SUBCRITICAL BIHARMONIC EQUATION

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SUBCRITICAL BIHARMONIC EQUATION

KHALIL EL MEHDI

Received 29 October 2004; Accepted 20 January 2005

We consider a biharmonic equation under the Navier boundary condition and with a nearly critical exponent (Pε):Δ²u=u⁹⁻^ε,u >0 inΩandu=Δu=0 on∂Ω, whereΩis a smooth bounded domain inR⁵,ε >0. We study the asymptotic behavior of solutions of (Pε) which are minimizing for the Sobolev quotient asεgoes to zero. We show that such solutions concentrate around a pointx0∈Ωasε→0, moreoverx0is a critical point of the Robin’s function. Conversely, we show that for any nondegenerate critical pointx0of the Robin’s function, there exist solutions of (Pε) concentrating aroundx0asε→0.

Copyright © 2006 Khalil El Mehdi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction and results

Let us consider the following biharmonic equation under the Navier boundary condition Δ²u=u^p⁻^ε, u >0 inΩ

Δu=u=0 on∂Ω, (Qε) whereΩis a smooth bounded domain inRⁿ,n≥5,εis a small positive parameter, and p+ 1=2n/(n−4) is the critical Sobolev exponent of the embeddingH²(Ω)∩H0¹(Ω) L²^n/⁽ⁿ⁻⁴⁾(Ω).

It is known that (Q_ε) is related to the limiting problem (Q0) (whenε=0) which ex- hibits a lack of compactness and gives rise to solutions of (Qε) which blow up asε→0.

The interest of the limiting problem (Q0) grew from its resemblance to some geometric equations involving Paneitz operator and which have widely been studied in these last years (for details one can see [4,6,10,12–14,17] and references therein).

Several authors have studied the existence and behavior of blowing up solutions for the corresponding second order elliptic problem (see, e.g., [1,3,9,18,21,22,24–26]

and references therein). In sharp contrast to this, very little is known for fourth order elliptic equations. In this paper we are mainly interested in the asymptotic behavior and

Hindawi Publishing Corporation Abstract and Applied Analysis

Volume 2006, Article ID 18387, Pages1–20 DOI10.1155/AAA/2006/18387

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the existence of solutions of (Qε) which blow up around one point, and the location of this blow up point asε→0.

The existence of solutions of (Q_ε) for allε∈(0,p−1) is well known for any domainΩ (see, e.g., [16]). Forε=0, the situation is more complex, Van Der Vorst showed in [28]

that ifΩis starshaped (Q0) has no solution whereas Ebobisse and Ould Ahmedou proved in [15] that (Q0) has a solution provided that some homology group ofΩis nontrivial.

This topological condition is suﬃcient, but not necessary, as examples of contractible domainsΩon which a solution exists show [19].

In view of this qualitative change in the situation whenε=0, it is interesting to study the asymptotic behavior of the subcritical solutionuεof (Qε) asε→0. Chou and Geng [11], and Geng [20] made a first study, whenΩis strictly convex. The convexity assumption was needed in their proof in order to apply the method of moving planes (MMP for short) in proving a priori estimate near the boundary. Notice that in the Laplacian case (see [21]), the MMP has been used to show that blow up points are away from the boundary of the domain. The process is standard if domains are convex. For nonconvex regions, the MMP still works in the Laplacian case through the applications of Kelvin transformations [21]. For (Qε), the MMP also works for convex domains [11]. How- ever, for nonconvex domains, a Kelvin transformation does not work for (Q_ε) because the Navier boundary condition is not invariant under the Kelvin transformation of biha rmonic operator. In [5], Ben Ayed and El Mehdi removed the convexity assumption of Chou and Geng for higher dimensions, that isn≥6. The aim of this paper is to prove that the results of [5] are true in dimension 5. In order to state precisely our results, we need to introduce some notations.

