Locating the boundary peaks of least-energy solutions to a singularly perturbed Dirichlet problem

Locating the boundary peaks of least-energy solutions to a singularly perturbed Dirichlet problem

TERESAD’APRILE ANDJUNCHENGWEI

Abstract.We consider the problem

ε²v−v−γ1Vv+ f(v)=0 V+γ2|v|²=0, v=V =0 on∂, where⊂R³is a smooth and bounded domain,ε, γ1, γ2>0, v,V :→R,

f : R → R. We prove that this system has aleast-energy solutionv_ε which develops, asε→0⁺, a single spike layer located near the boundary, in striking contrast with the result in [37] for the single Schr¨odinger equation. Moreover the unique peak approaches themost curvedpart of∂,i.e., where the boundary mean curvature assumes its maximum. Thus this elliptic system, even though it is a Dirichlet problem, acts more like a Neumann problem for the single-equation case. The technique employed is based on the so-called energy method, which consists in the derivation of an asymptotic expansion for the energy of the solu- tions in powers ofεup to sixth order; from the analysis of the main terms of the energy expansion we derive the location of the peak in.

Mathematics Subject Classification (2000):35B40 (primary); 35B45, 35J55, 92C15, 92C40 (secondary).

1.

Introduction

In this paper we study the following problem:







ε²v−v−γ1Vv+ f(v)=0 in , V+γ2v²=0 in ,

v, V >0 in, v=V =0 on∂

(1.1)

where ⊂

R

³is a smooth and bounded domain,ε, γ1, γ2 > 0, v,V : →

R

^, f :

R

^→

R

. Solutions of (1.1) correspond to the stationary waves for the following Schr¨odinger-Poisson system:

iε∂ψ

∂t = −ε²ψ+ψ+γ1Vψ− f(ψ), V+γ2|ψ|²=0.

The research of the second author is supported by an Earmarked Grant from RGC of Hong Kong.

Received February 2, 2006; accepted in revised form May 24, 2006.

This system, first proposed by Benci-Fortunato ([3]) and later studied in [6, 13], can be used as a model in Quantum Mechanics to describe a charged particle interacting with its own electrostatic field. The purpose of this paper is to construct a single spike for the system (1.1) located near the boundary, where by single spike we intend a solution whose shape has the form of a unique peak which becomes highly concentrated whenεis sufficiently small.

Whenγ2=0 we obtain the single Schr¨odinger equation:

ε²v−v+ f(v)=0, (1.2)

for which the existence of single and multiple spike solutions has been extensively studied. Concerning equation (1.2) in a bounded domain with Neumann boundary condition, Ni and Takagi in [34, 35] first proved that forεsufficiently small there is a least-energy solutionv_εwith the property thatv_εhas exactly one maximum point P_εin, andP_εmust be located on∂and near themost curved part of the∂,i.e.,

H

(P_ε)→max_P∈∂

H

(P), where

H

(P)denotes the mean curvature of the bound- ary∂. On the other hand, for equation (1.2) in a bounded domain with Dirichlet boundary conditions, Ni and Wei in [37] showed that the least-energy solution de- velops a spike layer at themost centeredpart of the domain,i.e., dist(P_ε, ∂) → max_P∈dist(P, ∂). Since then, there have been many papers looking for higher- energy solutions. More specifically, solutions with multiple boundary peaks as well as multiple interior peaks have been established. It turns out thata general guide- line is that while multiple boundary spikes tend to cluster around the critical points of the boundary mean curvature

H

^(P), the location of the interior spikes is gov- erned by the distance between the peaks as well as from the boundary ∂ (see [1, 2, 8, 9, 10, 18, 19, 25, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 39, 40] and the references therein). In particular, it was established by Gui and Wei ([28]), that given two arbitrary integersl₁andl₂there exist solutions for the Neumann problem associated to (1.2) withl₁peaks on the boundary andl₂peaks in the interior.

