Multi-bump standing waves with critical frequency for nonlinear Schrödinger equations

www.elsevier.com/locate/anihpc

Multi-bump standing waves with critical frequency for nonlinear Schrödinger equations

Jaeyoung Byeon

, Yoshihito Oshita

aDepartment of Mathematics, Pohang University of Science and Technology, Pohang, Kyungbuk 790-784, Republic of Korea bDepartment of Mathematics, Okayama University, Tsushimanaka 3-1-1, Okayama 700-8530, Japan

Received 8 May 2009; received in revised form 6 April 2010; accepted 6 April 2010 Available online 22 April 2010

Abstract

We glue together standing wave solutions concentrating around critical points of the potentialV with different energy scales.

We devise a hybrid method using simultaneously a Lyapunov–Schmidt reduction method and a variational method to glue together standing waves concentrating on local minimum points which possibly have no corresponding limiting equations and those con- centrating on general critical points which converge to solutions of corresponding limiting problems satisfying a non-degeneracy condition.

©2010 Elsevier Masson SAS. All rights reserved.

Résumé

Nous recollons des ondes stationnaires d’ordres différents en énergie, se concentrant autour de points critiques d’un potentielV. Nous introduisons une méthode hybride, utilisant à la fois une méthode de réduction de Lyapunov–Schmidt, et une méthode variationnelle pour recoller des ondes stationnaires, se concentrant en des minima locaux, éventuellement sans équation-limite correspondante, et d’autres se concentrant en des points critiques quelconques, convergeant vers des solutions de problèmes-limites correspondants, satisfaisant une condition de non-dégénérescence.

©2010 Elsevier Masson SAS. All rights reserved.

1. Introduction and statement of main results

We consider a standing wave solution for the nonlinear Schrödinger equation ih¯∂ψ

∂t + ¯h²

2 ψ−V (x)ψ+f (ψ )=0, (t, x)∈R×Rⁿ, (1.1)

whereh¯denotes the Plank constant,iis the imaginary unit andf (e^iθψ )=e^iθf (ψ ).A solution of the formψ (x, t )= exp(−iEt /h)v(x)¯ is called a standing wave solution of the nonlinear Schrödinger equation (1.1). Then, a function ψ (x, t )≡exp(−iEt /h)v(x)¯ is a standing wave solution of (1.1) if and only if the functionvsatisfies

* Corresponding author.

E-mail addresses:[email protected] (J. Byeon), [email protected] (Y. Oshita).

0294-1449/$ – see front matter ©2010 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2010.04.002

¯ h²

2 v−

V (x)−E

+f (v)=0, x∈Rⁿ. (1.2)

For the physical background, refer to [8,29] and [30].

In this paper, we study standing waves of (1.1) for smallh >¯ 0.For small h >¯ 0, these standing waves of the nonlinear Schrödinger equation (1.1) are referred to as semi-classical states. Thus we are concerned on the following equation

ε²u−V (x)u+f (u)=0, u >0 inRⁿ,

lim_|x|→∞u(x)=0. (1.3)

In this paper, we are interested in the situation whereE is a critical frequency in the sense that minx∈RⁿV (x)=0.

Since the pioneering work [21], there haven been many further papers for the case infx∈RⁿV (x) > E (refer to [1,2, 4,5,14–21,24,25,27,28,31–34,37–39] and references therein). When infx∈RⁿV (x) >0,we see via a transformation v(x)≡u(εx)that the following equations with constantc >0 serve as limiting equations of (1.3)

u−cu+f (u)=0, u >0 inRⁿ,

lim_|x|→∞u(x)=0. (1.4)

Thus, if infx∈RⁿV (x) >0,for any solutionu_ε of (1.3), lim infε→0u_εL^∞>0. On the other hand, in the case of minx∈RⁿV (x) =0, it was shown in [8] and [9] that there exists a locally minimal energy solution wε of (1.3) concentrating around an isolated component of global minimum points of V as ε→0.In contrast to the case of infx∈RⁿV (x) >0,the amplitudewεL∞ and energy of the localized solutionwεconcentrating around global min- imum points of V decay to 0 asε→0,and their decay rates depend subtly upon how the potentialV decays to 0 around the concentration points. Moreover, if the decaying behavior of potentialV to 0 is sufficientlyirregular, there will be no corresponding limiting problem; in such a case, an exact estimation of the amplitude and energy of the corresponding solution may not be possible. This makes the gluing of the localized solutions very difficult. If the decaying behavior of the potentialV to 0 around global minimum points isregularand the corresponding limiting problems have good properties, a gluing of solutions concentrating around global minimum points has been worked out in [7,10,12,11]. Recently, without requiring the existence of limiting problems, Sato [36] were able to glue local- ized solutions concentrating around local minimum points whenf (u)=u^p, p∈(1,2^∗), where 2^∗=(n+2)/(n−2) for n3 and 2^∗= ∞forn=1,2. He glues the solution via a minimization on a torus of finite codimension in a Sobolev space which depends strongly on the homogeneity of anf (u)=u^p.In this paper we devise a new approach to glue together the localized solution for a more general type of nonlinearity, where the solution cannot be obtained via a minimization argument. In fact, we glue together the localized solution concentrating around local minimum points for quite general nonlinearitiesf without a monotonicity assumption forf (t )/t.Furthermore, we use both a variational method and a Lyapunov–Schmidt reduction method to glue together the solutions concentrating around global minimum points (without requiring any existence of limiting problems) and the solutions concentrating around stable critical points of potentialV .We have never seen this approach using simultaneously both variational method and Lyapunov–Schmidt reduction method in the literature.