We consider the following problem

Δ²u=u⁹⁻^ε, u >0 inΩ

Δu=u=0 on∂Ω, (Pε) whereΩis a smooth bounded domain inR⁵andεis a small positive parameter.

Let us define onΩthe following Robin’s function

ϕ(x)=H(x,x), withH(x,y)= |x−y|⁻¹−G(x,y), for (x,y)∈Ω×Ω, (1.1) whereGis the Green’s function ofΔ², that is,

∀x∈Ω Δ²G(x,·)=cδx inΩ

ΔG(x,·)=G(x,·)=0 on∂Ω, (1.2) whereδxdenotes the Dirac mass atxandc=3ω5, withω5is the area of the unit sphere ofR⁵. Forλ >0 anda∈R⁵, let

δ_a,λ(x)= c0λ¹^/²

1 +λ²|x−a|²1/2, c0=(105)¹^/⁸. (1.3) It is well known (see [23]) thatδa,λare the only solutions of

Δ²u=u⁹, u >0 inR⁵ (1.4)

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and are also the only minimizers of the Sobolev inequality on the whole space, that is S=inf|Δu|²_L²₍_R⁵₎|u|⁻_L¹⁰²₍_R⁵₎, s.t.Δu∈L²,u∈L¹⁰,u=0. (1.5) We denote byPδa,λthe projection ofδa,λonᏴ(Ω) :=H²(Ω)∩H0¹(Ω), defined by

Δ²Pδ_a,λ=Δ²δ_a,λ inΩ, ΔPδa,λ=Pδ_a,λ=0 on∂Ω. (1.6) Let

θ_a,λ=δ_a,λ−Pδ_a,λ, u =

Ω|Δu|² 1/2

, u,v =

ΩΔuΔv, u,v∈H²(Ω)∩H0¹(Ω) uq= |u|L^q(Ω).

(1.7)

Thus we have the following result.

Theorem 1.1. Let (uε) be a solution of (Pε), and assume that uε ² uε ⁻²

10₋ε−→S asε−→0, (H)

whereSis the best Sobolev constant inR⁵defined by (1.5). Then (up to a subsequence) there exista_ε∈Ω,λ_ε>0,α_ε>0 andv_εsuch thatu_εcan be written as

uε=αεPδaε,λε+vε (1.8) withαε→1,vε →0,aε∈Ωandλεd(aε,∂Ω)→+∞asε→0.

In addition,aεconverges to a critical pointx0∈Ωofϕand we have limε→0εuε ²

L^∞(Ω)= c1c²0/c2

ϕx0

, (1.9)

wherec1=c¹⁰0

R⁵(dx/(1 +|x|²)⁹^/²),c2=c¹⁰0

R⁵(log(1 +|x|²)(1− |x|²)/(1 +|x|²)⁶)dxand c0=(105)¹^/⁸.

Our next result provides a kind of converse toTheorem 1.1.

Theorem 1.2. Assume thatx0∈Ωis a nondegenerate critical point ofϕ. Then there exists anε0>0 such that for eachε∈(0,ε0], (Pε) has a solution of the form

uε=αεPδaε,λε+vε (1.10) withα_ε→1,v_ε →0,a_ε→x0andλ_εd(a_ε,∂Ω)→+∞asε→0.

Our strategy to prove the above results is the same as in higher dimensions. However, as usual in elliptic equations involving critical Sobolev exponent, we need more refined estimates of the asymptotic profiles of solutions whenε→0 to treat the lower dimensional case. Such refined estimates, which are of self interest, are highly nontrivial and use in a crucial way careful expansions of the Euler-Lagrange functional associated to (Pε), and its gradient near a small neighborhood of highly concentrated functions. To perform such

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expansions we make use of the techniques developed by Bahri [2] and Rey [25,27] in the framework of the Theory of critical points at infinity.

The outline of the paper is the following: inSection 2we perform some crucial estimates needed in our proofs andSection 3is devoted to the proof of our results.

