Multiple clustered layer solutions for semilinear Neumann problems on a ball

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Multiple clustered layer solutions for semilinear Neumann problems on a ball

Solutions multi-singulières pour un problème de Neumann sur la boule unité

A. Malchiodi

, Wei-Ming Ni

, Juncheng Wei

aSector of Functional Analysis and Applications, SISSA Via Beirut 2-4, 34014 Trieste, Italy bSchool of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA cDepartment of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

Received 12 March 2003; received in revised form 20 April 2004; accepted 7 May 2004 Available online 11 September 2004

Abstract

We consider the following singularly perturbed Neumann problem ε²u−u+f (u)=0 inΩ;

u >0 inΩ and ^∂u_∂ν =0 on∂Ω,

whereΩ=B₁(0)is the unit ball inRⁿ,ε >0 is a small parameter andf is superlinear. It is known that this problem has multiple solutions (spikes) concentrating at some points ofΩ. In this paper, we prove the existence of radial solutions which concentrate¯ atNspheres_N

j=1{|x| =r_j^ε}, where 1> r₁^ε> r₂^ε>· · ·> r_N^ε are such that 1−r₁^ε∼εlog¹_ε, r_j^ε₋₁−r_j^ε∼εlog¹_ε, j=2, . . . , N.

2004 Elsevier SAS. All rights reserved.

Résumé

On considère le problème de Neumann singulièrement perturbé suivant ε²u−u+f (u)=0 dansΩ;

u >0 dansΩ et ^∂u_∂ν =0 sur∂Ω,

oùΩ=B₁(0)est la boule unité deRⁿ,ε >0 est un paramètre petit etf est surlinéaire. Il est bien connu que ce problème possède plusieurs solutions se concentrant en certains points deΩ. Dans cet article nous prouvons l’existence de solutions¯

* Corresponding author.

E-mail addresses: [email protected] (A. Malchiodi), [email protected] (W.-M. Ni), [email protected] (J. Wei).

0294-1449/$ – see front matter 2004 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpc.2004.05.003

radiales qui se concentrent enNsphères_N

j=1{|x| =r_j^ε}, où 1> r₁^ε> r₂^ε>· · ·> r_N^ε sont tels que 1−r₁^ε∼εlog¹_ε, r_j^ε₋₁−r_j^ε∼ εlog¹_ε, j=2, . . . , N.

2004 Elsevier SAS. All rights reserved.

MSC: primary 35B40, 35B45; secondary 35J40, 92C40

Keywords: Multiple clustered layers; Singularly perturbed Neumann problem

1. Introduction

The aim of this paper is to construct a family of multiple layered solutions to the following singularly perturbed elliptic problem

ε²u−u+f (u)=0 inΩ;

u >0 inΩ and ^∂u_∂ν =0 on∂Ω, (1.1)

where=_n

i=1 ∂²

∂x_i² is the Laplace operator, Ω =B₁(0) is the unit ball in Rⁿ, ε >0 is a small parameter, f is superlinear, andν(x) denotes the unit outer normal atx ∈∂Ω. A typical nonlinearity is f (u)=u^p, with p∈(1,+∞).

Problem (1.1) is known as the stationary equation of the Keller–Segel system in chemotaxis [19]. It can also be viewed as a limiting stationary equation of the Gierer–Meinhardt system in biological pattern formation, see [22]

for more details.

In the pioneering papers [19,23] and [24], Lin, Ni and Takagi established the existence of a least-energy solu- tionuεof (1.1) and showed that, forεsufficiently small,uεhas only one local maximum pointPε∈∂Ω. Moreover, H (Pε)→maxP∈∂ΩH (P )asε→0, whereH (·)is the mean curvature of∂Ω. Such a solution is called boundary spike-layer. Its energy, defined by

J_ε[u] =ε² 2

Ω

|∇u|²+1 2

Ω

u²−

Ω

F (u), u∈H¹(Ω), (1.2)

whereF (u)=_u

0 f (s) ds, satisfies the following estimate

J_ε[u_ε] ∼εⁿ. (1.3)

(Here and throughout the paper,Aε∼Bεmeans that there exists fixed constantsC₁, C₂such thatC₁Aε/BεC₂ forεsmall.)