Our paper deals with the system (1.1) whenγ2 =0 and is in striking contrast with the results for single-equation case (1.2), in particular with the above-quoted paper [37]: indeed we will establish that the least-energy solutions of the Dirich- let problem (1.1) exhibit a concentration behaviour at the boundary. Before going further in the analysis of this phenomenon, let us briefly outline the concentration results already known for the system (1.1). The asymptotic analysis of (1.1) has been started very recently in the papers [14]-[17] and [38]. The radially symmetric case has been investigated in [14, 16] and [38]. In [16] and [38] it is proved that for 1< p< ¹¹₇ there exists a family of positive radial solutions in

R

^{3which con-} centrates at a sphere. In [14] the concentration on all the boundary∂is produced for the problem (1.1) whenis the unit ball of

R

³. In the other recent papers [15]

and [17] multiple interior spikes have been shown to exist for (1.1) in the case of a generic bounded domain ⊂

R

³(near the harmonic centers of) and for the whole of

R

^N respectively. However peaked solutions approaching the boundary have not yet been observed for (1.1) neither forγ2=0 nor in the caseγ2=0. This paper seems to be the first attempt in this line.

In order to provide the exact formulation of our main result we first enumerate the assumptions on the function f that will be steadily assumed:

(f1) f ∈C²(

R

^{); f(t)=0 fort ≤0;}

(f2) ^f(t)

t³ is nondecresing int >0;

(f3) f(t)=O(t^p)ast → +∞, where 3< p<5;

(f4) there exists a constants θ > 4 such that 0 < θF(t) ≤ f(t)t for all t > 0, where F(t)=_t

0 f(s)ds;

(f5) the problem in the whole space

w−w+ f(w)=0, w >0 in

R

^3, w(0)= max

x∈R³w(x), lim

|x|→+∞w(x)=0, (1.3) has a unique solutionw, which is nondegenerate,i.e., denoting by L the lin- earized operator

L: H²(

R

^{3)→ L2(}

R

^{3), L[u] :=u−u+ f(w)u,} then

Kern(L)=span ∂w

∂x₁, ∂w

∂x₁, ∂w

∂x₃

.

By the well-known result of Gidas, Ni and Nirenberg ([23])wis radially symmetric and strictly decreasing inr = |x|. Moreover, by classical regularity results,w ∈ C²(

R

³⁾and the following asymptotic behavior holds:

w(r), w(r)= A

re^−r 1+O 1 r

, w(r)= −A

re^−r 1+O 1 r

, (1.4) whereA>0 is a suitable positive constant. Note that assumptions (f1)–(f3) imply

f(t)≤c₁|t|³+c₂|t|^p, F(t)≤C₁|t|⁴+C₂|t|^p+1 ∀t ≥0. (1.5) Typical examples of f satisfying (f1)-(f5) include f(t) = t₊^p where 3 < p < 5.

Other nonlinearities can be found in [7]. The uniqueness ofwis proved in [31] for the case of power-like f; for a general nonlinearity, see [5]. The nondegeneracy condition can be derived from the uniqueness argument (see [34]).

We recall the variational structure of the system (1.1): indeed for every v ∈ H₀¹()let(−)⁻¹[v²]∈ H₀¹()be the unique solution of the following problem

V +v²=0 in, V =0 on∂.

Then (1.1) is equivalent to

ε²v−v−γ (−)⁻¹[v²]v+ f(v)=0 in ,

v >0 in, v =0 on∂, (1.6)

whereγ =γ1γ2, and associated to (1.6) is the following energy functional:

J_ε[v] := 1 2

ε²|∇v|²+ |v|² dx+γ

4

(−)⁻¹[v²]v²dx−

F(v)dx. (1.7) Our aim is to establish the existence of a least-energy solutionvε for (1.6) and to show thatv_ε exhibits a point-condensation phenomenonasε → 0⁺. More pre- cisely, whenε is sufficiently small, v_ε has a single spike centered at a point P_ε located at a distance(1+o(1))εlog¹_ε from the boundary, whilev_ε vanishes every- where else. Hence the following natural question immediately arises: which part of the boundary are the pointsP_ε situated near? It is the purpose of this paper to answer this question and to give an accurate description of the profiles of the solu- tionsv_ε. Indeed we shall prove that this unique peak must be situated near themost curvedpart of∂,i.e. where the boundary mean curvature assumes its maximum;

more precisely any limiting pointP₀of the familyP_εis such that

H

(P₀), the mean curvature of∂at P₀, is a maximum value of

H

(P)over∂.