2. Some crucial estimates

In this section, we prove some crucial estimates which will play an important role in proving our results. We first recall some results.

Proposition 2.1 [8]. Leta∈Ωandλ >0 such thatλd(a,∂Ω) is large enough. Forθ(a,λ)= δ(a,λ)−Pδ(a,λ), we have the following estimates

0≤θ(a,λ)≤δ(a,λ), θ(a,λ)=c0λ⁻¹^/²H(a,·) +f(a,λ), (2.1) where f(a,λ)satisfies

f(a,λ)=O 1

λ⁵^/²d³

, λ∂ f(a,λ)

∂λ ⁼O 1

λ⁵^/²d³

, 1

λ

∂ f(a,λ)

∂a ⁼O 1

λ⁷^/²d⁴

, (2.2) wheredis the distanced(a,∂Ω),

θ(a,λ)

L¹⁰=O(λd)⁻¹^/², θ(a,λ) =O(λd)⁻¹^/², λ∂θ(a,λ)

∂λ

L¹⁰=O 1

(λd)¹^/²

, 1

λ

∂θ(a,λ)

∂a

L¹⁰=O 1

(λd)³^/²

. (2.3)

Proposition 2.2 [5]. Letuε be a solution of (Pε) which satisfies (H). Then, there exist a_ε∈Ω,α_ε>0,λ_ε>0 andv_εsuch that

uε=αεPδaε,λε+vε (2.4) withαε→1,λεd(aε,∂Ω)→ ∞,c⁻0²uε²_∞/λε→1,uε^ε_∞→1 andvε →0.

Furthermore,v_ε∈E(aε,λε)which is the set ofv∈Ᏼ(Ω) such that v,Pδaε,λε

=

v,∂Pδaε,λε/∂λε

=0, vε,∂Pδaε,λε/∂a=0. (V0) Lemma 2.3 [5]. λ^ε_ε=1 +o(1) asεgoes to zero implies that

δ_ε⁻^ε−c⁻0^ελ^ε_ε⁽⁴⁻ⁿ⁾^/²=Oεlog1 +λ²_εx−aε²

inΩ, (2.5)

whereδ_ε=δ_aε,λεandd_ε=d(a_ε,∂Ω).

Proposition 2.4 [5]. Let (u_ε) be a solution of (P_ε) which satisfies (H). Thenv_εoccurring inProposition 2.2satisfies

v_ε≤Cε+λ_εd_ε⁻¹, (2.6) whereCis a positive constant independent ofε.

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Now, we are going to state and prove the crucial estimates needed in the proof of our theorems. In order to simplify the notations, we setδε=δaε,λε,Pδε=Pδaε,λε,θε=θaε,λε

andd_ε=d(a_ε,∂Ω).

Lemma 2.5. Forεsmall, we have the following estimates

(i)_Ωδ_ε⁹(1/λε)(∂Pδε/∂a)= −(c1/2λ²_ε)(∂H/∂a)(aε,aε) +O(1/(λεdε)³),

(ii)_ΩPδ_ε⁹⁻^ε(1/λε)(∂Pδε/∂a)= −(c1/λ²⁺_ε ^ε/²)(∂H/∂a)(aε,aε) +O(1/(λεdε)³+ε/(λεdε)²), wherec1is the constant defined inTheorem 1.1.

Proof. Notice that

Ω\Bε

δ_ε¹⁰=O 1

(λεdε)⁵

. (2.7)

Thus, we have, for 1≤k≤5

Ωδ⁹_ε 1 λ_ε

∂Pδε

∂a_k ⁼

Ωδ_ε⁹1 λ_ε

∂δε

∂a_k⁻

Ωδ_ε⁹1 λ_ε

∂θε

∂a_k ^{= −}

Bε

δ⁹_ε 1 λ_ε

∂θε

∂a_k+O 1

(λ_εd_ε)⁵

, (2.8) whereBε=B(aε,dε). Expanding∂θε/∂akaroundaεand usingProposition 2.1, we obtain

