, asn_−→+∞. (3.17) This completes the proof of the claim in this case.

IfN/ p=2,

∂Ωρu_n^p−2u_n− |u|^p−2uvdx≤ ρru_n^p−2u_n− |u|^p−2u^p−1

p vq, (3.18) whereqis given by (3.11). By Sobolev’s trace embedding, there existsc >0 such that

vq≤cv2,p, ∀v∈W^2,p(Ω). (3.19) Thus,

sup

v∈W2,p(Ω) v2,p≤1

∂Ωρun^p−2un− |u|^p−2uv dx≤cρrun^p−2un− |u|^p−2u^p−1

p . (3.20)

From the continuity ofu→ |u|^p−1ufrom L^p(∂Ω) into L^p(∂Ω), and from the compact embedding ofW^2,p(Ω) inL^p(∂Ω), we have the desired result.

IfN/ p <2,r=1.W^2,p(Ω)C(∂Ω), we obtain sup

v∈W2,p(Ω) v2,p≤1

∂Ωρun^p−2un− |u|^p−2uv dx≤cρ1sup

∂Ω

un^p−2un− |u|^p−2u,

(3.21) wherecis the constant given by embedding ofW^2,p(Ω) inC(∂Ω)∩L^∞(∂Ω).

It is clear that sup

∂Ω

u_n^p−2u_n− |u|^p−2u−→0, asn−→+∞. (3.22)

HenceBis completely continuous also in this case.

(ii){u_n}is bounded inW^2,p(Ω). Hence without loss of generality, we can assume that unconverges weakly inW^2,p(Ω) to some functionu∈W^2,p(Ω) andun2,p→c. For the rest, we distinguish two cases. Ifc=0, thenunconverges strongly to 0 inW^2,p(Ω).

Ifc=0, the claim is to prove thatu_nis of Cauchy inW^2,p(Ω).

Set

Gun,um

= Aun

−Aum

,un−um , G1

un,um

=

∆²pun−∆²pum,un−um , G2

un,um

=un^p−2un−um^p−2um,un−um

.

(3.23)

We remark that

Gun,um

=G1

un,um

+G2

un,um

. (3.24)

On the other hand,

Gun,um

= Aun

−Aum

,un−um

=

n−m,un−um

+hn−hm,un−um

, (3.25)

withndefined as in (3.4), andhn= u2,p^p B(un).

Gu_n,u_m≤n−m

∗u_n−u_m2,p+h_n−h_m∗u_n−u_m2,p, (3.26) where · ∗is the dual norm associated to · 2,p.

This implies thath_n converges, for a subsequence if necessary, inW^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_n2,p→c_≥0. It follows that (h_n)n≥0is con- vergent in (W^2,p(Ω)). Then,

Gun,um

−→0, asn−→+∞. (3.27)

From [7], we have the inequality

t1−t2^p≤ct1^p−2t1−t2^p−2t2

· t1−t2

γ/2t1^p+t2^p1−γ/2

, (3.28) for anyt1,t2∈R, withγ=pif 1< p <2 andγ=2 ifp≥2. By applying H¨older’s inequal- ity, we deduce that

∆un−∆um^p

p≤cG1

un,umγ/2∆un^p

p+∆um^p

p)^1−γ/2, (3.29) un−um^p

p≤cG2

un,um

γ/2un^p

p+um^p

p

1−γ/2

, (3.30)

wherecis a positive constant independent ofnandm,γ=p if 1< p <2, andγ=2 if p≥2.

From [7], we have

un^p−2un−um^p−2um un−um

≥cun−um^(−γ+p+2) |u_n+u_m^(2−γ), ∆un^p−2∆un−∆um^p−2∆um

∆un−∆um

≥c ∆un−∆um^(−γ+p+2) ∆u_n+∆u_m^(2−γ),

(3.31)

whereγ=pif 1< p <2 andγ=2 ifp≥2. By integrating these two relations overΩ, we find

G1

u_n,u_m≥0, G2

u_n,u_m≥0. (3.32)

On the other hand,G1≤GandG2≤G. Then from (3.27) and (3.32), G1

u_n,u_m)−→0 asn−→ ∞, G2

u_n,u_m−→0 asn−→ ∞. (3.33) Then from (3.29) and (3.30),

∆u_n−∆u_mp−→0 asn−→ ∞, u_n−u_mp−→0 asn−→ ∞. (3.34) So

u_n−u_m2,p^p −→0 asn−→ ∞. (3.35) Therefore (u_n)_nis a Cauchy’s sequence inW^2,p(Ω). This achieves the proof of the lemma.

Set

Γk=

K⊂ᏹ:Kis symmetric, compact andγ(K)≥k, (3.36) whereγ(K)=kis the genus ofK, that is, the smallest integerksuch that there exists an odd continuous map fromKtoR^k− {0}.

Now, by the Ljusternick-Schnirelmann theory, see, for example, [9], we have our main result.