Since then, many papers investigated further the solutions of (1.1) concentrating at one or multiple points ofΩ¯ (spike-layers). A general principle is that the location of interior spikes is determined by the distance function from the boundary. We refer the reader to the articles [3,6,9–11,13,14,16,18,26–28,30,31] and references therein. On the other hand, boundary spikes are related to the mean curvature of∂Ω. This aspect is discussed in the papers [4,8, 15,17,25,29,32,33], and references therein. A review of the subject up to 1998 is to be found in [22]. We mention one result which is related to our work here: in [14], it was proved that for any two integersk0, l0, k+l >0, problem (1.1) possesses a solutionu_εwith exactlykinterior spikes andlboundary spikes.

However, in all the papers mentioned above, we still have the energy bound (1.3) and the concentration set is zero-dimensional. The question of constructing higher-dimensional concentration sets has been investigated only in recent years. It has been conjectured in [22] that for any 1kn−1, problem (1.1) has a solutionuε which concentrates on ak-dimensional subset ofΩ¯. We mention two recent results that support such a conjecture.

In [20], Malchiodi and Montenegro proved that forn=2 andf (u)=u^p, p2, there exists a sequence of numbersεk →0 such that problem (1.1) has a solutionuε_k which concentrates at the boundary of∂Ω (or any component of∂Ω). Such a solution has the following energy bound

J_εk[u_εk] ∼ε_kⁿ⁻¹=ε_k. (1.4)

In another recent paper [2], Ambrosetti, Malchiodi and Ni proved the existence of a radially symmetric solutionu_ε to (1.1) withf (u)=u^p, p >1 and Ω=B₁(0), such thatu_ε concentrates at a sphere{|x| =r_ε}with 1−r_ε ∼ εlog¹_ε. For this solution there holds

J_ε[u_ε] ∼ε. (1.5)

In this paper we show existence of solutions concentrating at multiple spheres, which cluster near the boundary ofB₁(0). Throughout the paper, we assume the following conditions onf

(f1) f (t )≡0 fort >0, f (0)=f(0)=0 andf ∈C^1+σ[0,∞)∩C²(0,∞), (f2) the following ODE has a unique solution

w−w+f (w)=0 inR, w(0)=max

y∈Rw(y), w(y)→0 as|y| → +∞. (1.6) Assumption (f2) implies that the following holds: the only solution of the linearization of (1.6) atw

v−v+f(w)v=0 inR, v(y)→0 for|y| → +∞ (1.7) is a multiple ofw. In fact, problemv−v+f(w)v=0 is a second order linear ODE and admits two linearly inde- pendent solutions. Sincewis a solution to (1.7), another solution tov−v+f (w)v=0 must grow exponentially as|y| → +∞.

Examples off (u)include:f (u)=u^p+l

i=1aiu^qi,1< qi< p <∞. (See [5] and Appendix C of [24].) When f (u)=u^p, the functionw(y)can be written explicitly and has the form

w(y)= p+1

2

_p−1¹ cosh

p−1 2 y

_−p−1² .

Our main result in this paper is the following.

Theorem 1.1. LetN be a fixed positive integer. Then there existsε_N>0 such that for allε < ε_N, problem (1.1) admits a radially symmetric solutionu_εwith the following properties

(1)uεconcentrates atNspheres{|x| =r_j^ε}, j =1, . . . , N, with 1−r₁^ε∼εlog1

ε, r_j^ε₋₁−r_j^ε∼εlog1

ε, j =2, . . . , N. (1.8)

More precisely, we haveu_ε(r_j^ε)→w(0), wherew(y)is the unique solution of (1.6), and there exist two con- stantsaandbsuch that

u_ε(r)ae^{−bmini=1,...,N|r−riε|/ε}. (1.9)

(2)u_εhas the following energy bound

Jε[uε] =ωn−1N εI[w] +o(ε), (1.10)

where

I[w] =1 2

R

(w)²+w² − 1 p+1

R

F (w),

andωn−1denotes the volume ofSⁿ⁻¹.

Theorem 1.1 can also be generalized to singularly perturbed Schroedinger equations of the form ε²u−V

|x| u+u^p=0, inRⁿ, u >0, u∈H¹(Rⁿ) (1.11) wherep >1.

As in [1], the following auxiliary potential M(r)=rⁿ⁻¹V^θ(r), θ=p+1

p−1−1 2

determines the location of the clustered layers. We state the following result, whose proof is similar to that of Theorem 1.1, see also the computations in [1].