Bε

δ_ε⁹1 λε

∂θ_ε

∂ak= c0

2λ³ε^/²

∂Haε,aε

∂a

Bε

δ⁹_ε+O

⎛

⎝ 1 λεdε

3

⎞

⎠. (2.9)

Estimating the integral on the right-hand side in (2.9) and using (2.8), we easily derive claim (i). To prove claim (ii), we write

ΩPδ_ε⁹⁻^ε 1 λε

∂Pδε

∂ak =

Ωδ_ε⁹⁻^ε1 λε

∂δε

∂ak−

Ωδ⁹_ε⁻^ε 1 λε

∂θε

∂ak−(9−ε)

Ωδ⁸_ε⁻^εθε1 λ

∂δε

∂ak

+(9−ε)(8−ε) 2

Ωδ_ε⁷⁻^εθ_ε²1 λ

∂δ_ε

∂ak+O

Ωδ⁸_ε⁻^εθε

1 λε

∂θ_ε

∂ak

+

δ_ε⁷⁻^εθ_ε³

(2.10) and we have to estimate each term on the right-hand side of (2.10).

UsingProposition 2.1andLemma 2.3, we have

Ωδ_ε⁷⁻^εθ_ε³≤cθε ³

∞

δ_ε⁷=O

1 (λεdε)³

,

Ωδ_ε⁸⁻^εθε

1 λ_ε

∂θε

∂a_k

≤cθε

∞

1

λ_ε

∂θε

∂a_k

∞

Ωδ_ε⁸=O 1

(λ_εd_ε)³

.

(2.11)

We also have

Ωδ⁹_ε⁻^ε 1 λε

∂δε

∂ak =

Ω\Bε

δ_ε⁹⁻^ε1 λε

∂δε

∂ak =O 1

(λεdε)⁵

. (2.12)

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Expandingθεaroundaεand usingProposition 2.1andLemma 2.3, we obtain 9

Bε

δ_ε⁸⁻^εθ_ε 1 λ_ε

∂δε

∂a_k ⁼ c1

2λ²⁺ε ^ε/²

∂H(aε,aε)

∂a +O 1

(λ_εd_ε)³+ ε (λ_εd_ε)²

,

Bε

δ_ε⁷⁻^εθ²_ε 1 λε

∂δ_ε

∂ak =O 1

(λεdε)³

.

(2.13)

In the same way, we find

Ωδ_ε⁹⁻^ε1 λ_ε

∂θε

∂a_k ⁼ c1

2λ²⁺ε ^ε/²

∂Ha_ε,a_ε

∂a +O

⎛

⎝ 1

λ_εd_ε³+ ε λ_εd_ε²

⎞

⎠. (2.14)

Combining (2.10)–(2.14), we obtain claim (ii).

To improve the estimates of the integrals involving vε, we use an idea of Rey [27], namely we write

vε=Πvε+wε, (2.15)

whereΠvεdenotes the projection ofvεontoH²∩H0¹(Bε), that is

Δ²Πvε=Δ²vε inBε; ΔΠvε=Πvε=0 on∂Bε, (2.16) whereB_ε=B(a_ε,d_ε). We splitΠvεin an even partΠv^e_εand an odd partΠv_ε^owith respect to (x−aε)_k, thus we have

v_ε=Πvê_ε+Πvô_ε+w_ε inB_εwithΔ²w_ε=0 inB_ε. (2.17) Notice that it is difficult to improve the estimate (2.6) of thev_ε-part of solutions. However, it is sufficient to improve the integrals involving the odd part ofvε with respect to (x− aε)_k, for 1≤k≤5 and to know the exact contribution of the integrals containing the w_ε-part ofv_ε. Let us start by the terms involvingw_ε.