To begin, we list some conditions forV andf:

(V1) V ∈C(Rⁿ); lim inf_|x|→∞V (x) >0=infx∈RⁿV (x);

(V2) there exist disjoint bounded open sets Ω_i with smooth boundary ∂Ω_i, i=1, . . . , k, satisfying 0m_i = infx∈ΩiV (x) <minx∈∂ΩiV (x);

(V3) there arex₁, . . . , x_m∈Rⁿ and disjoint bounded open setsΩ_k+1, . . . , Ω_k+m with smooth boundaries∂Ω_k+j such thatx_j∈Ω_k+j, V ∈C²(Ω_k+j),∇V (x) =0 forx∈Ω_k+j\ {x_j}, infx∈Ωk+jV (x) >0 andx_j is a non- degenerate critical point ofV, forj∈ {1, . . . , m};

(f1) f ∈C¹(R),f (t )=0 fort0 and there exist someμ₁>1 andC >0 satisfying|f (t )|Ct^μ1 fort∈(0,1);

(f2-1) there exists somep∈(1,ⁿ_n⁺₋²₂)forn3 andp∈(1,∞)forn=1,2 such that lim inft→0+ 1 t^p+1

_t

0f (s) ds >0;

(f2-2) there exists somep∈(1,ⁿ_n⁺₋²₂)forn3 andp∈(1,∞)forn=1,2 such that lim sup_t→∞^{|f (t )|+|}_tp^{f(t )t|}<∞;

(f3-1) there existsμ₂>1 such that(μ₂+1)_t

0f (s) dsf (t )tfort >0;

(f3-2) there existsμ₃>1 andt₁>0 such thatμ₃f (s)f(t )tfort∈ [0, t1];

(f3-3) there existsμ₃>1 such thatμ₃f (s)f(t )tfort >0;

(f4) for anya∈ {V (x₁), . . . , V (x_m)},the problem

u−au+f (u)=0, u >0 inRⁿ, u∈H^1,2 Rⁿ

(1.5) has a radially symmetric solutionUawhich is non-degenerate inH_r^1,2(Rⁿ)≡ {w∈H^1,2(Rⁿ); w(x)=w(|x|)}, andf∈C_loc^1,γ(R)for someγ∈(0,1).

It is proved in [8] that if (f1), (f2-1) and (f3-1) hold, there exists a positive solution of (1.3) concentrating around an isolated component of zeros ofV, and that if (f1), (f2-1), (f2-2) and (f3-1) hold, there exists a positive solution of (1.3) concentrating around an isolated component of local minimum points ofV .In this paper, we will glue together the solutions found in [8] under the same conditions.

Throughout this paper we assume (f1), (f2-1) and (f3-1). Then we see thatf (t ) >0 fort >0.Note that (f1), (f2-1) and (f3-1) hold forf (t )=t₊^p+t₊^q withp∈(1,2^∗), q >1.

Some remarks about the above conditions on f are in order. If we take any μ∈(1,min{μ₁, μ₂, μ₃}), then the conditions (f1), (f3-1), (f3-2) hold withμ instead of μ₁, μ₂, μ₃,respectively. Note that (f1) or (f3-2) implies lim sup_t→0^{f (t )}_tμ <∞, that (f3-1) implies lim sup_t→0 ¹

t^μ2+1

t

0f (s) ds <∞,and thatμ₁, μ₂, μ₃pif the conditions for μ₁, μ₂, μ₃ andp hold. For μ < μ₁,we can findt₀∈(0, t1)such thatμf (t₀)f(t₀)t₀. If not, there exists a constantC >0 such thatf (t )Ct^μfort∈(0,1); this contradicts (f1). Now, we define

f (t )˜ =

f (t ) tt₀,

f (t0)+^f(t0)

μt₀^μ−1(t^μ−t₀^μ) t > t0. Then we see that(μ+1)_t

0f (s) ds˜ f (t )t˜ for allt >0 if (f3-1) holds, and thatμf (t )˜ f˜(t )tfor allt >0 if (f3-2) holds. Refer to [26] for the result related to the non-degeneracy condition appearing in (f4).