Theorem 1.2. Assume thatM(r)has strict local maximum atr= ¯rsuch thatM(r) <¯ 0. Then forε >0 sufficiently small, problem (1.11) admits a radially symmetric solutionu_ε which concentrates onN spheres{|x| =r_j^ε}, j= 1, . . . , N, with

|¯r−r₁^ε| ∼εlog1

ε, r_j^ε−r_j^ε₋₁∼εlog1

ε, j =2, . . . , N. (1.12)

More precisely, we haveu_ε(r_j^ε)→w(0), wherew(y)is the unique solution of (1.6) (withf (u)=u^p), and there exist two constantsaandbsuch that

u_εae^{−bmini=1,...,N|r−riε|/ε}. (1.13)

Clustered spikes have been shown to exist for (1.1) in the case of general domainΩ. For example, if the bound- ary mean curvature has a local minimum pointP₀, then it is shown [15] that given arbitrary positive integerK, there exists a boundary spike solution withKspikes concentrating nearP₀. Interior clustered layers are shown to exist for one-dimensional Allen–Cahn equation with inhomogeneous potential [21] and for bistable nonlinearities with inhomogeneous potential in a unit ball [12]. However, we are not aware of any result on boundary clustered layers in higher dimensions. This paper seems to be the first in dealing with such phenomenon in higher dimensions.

Our approach mainly relies upon a finite dimensional reduction procedure combined with a variational approach.

Such a method has been used successfully in many papers, see e.g. [1,2,7,11,13,14]. In particular, we shall follow the one used in [13].

In the rest of section, we introduce some notations for later use.

By the following scalingx=εy, problem (1.1) is reduced to the ODE u_rr+ⁿ⁻_r¹u_r−u+f (u)=0, r∈(0,¹_ε),

u(0)=u(¹_ε)=0, u(r) >0. (1.14)

From now on, we shall work with (1.14).

Letw(y)be the unique solution to (1.6). Its energy is given by I[w] :=1

2

R

|w|²+1 2

R

w²−

R

F (w). (1.15)

Set

Ω_ε=1

εB₁(0)=B1

ε(0), I_ε=

0,1

ε

. (1.16)

Foru∈C²(Ω_ε)andu=u(r), we have u=u+n−1

r u. (1.17)

Fork∈N, we denote byH_r^k(Ω_ε)the space of radial functions inH^k(Ω_ε). OnH_r¹(Ω_ε), we define an inner product as follows:

(u, v)_ε=

1

ε

0

(uv+uv)rⁿ⁻¹dr. (1.18)

Similarly, the inner product onL²_r(Ω_ε)can be defined by u, vε=

1

ε

0

(uv)rⁿ⁻¹dr. (1.19)

We also introduce a new energy functional, which, up to a positive multiplicative constant, is equivalent toJε

Eε[u] =1 2

1

ε

0

|u|²+u² rⁿ⁻¹−

1

ε

0

F (u)rⁿ⁻¹dr, u∈H_r¹(Ω_ε). (1.20)

Throughout this paper, unless otherwise stated, the letterCwill always denote various generic constants which are independent ofε, forεsufficiently small. The notationA_εB_εmeans that lim_ε→0|B_ε|/|A_ε| =0, whileA_εB_ε means 1/A_ε1/B_ε.

2. Some preliminary analysis

In this section we introduce a family of approximate solutions to (1.14) and derive some useful estimates.

Letwbe the unique solution of (1.6) (see assumption (f2)), and lett∈(1/2,1). We define ρ_ε(t )= −w

1−t ε

; β_ε(r)=e^r−1ε, r∈

0,1 ε

, (2.1)

and

wε,t(r)=

w

r−t ε

+ρε(t )βε(r)

χ (εr), (2.2)

whereχ (s)is a smooth cut-off function such that





χ (s)=0 fors1/8, χ (s)=1 fors1/4, χ (s)∈ [0,1] fors∈ [1/8,1/4].

(2.3) Using ODE analysis, it is standard to see that

w(y)=A₀e^−y+O(e^{−(1+σ )y});

w(y)= −A0e^−y+O(e^{−(1+σ )y}); y0, (2.4)

whereA₀>0 is a fixed constant andσ >0 is given in (f1). This implies that for ¹⁻_ε^t 1

ρ_ε(t )=A₀e^−1−tε +O(e^{−(1+σ )1−tε} ). (2.5)

Note that forr_4ε¹, we have

wε,t(r)+w_ε,t(r)+w_ε,t(r)e^−4ε1. (2.6)

Observe also that, by construction,w_ε,t satisfies the Neumann boundary condition, i.e.,w_ε,t (1/ε)=0. Further- more,w_ε,t depends smoothly ontas a map with values inC²([0,1/ε]).