Now we proceed to state our main theorem.

Theorem 1.1. Assume that⊂

R

³is a smooth and bounded domain and that the hypotheses(f1)-(f5)hold. Then for everyε >0there exists a least-energy solution v_ε ∈ H₀¹() of (1.6). Furthermore, as ε → 0⁺, v_ε develops a spike near the maxima of the mean curvature; more precisely there exists P_ε ∈such that

(1) v_ε(x)=w_x−Pε

ε

+o(1)uniformly in; (2) dist(P_ε, ∂)=(1+o(1))εlog¹_ε.

Finally, for every sequenceεn→0⁺, up to a subsequence, (3) P_εn → P₀∈∂where

H

^(P0)=

H

^0:=maxP∈∂

H

^(P).

Remark 1.2. Notice that if, in addition, we assume the existence of a unique global maximumP₀of

H

(P), Part(3)of Theorem 1.1 holds for all the familiesP_ε, without need to pass to subsequences, and all the wavesv_ε concentrate at that point P₀ as ε→0⁺.

It is interesting to see how the geometry of the domain determines exactly the location of the spike-layers as well as how this result is in striking contrast with the result in [37] for the single Schr¨odinger equation (1.2) with Dirichlet boundary con- dition, in which the least-energy solutions are located at themost centeredpart of the domain. Furthermore, even among higher-energy solutions, it is also known that there arenopositive spike-layers concentrating near the boundary for the Dirichlet problem associated to (1.2) (see [10, 41]). On the contrary least-energy solutions with a single boundary peak close to the maxima of the mean curvature are known for (1.2) with Neumann conditions. So we are in presence of a Dirichlet problem which acts more like a Neumann problem for the single-equation case.

To our knowledge, the only other results concerning boundary-concentration occurring for a Dirichlet problem were established for the FitzHugh-Nagumo sys- tem in [11] and for changing-sign solutions of an elliptic equation in [12] by Dancer and Yan; however in [11] only the existence of such solutions is proved and the ex- act boundary limiting points are not determined, while in [12] a special kind of nonlinearity is considered such that the changing-sign solutions are obtained as mountain passes and blow-up at the boundary points which are maxima of the main curvature. Although the result in [12] looks similar to that in our paper, the loca- tions of the peaks for the two problems are different: indeed in [12] the solutions are constructed as approximation of a suitable mountain pass solution for an elliptic problem on a half-space (while our limiting problem (1.3) is in the whole space) and then the distance of the peaks from the boundary is O(ε)(and not, as in our case, O(εlog¹_ε)). This paper seems to be the first one that succeeds in locating exactly the boundary spikes forpositivesolutions of a Dirichlet problem.

The proof of Theorem 1.1 is based on theenergy method,i.e., on the derivation of an asymptotic formula for the smallest critical value J_ε^∗ := J_ε[v_ε] asε → 0⁺, in the spirit of [34, 35, 37]. However, here the technique is more complicated since we have to expand the energy up to sixth order. The first object is to apply the Mountain-Pass Lemma to obtain a critical pointv_εofJ_ε; furthermore we prove that v_ε is actually a least-energy solution of (1.6), by which it is meant thatv_ε has the smallest energyJ_ε^∗among all the solutions to (1.6), andJ_ε^∗can be characterized as

J_ε^∗= inf

v∈H₀¹()max

t≥0 J_ε[tv]. (1.8)

Then we show that forε sufficiently smallv_ε is a single spike solution which is localized in a ε-neighborhood of a maximum point P_ε with ^dist(P_ε^ε,∂) → +∞. Next, the critical step is to know the detailed structure ofv_ε aroundP_ε. To do this we first use the solutionwof the limiting problem (1.4) to construct a family of suitable functionsw˜_ε,P and then prove that the solution v_ε can be obtained as a suitable perturbation ofw˜_ε,P_ε. To perform such approximation we make extensive use of the nondegeneracy condition (f5). Once we have obtained the shape ofv_ε, we have to expandJ_ε[v_ε]= J_ε^∗ up to the order O(ε⁶). The first term in the expansion formula ofJ_ε^∗is given by I[w]ε³, whereI[w] is the energy ofw:

I[w]= 1 2

R³(|∇w|²+ |w|²)dx−

R³F(w)dx = 1 2

R³ f(w)wdx−

R³F(w)dx. The first correction term inJ_ε^∗contains the distance functionP_εfrom the boundary.

