Solutions with interior bubble and boundary layer for an elliptic Neumann problem with critical nonlinearity

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http://france.elsevier.com/direct/CRASS1/

Partial Differential Equations

Solutions with interior bubble and boundary layer for an elliptic Neumann problem with critical nonlinearity

Juncheng Wei

^a

, Shusen Yan

^b

aDepartment of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong

bSchool of Mathematics, Statistics and Computer Science, The University of New England, Armidale, NSW 2351, Australia Received 14 March 2006; accepted after revision 17 May 2006

Available online 14 August 2006 Presented by Pierre-Louis Lions

Abstract

We study positive solutions of the equation²u−u+uⁿⁿ⁻²⁺² =0, wheren=3,4,5 and >0 is small, with Neumann boundary condition in a unit ballB. We prove the existence of solutions with an interior bubble at the center and a boundary layer at the boundary∂B.To cite this article: J. Wei, S. Yan, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

Résumé

Solution explosant au centre et le long du bord pour un problème elliptique de Neumann avec non-linéarité critique.

Nous considérons le problème²u−u+uⁿⁿ⁺⁻²² =0,u >0,dans la boule unitéBdeRⁿoùn=3,4,5, >0 est petit etuvérifie les conditions au bord de Neumann. Nous montrons l’existence d’une solution radiale se concentrant au centre et le long de la frontière deBquandtend vers 0.Pour citer cet article : J. Wei, S. Yan, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

Version française abrégée

SoientΩ⊂Rⁿ,n3,p >1 et >0 petit. On considère le problème ²u−u+u^p=0, u >0 dansΩ, ∂u

∂ν =0 sur∂Ω. (1)

Si p <ⁿ_n⁺₋²₂, ou sait qu’il existe des solutions se concentrant en des points situés à l’intérieur ou sur la frontière du domaine quand tend vers 0 (voir [6,7]). Si p=ⁿ_n⁺₋²₂, l’existence de solutions explosant en un point [12] ou plusieurs points [5,8] du bord est counue. La possibilité d’explosions à l’intérieur du domaine demeure ouverte, même si l’existence de solutions se concentrant exclusivement loin du bord est exclue (voir [3,13]). Par ailleurs, Malchiodi et Montenegreto [9] ont établi l’existence de solutions qui explosent sur tout le bord, pour toutp >1 (au moins pour une suite_k tendant vers 0).

E-mail addresses:[email protected] (J. Wei), [email protected] (S. Yan).

doi:10.1016/j.crma.2006.07.010

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Dans cette Note, nous considérons le cas particulier avec exposant critique, ²u−u+n(n−2)uⁿ⁺²ⁿ⁻² =0, u >0 dansB, ∂u

∂ν =0 sur∂B (2)

oùB est la boule unité deRⁿavecn=3, 4, 5. Nous montrons l’existence d’une solution radialeu qui explose à la fois au centre deB et le long de sa frontière quandtend vers 0. Plus précisément, soitw la famille de solutions radiales de (2) qui explosent le long du bord [9]. Asymptotiquementwse comporte commew(¹^−|^y^|), oùwl’unique solution de l’équation différentielle :

w−w+n(n−2)wⁿⁿ⁺⁻²² =0, w >0 dansR, w(0)=max

t∈R¹w(t ), w(t )→0, quand|t| → +∞.

Ou sait d’autre part que les solutions deu+n(n−2)u⁽ⁿ⁺^2)/(n⁻²⁾=0,u >0 dansRⁿ s’écrivent sous la forme U_a,λ(x)=( ^λ

1+λ²|x−a|²)ⁿ⁻²² ,λ >0,a∈Rⁿ. Notre résultat principal s’énonce ainsi :

Théorème.Soitn=3,4ou5. Pour >0assez petit, le problème(2)admet une solution radiale telle que u=⁽ⁿ⁻^2)/2U0,λ+w+o(1), avecλ=e⁽²⁺^o(1))/sin=3etλ=e⁽⁶⁻ⁿ⁺^o(1))/2sin=4,5.

1. Introduction and statement of main result

In recent years, there have been many works devoted to the study of the following singularly perturbed Neumann problem:

²u−u+u^p=0, u >0 inΩ, ∂u

∂ν =0 on∂Ω (3)

whereΩ⊂Rⁿ,p >1 and >0 is small. Problem (3) arises in the study of many reaction–diffusion systems in chemistry or biology, see [11] and the references therein for backgrounds and progress up to 2004.