For simplicity of notation, let w_t(r):=w

r−t

ε

. (2.7)

Foru∈H_r²(Ω_ε), we define the operator Sε[u] :=u_rr+n−1

r u_r−u+f (u). (2.8)

We introduce the following set Λ=

t=(t1, . . . , tN)

t_N>¹₂,1−t₁ηεlog¹_ε,

tj−1−tj> ηεlog¹_ε, j=2, . . . , N

, (2.9)

whereη∈(0,1/8)is a fixed number.

For t∈Λ, we define w_ε,t(r)=

N j=1

w_ε,tj(r). (2.10)

Before studying the properties ofw_ε,t, we need a preliminary lemma.

Lemma 2.1. For|t−s| 1 andα > β >0, there holds w^α(y−t )w^β(y−s)=O

w^β

|t−s|

, (2.11)

R

w^α(y−t )w^β(y−s) dy=

1+o(1)w^β

|t−s|

R

w^α(y)e^−βydy, (2.12)

where o(1)→0 as|t−s| → +∞.

Proof. Lety−t=z. Formula (2.4) implies

w^β(y−s)=w^β(z+t−s)=A₀e^{−β|z+t−s|}+O(e^{−β(1+σ )|z+t−s|}).

Multiplying byw^α(y−t )and using the fact thatβ|z+t−s| +α|z|> β|t−s|, we obtain (2.11).

Regarding the second inequality, assumingt > s, we can write

R

w^α(y−t )w^β(y−s) dy=A₀e^−β(t−s)

R

w^α(z)e^−βzdz+

R

w^α(z)

w^β(z+t−s)−A₀e^{−β(z+t−s)} dz.

Then the conclusion follows from Lebesgue’s Dominated Convergence Theorem. The cases > tcan be handled similarly. 2

Lemma 2.2. Forεsufficiently small and t∈Λ, we have Sε[wε,t]

L^∞+εⁿ⁻¹

1

ε

0

Sε[wε,t]rⁿ⁻¹drC

ε+ N j=1

ρε(tj) ¹⁺

σ

2 +

i=j

e⁻

τ|ti−tj| ε

, (2.13)

whereτ satisfies ¹₂< τ <¹⁺₂^σ.

Proof. Using (1.6) it is easy to see that Sε[w_ε,t] =n−1

r w_ε,t+f (w_ε,t)− N j=1

f (w_tj)+O(e^−Cε1 ). (2.14)

The first term in right hand side of (2.14) can be estimated as follows 1

rw_ε,t=1 r

N j=1

w_t

j +ρε(tj)βε(r) +O(e^−Cε1 ).

From the decay ofwandβ_εwe deduce that 1

rw_ε,t ∞+

1

ε

0

1

r|w_ε,t|rⁿ⁻¹drCε. (2.15)

Next, we note that

f (wε,t)− N j=1

f (wt_j)

S₁+S₂, where

S₁= f

_N

j=1

w_tj

− N j=1

f (w_tj)

; S₂= f

_N

j=1

w_ε,tj

−f _N

j=1

w_tj .

To estimateS₁andS₂, we divide the domainI_ε=(0,1/ε)into theNintervalsI₁, . . . , I_Ndefined by I₁=

t₁+t₂ 2ε ,1

ε

, I_j=

tj+tj+1

2ε ,tj+tj−1

2ε

, j=2, . . . , N−1, I_N=

0,t_N+t_N−1 2ε

. (2.16)

Then we have

|wt_i||wt_j| onIjfori=j. (2.17)

We need to apply the following inequality, which can be proved with elementary calculus. Fixed a constantL >0, for each 0< τ1 and for anyδ∈(0,1)there exists a constantC=C(L, τ, δ)such that

f (x+y)−f (x)C|x|^1+σ−τ|y|^τ, providedδ|y|<|x|< L. (2.18) Using (f1), (2.17), (2.18) and recalling thatτ <¹⁺₂^σ, we deduce that forr∈I₁there holds

S1C

j=1

|wt₁|^1+σ−τw^τ_t

j+C

j=1

w_t^1+σ

j C

j=1

|wt₁|^1+σ−τw^τ_t

j.