The most delicate part is the computation of the nonlocal term

(−)⁻¹[v_ε²]v_ε² which is the crucial term for locating the peaks at the boundary: its effects are felt at the orderε⁶of the expansion where it interacts with the nonlinear part

F(v_ε) giving rise to a term involving the main curvature. Finally the location ofP_εis deter- mined by usingw˜_ε,P as comparison functions (for suitableP ∈),i.e., according to the characterization (1.8), we compareJ_ε^∗with max_t≥0J_ε[tw˜_ε,P]; such compar- ison gives information on the terms in the asymptotic expansion ofJ_ε^∗, in particular on dist(P_ε, ∂)as well as on which portion of the boundary P_εapproaches to.

We believe that using the asymptotic expansions derived in this paper it is possible to construct single boundary spikes at nondegenerate critical points of the mean curvature (as in [39]), or at topologically nontrivial critical points of the mean curvature (as in [19]). It may also be possible to show the existence of clustered spikes at a local minimum point of the mean curvature (as in [8] and [29]). Another interesting problem is to study the stability of such solutions. We believe that, as for the single-equation case (see [4]), under some conditions on the exponentpthe least-energy solution constructed in this paper should be stable.

The paper is organized as follows. Section 2 is devoted to introduce some no- tation and preliminaries. In Section 3 we construct the approximated solutionw˜_ε,P

and we determine its shape. Section 4 contains the expansion of the functionalJ_εon

˜

w_ε,P as a function ofεandP. In Section 5 we construct the least-energy solutions v_ε and prove that their shape can be approximated by w˜_ε,P_ε, for suitable P_ε ∈ , up to a certain orderε^τε; furthermore an upper bound for the critical values J_ε^∗ is derived by usingw˜_ε,P as comparison functions and computing max_t≥0J_ε[tw˜_ε,P].

Finally the proof of Theorem 1.1 is completed in Section 6.

ACKNOWLEDGMENTS. This paper was begun while the first author was visiting The Chinese University of Hong Kong in April 2005. She gratefully acknowledges the hospitality of the Department of Mathematics at CUHK.

Notation

- Given A ⊂

R

³an open subset, L^p(A)is the usual Lebesgue space endowed with the norm

u_L^pp :=

A

|u|^pdxfor 1≤ p<+∞, u_∞=sup

x∈A

|u(x)|.

Furthermore H₀¹(A)is the usual Sobolev space endowed with the norm u²_H1 =

A

|∇u|²+ |u|² dx.

– Ifu:

R

^{N →}

R

is a radially symmetric function, we will continue to denote by uthe real functionr >0→u(x)with|x| =r.

– We will often use the symbol c or C to denote different positive constants independent of ε. The value ofc,C is allowed to vary from line to line (and also in the same formula).

– o(1)denotes a vanishing quantity asε→0⁺.

– Given{a_ε}_ε>0and{b_ε}_ε>0two family of numbers, we writea_ε =o(b_ε) (resp.

a_ε =O(b_ε)) to mean that ^a_b^ε

ε →0 (resp.|a_ε| ≤C|b_ε|) asε→0⁺.

2.