Whenp < ⁿ_n⁺₋²₂, it is known that there are many solutions with point condensations in the interior or on the boundary: for example, Gui and Wei [6] proved that given any two positive integersl₁, l₂, there are solutions to (3) withl₁ interior spikes andl₂boundary spikes. Lin, Ni and Wei [7] showed that there are at leastn(|ln^C|)ⁿ number of interior spike solutions. Whenp=ⁿ_n⁺₋²₂, it is known that nonconstant solutions exist forsmall enough [1], and the least energy solution blows up, as→0, at a point which maximizes the mean curvature of the boundary [2]. Higher energy solutions have also been exhibited, blowing up at one [12] or several (separated) boundary points [5,8]. However, the question of existence ofinterior blow-upsolutions is still open. Under some assumptions, it is proved in [3] and [13]

that there are no interior bubble solutions. In another direction, Malchiodi and Montenegro [9] proved that there exists solutions concentrating on the whole boundary (at least along a subsequence_k→0). This boundary layer solution exists foranyp >1.

In this Note, we show that in the critical casep=ⁿ_n⁺₋²₂, one can build up an interior bubble solution on the top of the boundary layer solutions, at least when the domain is the unit ball and the dimensionn=3, 4 or 5. Namely, we consider the following

²u−u+n(n−2)uⁿⁿ⁺⁻²² =0, u >0 inB, ∂u

∂ν =0 on∂B (4)

whereB is the unit ball inRⁿ centered at the origin andn=3, 4, 5. Note that interior bubble has zero dimension concentration set while the boundary layer hasn−1 dimensional concentration set. Our solution constructed in this paper has both point condensation and(n−1)-dimensional concentration. This type of solution is new.

To state our results, we need to introduce two functions. First, it is known that problem (4) has radial symmetric solution concentrating atr=1. (This is actually true forgeneral domain. See [9].) This boundary layer solution is denoted asw. Asymptotically,w≈w(¹^−|^y^|), wherewis the unique solution satisfying

w−w+n(n−2)wⁿ⁺²ⁿ⁻² =0, w >0 inR¹, w(0)=max

t∈R¹w(t ), w(t )→0, as|t| → +∞. (5) On the other hand, it is well-known that the functionsUa,λ(x)=( ^λ

1+λ²|x−a|²)ⁿ⁻²²,λ >0, a∈Rⁿ are the only solutions to the problemu+n(n−2)u⁽ⁿ⁺^2)/(n⁻²⁾=0,u >0 inRⁿ.

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The main result in this Note is:

Theorem 1.Letn=3,4or5. There exists an₀>0such that for∈(0, 0), problem(4)has a radially symmetric solution of the form

u=⁽ⁿ⁻^2)/2U_0,λ+w+o(1), withλ=e⁽²⁺^o(1))/when n=3andλ=e⁽²⁺^o(1))/((6⁻ⁿ⁾⁾whenn=4,5.

Remarks.1. From the calculations in this paper, we can see that (4) does not have a solution which just has an interior bubble at the origin. Our main result shows that it is the boundary layer that creates a solution with a bubble at the origin. Whenn6, our computations suggest that such kind of solutions don’t exist.

2. In [10], it is proved that problem (4) has radial solutions concentrating on arbitrarily many spheres nearr=1.

We can also show the existence of an interior bubble solution on the top of clustered boundary interface solutions.

See [16].

3. We believe that Theorem 1 also holds in general domains. We conjecture that one can add one (maybe many) interior bubbles to the boundary layer solution constructed in [9], in the lower dimension casen=3,4,5.

The present Note concerns partial results obtained in [16], where the procedure and the proof of more general theorems is carried out in full detail.

2. Error estimates and energy computations for approximate solutions

We first analyze the boundary layer solutionw. We have

Lemma 2.As→0,−logw(|x|)→1−|x|uniformly for|x|¹₂. Furthermore, the linearized operatorL_,1(φ):=

²φ−φ+n(n+2)w

n−24

φis an invertible operator fromH_r,ν² (B)=H²(B)∩ {u=u(r),^∂u_∂ν =0onB}toL²(B).

Letλbe such thatλ∈Λ:=(e^an−^δ,e^an+^δ)wherea₃=2,a_n=₆₋²_n forn=4 or 5, and 0< δ <₁₀₀¹ is a fixed small number. Then by the following scalingu(x)=(λ)ⁿ⁻²²u(y), xˆ =_λ¹y, problem (4) becomes

Sλ,[u] :=u−(λ)⁻²u+n(n−2)u

n+2 n−2

+ =0 inBλ, ∂u

∂ν =0 on∂Bλ (6)

whereB_λ=^B_λ. Solutions to (6) can be found as critical points of the following energy functional J(u)=1

2

B_λ

|∇u|²+(λ)⁻²u²

−(n−2)² 2

B_λ

u

n−22n

+ . (7)

We consider a linear Neumann problem whose solution can be viewed as a projection ofU_a,λontoH_r,ν² (B):

V_a,λ−⁻²V_a,λ+n(n−2)U

n+2 n−2

a,λ =0 inB, ∂V_a,λ

∂ν =0 on∂B. (8)

DefineW₁:=λ⁻ⁿ⁻²²V_0,λ(^y_λ),W₂:=(λ)⁻ⁿ⁻²²w(^y_λ),W:=W₁+W₂.