From (2.11) and (2.12) it follows that S₁L∞(I₁)+εⁿ⁻¹

I₁

S₁(r)rⁿ⁻¹drC

j=1

e^{−τ|t1−tj|/ε}. (2.19)

On the other hand, using (2.18) withτ =1 and the inequalityw^σ/2_t1 (r)βε(r)Cρ^σ/2ε (t₁), we find

S₂Cw_t^σ

1

_N

j=1

ρ_ε(t_j)

β_ε(r)C _N

j=1

ρ_ε(t_j)

w

σ

t2₁β_ε(r) w

σ

t2₁

C

_N

j=1

ρ_ε(t_j)

ρ_ε(t₁)

σ 2w

σ

t2₁ C

ρ_ε(t₁) ¹⁺

σ 2w

σ

t2₁.

Hence we also get S₂L^∞(I₁)+εⁿ⁻¹

I1

S₂(r)rⁿ⁻¹drCρ_ε(t₁)^1+σ2. (2.20)

Forr∈I_j, j=2, . . . , N−1, we can estimate similarly S₁C

i=j

|w_tj|^1+σ−τw_t^τ

i, (2.21)

and

S₂Cw_t^σ

j

_N

i=1

ρ_ε(t_i)

β_ε(r)C

ρ_ε(t₁)¹⁺

σ 2w

σ

t2_j (2.22)

sincew_t^σ/2_j (r)β_ε(r)ρ_ε^σ/2(t_j)ρ_ε^σ/2(t₁). OnI_None can use similar estimates and, adding an error term of order e^−1/(Cε). Combining (2.15), (2.19), (2.20) and similar estimates onI_j,j2, we obtain (2.13). 2

The proof of the next lemma is postponed to Appendix A.

Lemma 2.3. Let t∈Λ. Then forεsufficiently small we have Eε

_N

j=1

wε,t_j

= N i=1

t_i ε

n−1

I[w] −

A²₀+o(1)e⁻²

1−ti

ε

−

i=j

t_i ε

n−1

A²₀+o(1) e⁻

|ti−tj|

ε +O(ε²⁻ⁿ), (2.23)

whereA₀>0 is defined in (2.4).

3. An auxiliary linear problem

In this section we study a linear theory which allows us to perform the finite-dimensional reduction procedure.

Fix t∈Λ. Integrating by parts, one can show that orthogonality to^∂w_∂t^ε,tj

j inH_r¹(Ω_ε),j=1, . . . , N, is equivalent to orthogonality inL²(Ωε)to the following functions

Z_ε,tj = ∂w_ε,tj

∂tj

−∂w_ε,tj

∂tj

, j=1, . . . , N. (3.1)

By elementary computations, we obtain

∂w_ε,tj

∂t_j = −1 εw

r−t_j

ε

+1 εw

1−t_j ε

β_ε(r)+O(e^−Cε1 ), (3.2)

Z_ε,tj = −f(w_tj)∂w_ε,tj

∂t_j −n−1 r

∂w_ε,tj

∂t_j

+o 1

ε

= −1 εw_t

jf(w_tj)+o 1

ε

, (3.3)

where O(e^−1/(Cε))and o(1/ε)are intended both in theC¹andH_r¹sense.

In this section, we consider the following linear problem. Givenh∈L^∞(Ω_ε), find a functionφsatisfying L_ε[φ] :=φ+ⁿ⁻_r¹φ−φ+f(w_ε,t)φ=h+_N

j=1c_jZ_ε,tj;

φ(0)=φ(¹_ε)=0; φ, Z_ε,tjε=0, j=1, . . . , N, (3.4) for some constantsc_j, j=1, . . . , N. To this purpose, define the norm

φ_∗= sup

r∈(0,¹_ε)

φ(r). (3.5)

We have the following result.

Proposition 3.1. Letφsatisfy (3.4). Then forεsufficiently small, we have

φ_∗Ch_∗ (3.6)

whereCis a positive constant independent ofεand t∈Λ.