Preliminaries

In this section we collect some preliminary results concerning the variational struc- ture of the system (1.6). In particular we recall some well-known facts on the rep- resentation formula for the Poisson equation: for a smooth domain there exists a unique Green’s functionG(x,z)of the Laplace operator with Dirichlet boundary condition (see [30]). FurthermoreGis symmetric inxandzand

0<G(x,z) < 1

4π|x−z| ∀x,z∈×, x =z. (2.1) Proposition 2.1. Let be a smooth and bounded domain of

R

³. For every g ∈ L²()denote by(−)⁻¹[g]the unique solution in H₀¹()of

−ψ =g. (2.2)

Then the following representation formula holds:

(−)⁻¹[g](x)=

G(x,z)g(z)dz. (2.3) Furthermore

a)

(−)⁻¹[g]hdx=

(−)⁻¹[h]gdx for every g,h∈L²(); b) (−)⁻¹[g]_∞≤Cg_L²for every g∈L²();

c) (−)⁻¹[g]_∞≤ε²g_∞+ ¹_εg_L¹ for every g∈L^∞(); d) the functional

J

^:u∈ H₀¹()→

u²(−)⁻¹[u²]dx is C¹and

J

^[u], v =4

uv(−)⁻¹[u²]dx ∀u, v ∈H₀¹().

Proof. By Lax-Milgram’s lemma we get the existence of a unique solution inH₀¹() of (2.2). The representation formula (2.3) holds foru ∈C₀^∞()(see, for example, [22, page 23, Theorem 1]); by density (2.3) can be extended to anyg∈ L²(). a) follows immediately from (2.3) and Fubini-Tonelli’s theorem. By (2.1) for every g∈L²(), by using H¨older’s inequality, we have

|(−)⁻¹[g](x)| ≤ 1 4π

|g(z)|

|z−x|dz

≤ 1 4πgL²

|z|≤2diam()

1

|z|²dz 1/2

≤CgL², while, forg∈L^∞(),

|(−)⁻¹[g](x)| ≤ 1 4π

|z−x|≤ε

|g(z)|

|z−x|dz+ 1 4πε

|g(z)|dz

≤ g∞

4π

|y|≤ε

1

|z|dz+ 1

4πεgL¹ = ε²g∞

2 + 1

4πεgL¹

and we obtain b)-c). Part d) is a direct computation.

In view of d) of Proposition 2.1 the energy functional J_ε defined in (1.7) is of classC¹(H₀¹(),

R

⁾and its critical points correspond to the solutions of (1.6).

FurthermoreJ_ε can be rewritten as J_ε[v]=1

2

ε²|∇u|²+u² dx−

F(u)dx+ γ 4

G(x,z)u²(x)u²(z)dxdz. 3.

Computation of

w˜_ε,P

In this section we introduce some suitable approximated solutions and derive some crucial estimates: first set

w_ε,P(x)=w x−P ε

, x, P∈

R

^3.

Next for everyP∈

R

^3definew˜ε,P to be the unique solution of the problem ε²w˜_ε,P− ˜w_ε,P+ f(w_ε,P)=0 in, w˜_ε,P =0 on∂. (3.1) From the comparison principle it is immediate that

0<w˜_ε,P(x) < w_ε,P(x) ∀x ∈,∀P∈

R

^{3. (3.2)} The goal is to obtain an asymptotic expansion of the approximationsw˜ε,P. To this aim some preparations are needed. First define the distance functiond_P from the boundary∂by

d_P =dist(P, ∂), P∈

R

^3.

The regularity ofimplies that∂satisfies the uniform interior and exterior sphere condition; that is, at each point Q ∈ ∂there exist two balls B₁, B₂ such that B₁∩ ¯ = {Q}, B₂∩(

R

^{3\) = {Q}}, and the radii of the balls B₁ and B₂are bounded from below by a positive constant; taking such constant asµ, we obtain that, set

_µ:= {P∈|d_P ≤µ},

for everyP∈µthere exists a uniqueP ∈∂such that|P−P| = |P−P^∗| = d_P = d_P∗, (see, for example, [24], page 355), where P^∗ = 2P − P (i.e. P^∗ is the symmetric ofPwith respect to the tangent plane at∂inP). Notice that by construction, using (1.4),

w_ε,P^∗(x)≤C ε

d_Pe^{−|x−P∗|ε} ≤C ε

d_Pe^−dPε ∀x∈,∀P∈_µ. (3.3) w_ε,P(x)≤C ε

d_Pe^−|x−P|ε ≤C ε

d_Pe^−dPε ∀x∈

R

^3\,∀P∈

R

^{3. (3.4)}

For every P ∈_µ let

H

1(P),

H

2(P)be the principal curvatures of∂atP, so that the mean curvature

H

(P)of∂atP is given by the average:

H

(P)=

H

1(P)+

H

2(P)

2 .