Lemma 2.6. Forεsmall, we have that

Bε

δ_ε⁸

δ_ε⁻^ε− 1 c^ε0λ^ε/2ε

1 λ_ε

∂δε

∂a_kw_ε=O

⎛

⎝ εv_ε λεdε

_1/2

⎞

⎠. (2.18)

Proof. Letψbe the solution of Δ²ψ=δ⁸_ε

δ_ε⁻^ε− 1 c0^ελ^ε/ε²

1 λε

∂δ_ε

∂ak inBε; Δψ=ψ=0 on∂Bε. (2.19) Thus we have

Iε:=

Bε

Δ²ψwε=

∂Bε

∂ψ

∂νΔwε+

∂Bε

∂Δψ

∂ν wε. (2.20)

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LetGεbe the Green’s function for the biharmonic operator onBεwith the Navier boundary conditions, that is,

Δ²Gε(x,·)=cδx inBε; ΔGε(x,·)=G(x,·)=0 on∂Bε, (2.21) wherec=3w5. Thereforeψis given by

ψ(y)=

Bε

G_ε(x,y)δ_ε⁸

δ_ε⁻^ε− 1 c0^ελ^ε/ε²

1 λ_ε

∂δε

∂a_k, y∈B_ε (2.22) and its normal derivative by

∂ψ

∂ν(y)=

Bε

∂G_ε

∂ν (x,y)δ_ε⁸

δ_ε⁻^ε− 1 c^ε0λ^ε/ε²

1 λε

∂δ_ε

∂ak, y∈∂Bε. (2.23) Notice that fory∈∂B_εwe have the following estimates: forx∈B_ε\B(y,d_ε/2), we have

∂Gε

∂ν (x,y)=O 1

d²_ε

; ∂ΔGε

∂ν (x,y)=O 1

d⁴_ε

(2.24) forx∈Bε∩B(y,dε/2), we have

∂Gε

∂ν (x,y)≤ c

|x−y|²; ∂ΔGε

∂ν (x,y)≤ c

|x−y|⁴ (2.25) forx∈Bε∩B(y,dε/2), we have

δ_ε⁸

δ_ε⁻^ε− 1 c^ε0λ^ε/ε²

1 λε

∂δε

∂ak =O

εlogλεdε

(λεdε)⁹

, (2.26)

forx∈Bε\B(y,dε/2), we have δ_ε⁸

δ_ε⁻^ε− 1 c^ε0λ^ε/ε²

1 λε

∂δ_ε

∂ak =Oδ_ε⁹εlog1 +λ²_εx−aε²

. (2.27)

Therefore

∂ψ

∂ν(y)=O ε

λ¹ε^/²d²_ε

. (2.28)

In the same way, we have

∂Δψ

∂ν (y)=O ε

λ^1/2ε d_ε⁴

. (2.29)

Using (2.20), (2.28), (2.29), we obtain Iε=O

ε λ¹ε^/²d_ε²

∂Bε

Δwε+ ε λ¹ε^/²d_ε⁴

∂Bε

wε

. (2.30)

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To estimate the right-hand side of (2.30), we introduce the following function

w(X)¯ =d_ε¹^/²w_εa_ε+d_εX, v(X)¯ =d¹_ε^/²v_εa_ε+d_εX forX∈B(0, 1). (2.31) w¯satisfies

Δ²w¯=0 inB:=B(0, 1); Δw¯=Δv,¯ w¯=v¯on∂B. (2.32) We deduce that

∂B|Δw¯|+

∂B|Δw¯| ≤C

B|Δv¯|² 1/2

=C

Bε

Δvε²1/2

. (2.33)

But, we have

∂B|Δw¯|+

∂B|Δw¯| = 1

dε

3/2

∂Bε

Δwε+ 1

dε

7/2

∂Bε

wε. (2.34)

Using (2.30), (2.33) and (2.34), the lemma follows.

Lemma 2.7. Forεsmall, we have

(i)_B_εΔ((1/λε)(∂Πδε/∂ak))Δwε=O(vε/(λεdε)³^/²), (ii)_B_εδ_ε⁸⁻^εΠv^o_εw_ε=O(v_εΠv_ε^o/(λ_εd_ε)¹^/²).