Proof. Arguing by contradiction, assume that

φ_∗=1; h_∗=o(1). (3.7)

We multiply (3.4) by^∂w_∂t^ε,tj

j and integrate overΩεto obtain N

i=1

c_i

Z_ε,ti,∂w_ε,tj

∂t_j

ε

= −

h,∂w_ε,tj

∂t_j

ε

+

φ−φ+f(w_ε,t)φ,∂w_ε,tj

∂t_j

ε

. (3.8)

From the exponential decay ofwone finds

h,∂w_ε,tj

∂t_j

ε

=

1

ε

0

h∂w_ε,tj

∂t_j rⁿ⁻¹dr=O

h∗ε⁻ⁿ .

Moreover, integrating by parts, using (3.2) and (3.3) we deduce

φ−φ+f(w_ε,t)φ,∂w_ε,tj

∂t_j

ε

=

Z_ε,tj+f(w_ε,t)∂w_ε,tj

∂t_j , φ

ε

=o

ε⁻ⁿφ_∗ . From (3.2) and (3.3), we see that

Z_ε,ti,∂w_ε,tj

∂tj

ε

= −ε⁻ⁿ⁻¹

t_iⁿ⁻¹δ_ij

R

f(w)(w)²+o(1)

, (3.9)

whereδ_ijdenotes the Kronecker symbol. Note that, using the equationw−w+f(w)w=0 we find

R

f(w)(w)²=

R

(w)²+(w)² >0.

This shows that the left hand side of Eq. (3.8) is diagonally dominant in the indexesi, j, and hence by (3.7) we have

ci=O

εh∗ +o

εφ∗ =o(ε), i=1, . . . , N. (3.10)

Also, since we are assuming thath_∗=o(1)and sinceZ_ε,tj∗=O(1/ε), there holds

h+ N j=1

cjZε,t_j

∗

=o(1). (3.11)

Thus (3.4) yields

φ+ⁿ⁻_r¹φ−φ+f(w_ε,t)φ=o(1);

φ(0)=φ(¹_ε)=0; φ, Z_ε,tjε=0, j=1, . . . , N. (3.12) We show that (3.12) is incompatible with our assumptionφ_∗=1. First we claim that, fixedR >0, there holds

|φ| →0 on y∈ N j=1

tj

ε −R,tj

ε +R

asε→0, (3.13)

whereRis any fixed positive constant.

Indeed, assuming the contrary, there existδ₀>0,j∈ {1, . . . , N}and sequencesε_k, φ_k, y_k∈(t_j/ε−R, t_j/ε+R) such thatφ_ksatisfies (3.4) and

φ_k(y_k)δ₀. (3.14)

Letφ˜_k=φ_k(y−t_j/ε_k). Then using (3.12) andφ_∗=1, asε_k→0φ˜_kconverges weakly inH_loc² (R)and strongly inC¹_loc(R)to a bounded functionφ₀which satisfies

φ₀−φ₀+f(w)φ₀=0 inR.

Henceφ₀must tend to zero at infinity and so, by (1.7),φ₀=cw for somec. Sinceφ˜_k⊥Z_ε,tj, we conclude that

Rφ₀f(w)w(y)=0, which yieldsc=0. Henceφ₀=0 andφ˜_k→0 inB_2R(0). This contradicts (3.14), so (3.13) holds true.

Givenδ >0, the decay ofwand (3.13) (withRsufficiently large) imply f(w_ε,t)φ_∗δ+1

2φ_∗. (3.15)

Using (3.12) and the Maximum Principle one finds φ_∗f(wε,t)φ

∗+ N j=1

|cj|Zε,t_j∗+ h_∗2δ+1 2φ_∗, and hence

φ∗4δ <1

if we chooseδ <1/4. This contradicts (3.7). 2

Proposition 3.2. There existsε₀>0 such that for anyε < ε₀the following property holds true. Givenh∈L^∞(Ω_ε), there exists a unique pair(φ, c₁, . . . , c_N)such that

L_ε[φ] =h+ N j=1

c_jZ_ε,tj, (3.16)

φ(0)=φ 1

ε

=0; φ, Z_ε,tjε=0, j =1, . . . , N. (3.17)

Moreover, letting c= {c_j}j=1,...,N, we have φ∗+1

ε|c|Ch∗ (3.18)

for some positive constantC.

Proof. The bound in (3.18) follows from Proposition 3.1 and (3.10). Let us now prove the existence part. Set H=

u∈H¹(Ω_ε) u,∂w_ε,t

∂t

ε

=0

.

Note that, integrating by parts, one has

ψ∈H if and only if ψ, Z_ε,tjε=0, j=1, . . . , N.