We introduce a diffeomorfism which straightens a boundary portion nearP: con- siderT_P(x)the rotation and translation of coordinates which mapPin 0, the inner normal to∂atP in the positive3coordinate axis and the principal directions corresponding to

H

1(P),

H

2(P)in the1, 2axes. ThenT_P(P)= (0,0,d_P), T_P(P^∗) = (0,0,−d_P)and in some neighborhood of 0 the boundary∂(T_P)can be represented by

y₃=1 2

i=1,2

H

^j(P)yi²+ωP(y), lim

y→0

ωP(y)

|y|² =0, wherey=(y₁,y₂). (3.5) Before providing in Proposition 3.2 the asymptotic expansion of the approximated solutionsw˜_ε,P we state first the following useful result.

Lemma 3.1. Fix a > 0,b ≥0. For P ∈_µsuch that ^d_ε^P is sufficiently large the following holds

w_ε,^a_P∗|y|^b ε^b

L^∞()+ε⁻³

w_ε,^a_P∗|y|^b

ε^b dx ≤Ce^−2adP3ε (y=T_P(x)).

Proof. According to (3.3), for ^d_ε^P sufficiently large we get w^a_ε,P∗|y|^b

L^∞()+ε⁻³

w_ε,^a_P∗|y|^bdx

≤Ce^−2adP3ε w^a_ε,^/_P³∗|y|^b

L^∞()+ε⁻³

w_ε,^a/_P³∗|y|^bdx

≤Cε^be^−2adP3ε w^a/3 y+d_P3

ε

|y|^b ε^b

L^∞(R³)

+ε⁻³

R³w^a/3 y+d_P3

ε

|y|^b ε^b dy

≤Cε^be^−23εdP w^a/3(y)|y|^b_L∞(R³)+

R³w^a/3(y)|y|^bdy

≤Cε^be^−2adP3ε .

Proposition 3.2. For P ∈ such that ^d_ε^P is sufficiently large the following esti- mates hold:

i) w˜_ε,P(x)=w_ε,P(x)+O(ε⁴)uniformly for x∈and P ∈\_µ;

ii) w˜ε,P(x)=wε,P(x)−wε,P^∗(x)+O(ε)h_ε,P(x)+k_ε,P(x)uniformly for x ∈ and P ∈_µwhere h_ε,P and k_ε,P solve

ε²h_ε,P−h_ε,P=0in, h_ε,P=w_ε,P^∗|y|²

ε² +ε⁴on∂, y=T_P(x), (3.6) ε²k_ε,P−k_ε,P = −f(w_ε,P^∗), k_ε,P =0on∂. (3.7) Furthermore

h_ε,P²_∞+ε⁻³

h²_ε,Pdx =O(ε⁸+e^−43εdP), k_ε,P²_∞+ε⁻³

k_ε,²_Pdx =O(e^−4dPε )

(3.8)

uniformly for P ∈_µ. iii) ε∇ ˜w_ε,P(P)=O

ε⁴+εe^−23εdP +e^−2dPε .

Proof. The proof of Part i) is immediate: indeedw_ε,P − ˜w_ε,P satisfies

ε²(w_ε,P − ˜w_ε,P)−(w_ε,P− ˜w_ε,P)=0 in, w_ε,P− ˜w_ε,P =w_ε,P on∂.

On the other hand by the definition of_µ w_ε,P ≤ Ce^−µε = O(ε⁴)uniformly for x ∈∂andP ∈\_µ. The maximum principle impliesw_ε,P − ˜w_ε,P =O(ε⁴) uniformly forx ∈andP∈\_µ.