To state our main results we give some definitions. We defineAi ≡ {x∈Ωi|V (x)=mi}fori=1, . . . , k,Z≡ {x∈Rⁿ|V (x)=0},F (t )≡t

0f (s) ds,F (t )˜ ≡t

0f (s) ds˜ and La(u)≡1

2

Rⁿ

|∇u|²+au²dx−

Rⁿ

F (u) dx, u∈H¹ Rⁿ

. For anya >0, letSabe the set of all least energy solutions of the problem

u−au+f (u)=0, u >0 inRⁿ, u(0)=max

x∈Rⁿu(x). (1.6)

It is known in [37] that for eacha >0, Sais nonempty and if (f1), (f2-2) and (f3-1) are satisfied, and in [22] that any u∈Sais radially symmetric. Moreover, any solutionU∈Sasatisfies

U (x)Ce^−c|x|, for some constantsC, c >0,

n−2 2

Rⁿ

|∇U|²dx+na 2

Rⁿ

U²dx−n

Rⁿ

F (U ) dx=0, and

L_a(U )=1 2

Rⁿ

|∇U|²dx+a 2

Rⁿ

U²dx−

Rⁿ

F (U ) dx=1 n

Rⁿ

|∇U|²dx.

Fori=1, . . . , k, we consider the following localized problem ε²u−V (x)u+h(u)=0, u >0 inΩ_i,

u(x)=0 on∂Ω_i, (1.7)

whereh=f ifmi>0 andh= ˜f ifmi=0.We define J_ε(u;Ω_i)≡1

2

Ωi

ε²|∇u|²+V u²dx−

Ωi

H (u) dx,

where H (t )=

t 0

h(s) ds=

F (t ) ifmi>0,

F (t )˜ ifmi=0. (1.8)

Then a solutionu_εof (1.7) corresponds to a critical point of the energy functionalJ_ε(u;Ω_i)onH₀¹(Ω_i).

If (f3-1) holds, for each nonnegative functionh^ε_i ∈H₀¹(Ω_i)\ {0},we can findt (h^ε_i) >0 such that fortt (h^ε_i), J_ε(t h^ε_i;Ω_i) <0.Then we define

C_εⁱ≡ inf

γ∈Φ_εⁱ

max

t∈[0,1]Jε

γ (t );Ωi

,

whereΦ_εⁱ≡ {γ∈C([0,1], H₀¹(Ω_i))|γ (0)=0, γ (1)=t (h^ε_i)h^ε_i}. Then, it follows from the Mountain Pass Theorem (refer to [35]) that if (f1) and (f3-1) are satisfied whenm_i =0,and (f1), (f2-2) and (f3-1) whenm_i>0,there exists a mountain pass solutionuⁱ_εof (1.7) withJ_ε(uⁱ_ε, Ω_i)=C_εⁱ.The main result in [9] implies that in casem_i=0,if we further assume (f2-1), limε→0ε⁻ⁿC_εⁱ=0,

εlim→0

uⁱ_ε

L^∞(Ωi)=0, lim inf

ε→0 ε^−μ2−1uⁱ_ε

L^∞(Ωi)>0

and in case m_i >0, limε→0ε⁻ⁿC_εⁱ =L_mi(U ) for U ∈ S_mi and there exists a maximum point xⁱ_ε of uⁱ_ε with limε→0dist(xⁱ_ε, A_i)=0 such that for some C, c >0, u(x)Cexp(−^c|x−_ε^xεi|)and uⁱ_ε(· +x_εⁱ) converges (up to a subsequence) uniformly to a functionU∈S_mi.

For any setA⊂Rⁿandd >0,we defineA^d≡ {x∈Rⁿ|dist(x, A) < d}.