We first need to analyze the projectionV0,λ(x). Forn=3, letV0,λ(x)=U0,λ(x)−_λ1/2¹

|1x|(1−e^−|^x^|^/)+ϕλ,(x).

Then it is easy to see that (see e.g. estimate (2.10) of [14])ϕλ,(x)=O(2λ^3/2(1¹+λ|x|)+^e_λ^−1/1/2), which gives

W1(y)=U0,1(y)− 1

|y|

1−e⁻^|^λ^y^| +O

1

²λ²(1+ |y|)+e⁻¹ λ

, whenn=3. (9)

Observe that whenn=3,|y|> δλor|x|> δ, (9) gives W1(y)=O

1

λ² +e⁻^|^λ^y^| λ

, V0,λ(x)=O 1

λ^3/2+Ye⁻^|^x^| λ¹²

. (10)

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Whenn=4 or 5, we have (similar to estimates in Lemma A.1 of [15]) W1(y)=U0,1(y)−ϕ2

y λ

+O

1 λⁿ⁻²

(11) whereϕ₂(x)satisfiesϕ₂−⁻²ϕ₂+⁻²U_0,λ=0 inB,^∂ϕ_∂ν² =0 on∂B. Hence|ϕ²(x)|_λ2³(1+^Cλ|x|)ⁿ⁻⁴.

Next we define two Sobolev norms. (See [4] and [15].) Let φ_∗:=sup_y_∈_B_λ(1+ |y|)^n−2+δ² |φ(y)|, f_∗∗:=

sup_y_∈_B_λ((1+ |y|)^n+2+δ² |f (y)|). Then we compute that S_λ,[W]=n(n−2)(W1+W2)

n+2

+n−2 −W

n+2 n−2

2 −U

n+2 n−2

0,1 CW

n−24

1 W2+CW

n−24

2 W1+CU

n−24

0,1 |W1−U0,1|.(12) The most difficult term in (12) isW

4 n−2

2 W₁, which we now estimate: whenn=3 and|y|< δλ, we have W

4 n−2

2 W₁Cλ⁻²W₁w⁴Cλ⁻²

1+ |y|₋1

e⁻³ Cλ⁻^3/2

1+ |y|₋3

while whenn=3 and|y|δλ, we use (10) to obtainW

4 n−2

2 W₁Cλ⁻²W₁w⁴Cλ⁻²λ⁻^3/2Cλ⁻¹(1+ |y|)⁻^5/2. Thus whenn=3 we have W

4 n−2

2 W₁_∗∗< λ⁻⁽¹⁻^δ). Similarly, using (9) and (11), a straightforward computation shows that

S_λ,[W]

∗∗Cλ⁻^βn+²^2δ whereβ_n=1 whenn=3,andβ_n=2 whenn4. (13) Next, we compute the energy expansion-J[W]. Observe that

J[W₁] = −(n−2)² 2

Bλ

(U_0,1+W₁−U_0,1)ⁿ²ⁿ−2 +n(n−2) 2

Bλ

U

n+2n−2

0,1 (U_0,1+W₁−U_0,1).

Using (9) and (11), we obtain

J(W₁)=(n−2)

Rⁿ

U_0,1^2n/(n⁻²⁾+

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ 3 2

R³

U_0,1⁵ 1 λ+O

e⁻⁽³⁻^δ)/

, whenn=3, 1

2⁻²

B

U_0,λ² +O

⁻¹λ⁻⁽ⁿ⁻²⁾

, whenn4.

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Using (14), we have the following asymptotic behavior of the energy expansionJ(W ):

J(W )−J(W₂)=(n−2)

Rⁿ

U_0,1^2n/(n⁻²⁾

+

⎧⎪

⎪⎨

⎪⎪

⎩

2π+o(1)

⁻¹λ⁻¹−B₃+o(1)

⁻^1/2e⁻^1/λ⁻^1/2+O

e⁻⁽²⁺^4δ)/

, whenn=3, ⁻²

2

B

U_0,λ² −B_n+o(1)

⁽²⁻^n)/2e⁻^1/λ⁻⁽ⁿ⁻^2)/2+O

⁻¹λ⁻²

, whenn4, (15)

whereB_n>0 is a positive constant.