We go on with the proof of Part ii), which is more technical. During its proof it is understood, even though not stated plainly, that all the estimates hold uniformly forP∈_µ. First decompose

˜

w_ε,P =w_ε,P−w_ε,P^∗−εhˆ_ε+k_ε wherehˆ_εsolves the following problem

ε²hˆ_ε− ˆh_ε =0 in, hˆ_ε = wε,P −wε,P^∗

ε on∂,

andk_ε =k_ε,P solves (3.7).

The first object is to prove the following estimate for the boundary points:

|w_ε,P(x)−w_ε,P^∗(x)|

ε ≤Cw_ε,P^∗(x)|y|²

ε² +ε⁴ (3.9)

uniformly forx ∈∂. Indeed

d_P ≤ |x−P| = |y−d_P3| =

d²_P + |y|²−2d_Py₃≥ |y|, d_P ≤ |x−P^∗| = |y+d_P3| =

d_P² + |y|²+2d_Py₃≥ |y|

(3.10)

uniformly forx ∈∂. Using (3.5) we havey₃=O(|y|²)on∂; consequently

||x− P| − |x−P^∗|| = ||x−P|²− |x−P^∗|²|

|x−P| + |x−P^∗|

= 4d_P|y₃|

|x−P| + |x−P^∗| ≤2|y₃| ≤C|y|²

(3.11)

uniformly for x ∈ ∂. Take x ∈ ∂ and distinguish two cases: first assume

|y| ≥√

ε; then, by (3.10),|x−P|, |x−P^∗| ≥√

ε, and, by (1.4),

|w_ε,P(x)|, |w_ε,P^∗(x)| ≤C√ εe⁻

√1ε ≤ε⁵.

Next assume|y| ≤ √

ε; then for every r ∈ [min{|x − P|,|x − P^∗|},max{|x− P|,|x−P^∗|}] by (3.11) we have ^|r−|x_ε^−P∗|| ≤C, by which, using again (1.4),

wr

ε ≤Cε

re^−|r|ε ≤C ε

|x−P^∗|e^{−|x−εP∗|} ≤Cw_ε,P^∗(x);

hence, by applying the mean value theorem, we get

w_ε,P(x)−w_ε,P^∗(x)≤Cw_ε,P^∗(x)||x−P| − |x− P^∗||

ε ≤Cw_ε,P^∗(x)|y|² ε uniformly forx ∈∂with|y| ≤√

ε. Hence (3.9) holds. The maximum principle applies and gives| ˆh_ε| ≤C h_ε whereh_ε :=h_ε,P.

By multiplying both members of (3.6) byh_ε(x)−w_ε,P^∗(x)^|y_ε2^|2−ε⁴and since, using (1.4),

∇

w_ε,P^∗(x)|y|²

ε² ≤ C

εw_ε,P^∗(x)

1+ |y|² ε²

integrating by parts we get

(ε²|∇h_ε|²+h²_ε)dx ≤C

w_ε,P^∗

1+ |y|² ε²

(ε|∇h_ε| +h_ε)dx+ε⁴

h_εdx

≤C





w_ε,²_P∗

1+|y|² ε²

₂ dx





1/2

(ε²|∇h_ε|²+h²_ε) 1/2

+Cε⁴

h²_{ε 1/2}

≤Cε^3/2e^−2dP3ε

(ε²|∇h_ε|²+h²_ε)dx 1/2

+Cε⁴

h²_ε 1/2

where, in the last inequality, we have used Lemma 3.1. In the same way, by multi- plying both members of (3.7) byk_ε, since by (1.5) f(w)≤Cw³, we have

(ε²|∇k_ε|²+k_ε²)dx≤C

w_ε,³_P∗k_εdx ≤Cε^3/2e^−2dPε

k²_εdx _1/2

and a first part of (3.8) follows. In order to complete the proof, first notice that by the maximum principle we derive h_ε,k_ε ≥ 0 in . Furthermore according to Lemma 3.1w_ε,P^∗|y|²

ε² ≤ Ce^−23εdP on∂, hence from the maximum principle, h_ε,P∞ = O(e^−23εdP +ε⁴). In the same way f(w_ε,P^∗)≤ Cw_ε,³_P∗ ≤Ce^−2dPε on , then the comparison principle givesk_ε,P∞ =O(e^−2dPε ).