Theorem 1.Suppose that(V1)and(V2)hold. Assume thatm₁= · · · =m_l=0< m_l+1, . . . , m_k.Suppose that(f1), (f2-1), (f3-1)hold ifl=k, and that(f1), (f2-1), (f2-2), (f3-1)hold ifl < k. Then, for sufficiently smallε >0, there exists a positive solutionu_εof (1.3)such that

(i) for any sufficiently smalld >0, there existC, c >0satisfying u_ε(x)Cexp

−cdist

x, (A₁∪ · · · ∪A_k)^d /ε

; (ii) fori=1, . . . , l,

εlim→0u_εL^∞(Ωi)=0, lim

ε→0ε^{−2/(μ−1)}u_εL^∞(Ωi)>0;

and fori=l+1, . . . , k,a least energy solutionU∈S_miand somexⁱ_ε∈Rⁿwithlimε→0dist(x_εⁱ, A_i)=0, a trans- formed solutionuε(εx+x_εⁱ)converges(up to a subsequence)uniformly toU (x)on each bounded set inRⁿ. Moreover, if (f3-2)and(f3-3)are also satisfied whenl=kandl < k,respectively, there exist someC, c >0such that fori=1, . . . , k,

J_ε(u_ε;Ω_i)−C_i^ε Ce^−c/ε.

Theorem 2.Assume that(V1), (V2)and(V3)hold. Suppose that(f1), (f2-1), (f3-1)and(f4)hold. Then for sufficiently smallε >0, there exists a positive solutionuεof (1.3)such that

(i) for any sufficiently smalld >0, there existC, c >0satisfying u_ε(x)Cexp

−cdist x,

A₁∪ · · · ∪A_k∪ {x₁} ∪ · · · ∪ {x_m}d /ε

; (ii) form_i=0,

εlim→0u_εL^∞(Ωi)=0, lim

ε→0ε^{−2/(μ−1)}u_εL^∞(Ωi)>0,

and form_i>0, there existsx_εⁱ∈Rⁿsuch thatlimε→0dist(x_εⁱ, A_i)=0and thatu_ε(εx+x_εⁱ)converges uniformly (up to a subsequence)to someU_mi∈S_mion each bounded set inRⁿ;

(iii) for eachi=1, . . . , m,there existsy_εⁱ∈Rⁿsuch thatlimε→0y_εⁱ =x_i and thatu_ε(εx+y_εⁱ)converges uniformly (up to a subsequence)toU_{V (xi)}on each bounded set inRⁿ. HereU_{V (xi)}is a function given in(f4).

Moreover, if (f3-2)and(f3-3)are also satisfied whenl=kandl < k,respectively, there exist someC, c >0such that fori=1, . . . , k,

J_ε(u_ε;Ω_i)−C_i^ε Ce^−c/ε.

We conclude the introduction with an outline of our proof of the main results.

We look for a critical point of some energy functionalΓ_ε. First we choose a bounded open set Ω₀ enclosing Z\_k

i=1A_i (see Section 2 for a precise condition forΩ₀). Then we consider a modified problem outside_k+m

i=0 Ω_i for each given function u_i on Ω_i, and show the existence of a solution ϕ(u₀, . . . , u_k+m)for the external problem solving by a minimization problem. We also show the existence of a solutionP_i(u_i)of the modified problem by a minimization such thatP_i(u_i)(x)=0 for dist(x, Ωi)δ >0 andP_i(u_i)=u_i onΩ_i.Then, we will show that finding a solution is reduced to finding a critical point of a reduced functionalI_ε(u0, . . . , u_k+m).Finding a good estimate for

˜

ϕ(u₀, . . . , u_k+m)≡ϕ(u₀, . . . , u_k+m)−

k+m i=0

P_i(u_i)

is important in our proof. In fact, we show thatϕ˜ is exponentially small with respect to small ε >0 (see Proposi- tion 3). That enables us to regard the sumk+m

i=0 Γε(Pi(ui))of localized functionals depending only on eachui as an exponentially small perturbation of the reduced functionalIε. This is a novelty of our argument.

To prove Theorem 1, we consider an energy gradient flow in a product of an appropriate small ball inH¹(Ω₀)and appropriate annuli inH¹(Ω_i), i=1, . . . , k. To take appropriate radii of the ball and annuli is also important in our proof. If there exist no solutions in the product of the ball and the annuli, we show that via a gradient estimation near the boundary of the product of the ball and the annuli, we can deform a product of localized mountain paths into a surface where the maximum energy is less than a sum of independent local mountain pass levels by an algebraic order ofε >0.Then, from the exponential smallness ofϕ(u˜ 0, . . . , u_k+m),we will get a contradiction.

For the proof of Theorem 2, we use a Lyapunov–Schmidt reduction method in a regionΩk+1∪ · · · ∪Ωk+mbefore we use the variational argument in a regionΩ₀∪ · · · ∪Ω_kas in the proof of Theorem 1. For the typical casem=1 we will find a critical point of the functionalu_k+1→I (u₀, . . . , u_k+1)by the reduction method which depends smoothly on (u₀, . . . , u_k), and then use the same argument as in the proof of Theorem 1. So we will skip the variational procedure in the proof of Theorem 2 since the required variational argument after the reduction is exactly the same as that of the proof of Theorem 1.