Proof of (15). We first prove (15) whenn=3. Sincew is a solution to (4), we have J(W₁+W₂)=J(W₁)+ J(W₂)−I, where

I:=1 2

Bλ

(W₁+W₂)⁶−W₁⁶−W₂⁶−6W₂⁵W₁−6W2W₁⁵ +3

Bλ

W₂W₁⁵=:I_,1+I_,2

and

I,2=3⁻^1/2

|x|δ

wV_0,λ⁵ +O

^1/2λ⁻^5/2

=B3+o(1)

⁻^1/2e⁻^1/λ⁻^1/2+O

^1/2λ⁻^5/2

. (16)

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On the other hand, by (10),V_0,λ=O( ¹

λ^3/2 +^e_λ⁻^1/1/2)for|x|> δ. Thus I_,1=⁻²15

2

B

w⁴V_0,λ² +O

⁻^3/2

B

w³V_0,λ³ +⁻¹

B

w²V_0,λ

=O

⁻¹λ⁻³+e⁻^2/λ⁻¹

. (17)

Substituting (17) and (16) intoIand using the fact that

R³U_0,1⁵ =^4π₃ , we obtain (15) forn=3.

Whenn=4, the proof of (15) is similar and easier by noting that forn4, we have

B

U_0,λ² ∼lnλ

λ², ifn=4;

B

U_0,λ² ∼ 1

λ², ifn5, (18)

and hence

BλW

4 n−2

2 W₁²=O(⁻¹λ⁻⁽ⁿ⁻²⁾)is a higher order term. 2 3. Finite-dimensional reduction

In this section, we perform a finite-dimensional reduction procedure similar to that of [4] and [15].

We first consider the following linear problem: Let Z=U0,1+ⁿ⁻₂²y∇U1,0. Givenh=h(r), find a pair (φ, c) satisfying

φ−(λ)⁻²φ+n(n+2)W

4 n−2

+ φ=h+cZ, inB_λ, ∂φ

∂ν =0 on∂B_λ,

Bλ

φZ=0. (19)

We have the following a priori estimates.

Lemma 3.Let(φ, c)satisfy(19). Then forsufficiently small, there holdsφ_∗Ch_∗∗.

Proof. Observe first that the Green function ofG−(λ)⁻²G+δ_y=0 inB_λ, ^∂G_∂ν =0 on∂B_λhas the following decay property:G(x, y)_|_x₋^C_y_|n−2. Secondly, the operator−(λ)⁻²+n(n+2)W

4 n−2

2 is uniformly invertible by Lemma 2. The rest of the proof is similar to Proposition 3.1 of [15]. We omit the details. 2

By estimate (13), Lemma 3 and a contraction mapping principle, we derive the following reduction lemma:

Lemma 4.There exists₀>0such that for < ₀, the following problem (W+φ)−(λ)⁻²(W +φ)+n(n−2)(W+φ)

n+2

+n−2 =cλZ inBλ, ∂φ

∂ν =0 on∂Bλ,

B_λ

φZ=0 (20)

has a unique solution(φλ, cλ). Moreover the mapλ→φλisC¹andφλ∗Cλ⁻^βn+δ² . Now we defineM(λ)=J(W +φλ)−J(W2)−(n−2)

RⁿU_0,1^2n/(n⁻²⁾, whereJ(W2)is independent ofλ. Then we also have

Lemma 5.Ifλ=λis a critical point ofM(λ)inΛ, thenu=W+φ_λ is a solution to(6).

Thus we are reduced to finding a critical point ofM(λ).

4. Proof of Theorem 1

We first expandM(λ): using (13) and Lemma 4, we deduce that M(λ)=J(W )+

Bλ

S_λ[W]φ_λ+O φ_λ²_∗

−J(W₂)−(n−2)

Rⁿ

U_0,1^2n/(n⁻²⁾.

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Whenn=3, we use (15) to derive thatM(λ)=(2π+o(1))⁻¹λ⁻¹−(B₃+o(1))⁻^1/2e⁻^1/λ⁻^1/2+O(e⁻^3/).

Observe that the function(2π+o(1))²λ⁻¹−(B3+o(1))⁻^1/2e⁻^1/λ⁻^1/2attains its minimum atλ=e²⁺^o(1) ∈Λ.

On the other hand, whenn=4 or 5, we have M(λ)=1

2⁻²

B

U_0,λ² −B_n+o(1)

⁽²⁻^n)/2e⁻^1/λ⁻⁽ⁿ⁻^2)/2+O

⁻¹λ⁻²

. (21)

Using (18), we find that the function ¹₂⁻²

BU_0,λ² − B_n⁽²⁻^n)/2e⁻^1/λ⁻⁽ⁿ⁻^2)/2has a critical pointλ¯=e

2+o(1) (6−n). Thus, the reduced energy functionalM(λ)also has a critical pointλ=e

2+o(1)

(6−n). By Lemma 5,W1+W2+φλ is a solution to (6). Letu(x)=W1(λx)+W2(λx)+φλ(λx). Thenusatisfies all the properties of Theorem 1. 2 Acknowledgements

The second author is supported by an Earmarked Grant from RGC of Hong Kong. We thank the referee for careful reading and rewriting the French part.

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