Observe thatφsolves (3.16) and (3.17) if and only ifφ∈Hsatisfies

Ωε

(∇φ∇ψ+φψ )− f(w_ε,t)φ, ψ!

ε= h, ψε, ∀ψ∈H. This equation can be rewritten as

φ+S(φ)= ¯h inH, (3.19)

whereh¯is defined by duality andS:H→His a linear compact operator.

Using Fredholm’s alternative, showing that Eq. (3.19) has a unique solution for eachh, is equivalent to showing¯ that the equation has a unique solution forh¯=0, which in turn follows from Proposition 3.1 and our proof is complete. 2

In the following, ifφis the unique solution given in Proposition 3.2, we set

φ=Aε(h). (3.20)

Note that (3.18) implies

Aε(h)_∗Ch_∗. (3.21)

4. Construction of a natural constraint

In this section we reduce problem (1.14) to a finite-dimensional one. Let M_ε be the N-dimensional manifold defined as

M_ε= {w_ε,t: t∈Λ}.

Forεsmall and for t∈Λ, we are going to find a functionφ_ε,tsatisfying the two conditions

φ_ε,t⊥H_r¹(Ω_ε)T_wε,tM_ε; Eε(w_ε,t+φ_ε,t)∈T_wε,tM_ε, (4.1) whereT_wε,tM_εdenotes the tangential space ofM_εatw_ε,t. This amounts to finding a functionφsuch that for some constantsc_j, j=1, . . . , N, the following equation holds true

(w_ε,t+φ)−(w_ε,t+φ)+f (w_ε,t+φ)=_N

j=1c_jZ_ε,tj inΩ_ε, φ(0)=φ(¹_ε)=0, φ, Zε,t_jε=0, j=1, . . . , N.

(4.2)

Letting

M˜_ε= {w_ε,t+φ_ε,t: t∈Λ}, (4.3)

we will show thatM˜εis a natural constraint forEε, in the sense that a critical point ofEε|_M˜ε is a true critical point ofEε.

The first equation in (4.2) can be written as φ+n−1

r φ−φ+f(wε,t)φ=

−Sε[wε,t] +Nε[φ] + N j=1

cjZε,t_j,

where

Nε[φ] = −

f (wε,t+φ)−f (wε,t)−f(wε,t)φ

. (4.4)

Lemma 4.1. For t∈Λandεsufficiently small, we have forφ_∗+ φ_1∗+ φ_2∗1, N_ε[φ]

∗Cφ¹_∗^+σ; (4.5)

N_ε[φ₁] −N_ε[φ₂]

∗C

φ₁^σ_∗+ φ₂^σ_∗ φ₁−φ_2∗, (4.6)

whereσis defined in (f1).

Proof. Inequality (4.5) follows from the mean-value theorem. In fact, for every point in[0,1/ε]there holds f (w_ε,t+φ)−f (w_ε,t)=f(w_ε,t+θ φ)φ, θ∈ [0,1].

Sincefis Holder continuous with exponentσ, we deduce f (w_ε,t+φ)−f (w_ε,t)−f(w_ε,t)φC|φ|^1+σ,

which implies (4.5). The proof of (4.6) goes along the same way. 2

Proposition 4.2. For t∈Λandεsufficiently small, there exists a uniqueφ=φ_ε,tsuch that (4.2) holds. Moreover, t→φε,tis of classC¹as a map intoH_r¹(Ωε), and we have

φ_ε,t∗C

ε+ N j=1

e^−(1+σ2)

1−tj

ε +

i=j

e⁻

τ|ti−tj| ε

, (4.7)

whereτ∈(¹₂,¹⁺₂^σ).

Proof. LetAεbe as defined in (3.20). Then (4.2) can be written as φ=Aε

−Sε[w_ε,t] +N_ε[φ]

. (4.8)

Letrbe a positive (large) number, and set Fr =

φ∈H_r¹(Ωε): φ∗r

ε+ N j=1

e^−(1+σ2)

1−tj

ε +

i=j

e⁻

τ|ti−tj| ε

.

Define now the mapBε:Fr→H_r¹(Ω_ε)as Bε(φ)=Aε

−Sε[w_ε,t] +N_ε[φ] .

Solving (4.2) is equivalent to finding a fixed point forBε. By Lemma 4.1, forεsufficiently small andr large we have