Finally to prove Part iii), observe thatz_ε,P =(ε⁴+εe^−23εdP +e^−2dPε )⁻¹ w−

˜

w_ε,P(εx+P) solves

z_ε,P =z_ε,Pin B(0,1)

(note that for ^d_ε^P is sufficiently large we have B(0,1) ⊂ ⁻_ε^P). Furthermore by Parts i)-ii) it follows thatz_ε,P is uniformly bounded onB(0,1)(note thatw_ε,P^∗(εx+ P) ≤ Ce^{−|P−P∗|ε} ≤ Ce^−2dPε on B(0,1)). Then from the well-known Schauder interior estimatez_ε,P and its first and second derivatives are uniformly bounded on the compact sets ofB(0,1). Then iii) follows.

An easy consequence of Proposition 3.2 is the following corollary.

Corollary 3.3. Settingw˜_ε,P = 0for x ∈ , we have w˜_ε,P(εx + P) → win H¹(

R

³)and L^∞(

R

³)as ^d_ε^P → +∞uniformly for P ∈.

Proof. First observe that w_ε,P(εx)_H1(R³\_ε), w_ε,P(εx)_L∞(R³\_ε) → 0⁺ as

dP

ε → +∞uniformly for P∈. On the other hand for P∈_µ, since ^dP_ε^∗ = ^d_ε^P, then we deducew_ε,P^∗(εx)_H1(_ε),w_ε,P^∗(εx)_L∞(_ε) →0⁺as^d_ε^P → +∞. Then by i) and ii) of Proposition 3.2 this implies thatw˜_ε,P(εx)−w_ε,P(εx)→0 inL²(

R

³⁾ andL^∞(

R

³)as ^d_ε^P → +∞uniformly forP ∈ . By multiplying equation (3.1) byw˜_ε,P and integrating by parts we get

˜w_ε,P(εx)²_H1(R³) =

R³ f(w)w˜_ε,P(εx+P)dx →

R³ f(w)wdx= w²_H1(R³), by whichw˜ε,P(εx+P)→winH¹(

R

³)as^d_ε^P → +∞uniformly forP∈.

Our next lemma provides an estimate of the error up tow˜_ε,P satisfies the sys- tem (1.6). To this aim set

S

ε[v]=ε²v−v−γ (−)⁻¹[v²]v+ f(v), v∈ H²().

Lemma 3.4. For P ∈such that ^d_ε^P is sufficiently large the following holds:

S

ε[w˜_ε,P]≤C

e^−74εdP +εe^−23εdP +ε² w¹_ε,^/_P⁴. Proof. According to c) of Proposition 2.1 and (3.2)

|(−)⁻¹[w˜_ε,²_P]w˜_ε,P| ≤ε²

w²_ε,PL^∞()+ε⁻³w²_ε,PL¹₍₎

w_ε,P ≤Cε²w_ε,P. We just need to estimate the local term: by (3.1) and assumption (f1) we deduce

ε²w˜_ε,P − ˜w_ε,P+ f(w˜_ε,P)=f(w˜_ε,P)− f(w_ε,P)≤Cw_ε,P

w_ε,P − ˜w_ε,P

.

If P ∈ \_µ the thesis follows from Part i) of Proposition 3.2. Now assume P∈_µ;By using Part ii) of Proposition 3.2 we get

ε²w˜_ε,P− ˜w_ε,P+ f(w˜_ε,P)≤Cw_ε,P

w_ε,P^∗+εh_ε,P +k_ε,P

≤Cw_ε,P

w_ε,P^∗+εe^−2dP3ε +ε⁴+e^−2dPε . In order to conclude by (1.4) and (3.3) we compute

w_ε,Pw_ε,P^∗ ≤Cw¹_ε,^/_P⁴e^{−3|x−4εP|−3|x−P∗|4ε} e^−dP4ε

≤Cw¹_ε,^/_P⁴e^{−3|P−P∗|4ε} e^−dP4ε =Cw_ε,^1/_P⁴e^−74εdP uniformly forx ∈andP∈_µ.