Proof. Using (2.17), we obtain

Bε

Δ 1

λε

∂Πδ_ε

∂ak

Δwε=

∂Bε

∂ψk

∂ν Δwε, withψk= 1 λε

∂Πδ_ε

∂ak . (2.35) Using an integral representation forψkas in (2.23), we obtain fory∈∂Bε,

∂ψ

∂ν(y)=

Bε

∂Gε

∂ν (x,y)Δ²ψ_k, (2.36)

whereGεis the Green’s function defined in (2.21). Clearly, we have Πδε(x)=δ_ε(x)− c0λ¹_ε^/²

1 +λ²_εd²_ε¹^/²⁻

c_εa_ε,d_ε

10 x−a_ε²−d²_ε, (2.37) withc_ε(a_ε,d_ε)=Δδε|∂Bε. Thus we deduce that

∂ψ

∂ν(y)=9

Bε

∂Gε

∂ν (x,y)δ_ε⁸1 λ_ε

∂δε

∂a_k. (2.38)

InB_ε\B(a_ε,d_ε/2), we argue as in (2.28) and (2.25), we obtain

Bε

∂Gε

∂ν (x,y)δ⁸_ε 1 λε

∂δε

∂ak =O 1

λ⁹ε^/²d_ε⁶

. (2.39)

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Furthermore, since ∇∂Gε

∂ν (x,y)=O 1

d³_ε

for (x,y)∈Ba_ε,d_ε/2×∂B_ε, (2.40) we obtain

B(aε,dε/2)

∂G_ε

∂ν (x,y)δ_ε⁸1 λε

∂δ_ε

∂ak

≤ c d³_ε

B(aε,dε/2)δ_ε⁹x−aε=O 1

λ³ε^/²d_ε³

, (2.41) where we have used the evenness ofδεand the oddness of its derivative. Thus

∂ψ_k

∂ν (y)=O 1

λ^3/2ε d_ε³

. (2.42)

Using (2.35) and (2.42), we obtain

Bε

Δ 1

λ_ε

∂Πδε

∂a_k

Δwε≤ c λ³ε^/²d³_ε

∂Bε

Δwε. (2.43)

Arguing as in (2.34), claim (i) follows. To prove claim (ii), letψbe such that

Δ²ψ=δ_ε⁸⁻^εΠv^o_ε inB_ε; Δψ=ψ=0 on∂B_ε. (2.44) We have

Bε

δ⁸_ε⁻^εΠv_ε^owε=

∂Bε

∂Δψ

∂ν wε+

∂Bε

∂ψ

∂νΔwε. (2.45)

As before, we prove that, fory∈∂Bε

∂ψ

∂ν(y)=OΠv_ε^o λ¹ε^/²d_ε²

, ∂Δψ

∂ν (y)=OΠv^o_ε λ¹ε^/²d⁴_ε

. (2.46)

Therefore

Bε

δ_ε⁸⁻^εΠv^o_εwε≤cΠv_ε^o λ¹ε^/²d_ε⁴

1 δε³^/²

∂Bε

wε+ 1 δε⁷^/²

∂Bε

Δwε

≤cvε Πv_ε^o λεdε

1/2 . (2.47)

The proof of the lemma is completed.

Lemma 2.8. Forεsmall, we have

(i)_B_εδ_ε⁷⁻^εvε(1/λε)(∂δε/∂ak)=O(Πv^o_ε/λ¹_ε^/²+vε/λεd_ε¹^/²), (ii)_B_εδ_ε⁷⁻^εθεvε(1/λε)(∂δε/∂ak)=O(Πv_ε^o/λεdε+vε/(λεdε)³^/²).

Proof. Claim (i) can be proved in the same way asLemma 2.6, so we omit its proof. Claim

(ii) follows fromProposition 2.1and claim (i).