This paper is organized as follows. In Section 2, some preliminaries about the above reduction are given. Theo- rems 1 and 2 will be proved in Sections 3 and 4, respectively.

2. Preliminaries

We defineA₀:=Z\_k

i=1A_i andΩ₀⊃A₀a bounded open set with a smooth boundary such thatΩ₀∩Ω_i= ∅ fori=1, . . . , k+m.Forδ >0, letΩ_i^δ= {x∈Rⁿ|dist(x, Ωi)δ}, andΩ_i^−δ= {x∈Ω_i |dist(x, ∂Ωi)δ}. Taking sufficiently smallδ >0,we may assume thatA_i⊂Ω_i^−2δ, Ω_i^2δ∩Ω_j^2δ= ∅for each 0i =jk+m,and that∂Ω_i^δ is smooth for each 0ik+mandδ∈ [−2δ,2δ].Reordering the index, we may assume thatm₁= · · · =m_l=0 andm_l+1, . . . , m_k>0 for somel∈ {1, . . . , k}.In this section, we assume that (f1), (f2-1) and (f3-1) are satisfied and (f2-2) are also satisfied ifl < k.As in the previous section, we can findt0>0 such thatμf (t0)f(t0)t0.Then, we define

f (t )˜ =

f (t ) tt₀,

f (t₀)+^f(t0)

μt₀^μ−1(t^μ−t₀^μ) t > t₀. Then we see that(μ+1)t

0f (s) ds˜ f (t )t˜ for allt >0 if (f3-1) holds, and thatμf (t )˜ f˜(t )tfor allt >0 if (f3-2) holds. We define

b≡inf

V (x) x /∈Ω₀^−δ∪ · · · ∪Ω_k^−δ

>0.

We define V_ε(x):=V (εx), and H_ε the completion of C₀^∞(Rⁿ) with respect to a norm uε =(

Rⁿ|∇u|²+ V_εu²dx)^1/2. Note that H_ε ⊂H¹(Rⁿ)by (V1). We also define Ω_i,ε = {x|εx∈Ω_i},Ω_i,ε^δ = {x|εx∈Ω_i^δ}. Then we see that

inf

V_ε(x) x /∈Ω_0,ε^−δ∪ · · · ∪Ω_k,ε^−δ

=b >0.

From (f3-1), there exist someC1, C2>0 such thatf (t )C1t^μ2 fort∈ [0,1]andf (t )C2t^μ2fort∈ [1,∞).Thus, taking sufficiently smallλ∈(0,min{b/10, b(1/2−1/(μ+1))}), we can construct a functionfλ∈C¹(R)for small λ0and largeλ1>0 such that

f_λ(t )=

f (t ) tλ₀, λt tλ₁

with 0< λ0< λ1,andfλ(t )min{f (t ),f (t ), λt˜ },|f_λ(t )|2λfort0. Note thatfλ(t )=0 fort0 and

|f_λ(t₁)−f_λ(t₂)|

|t₁−t₂| 2λ fort1 =t2.

We find a functionχ∈C¹(Rⁿ)such that 0χ1, χ (x)=1 forx∈_k+m

i=0 Ω_i^−δandχ (x)=0 forx /∈_k+m

i=0 Ω_i. We define

g(x, t )=

⎧⎪

⎨

⎪⎩

χ (εx)f (t )˜ +(1−χ (εx))f_λ(t ) x∈_l

i=0Ω_i,ε, χ (εx)f (t )+(1−χ (εx))fλ(t ) x∈k+m

i=l+1Ωi,ε,

f_λ(t ) x /∈_k+m

i=0 Ω_i,ε, G(x, t )≡_t

0g(x, s) ds,and foru∈H_ε, Γ_ε(u)≡1

2

Rⁿ

|∇u|²+V_ε(x)u²dx−

Rⁿ

G(x, u) dx.

ThenΓ_ε∈C²(H_ε). For a measurable subsetU⊂Rⁿ, we define Γε(u;U )=1

2

U

|∇u|²+Vε(x)u²dx−

U

G(x, u) dx, u²_ε,U=

U

|∇u|²+V_ε(x)u²dx

foru∈Hε. Foru=(u0, . . . , uk+m)∈H¹(Ω0,ε)× · · · ×H¹(Ωk+m,ε), we define a norm u²_ε=

k+m i=0

u_i²_ε,Ωi,ε. Foru_i∈H¹(Ω_i,ε), let

X^ε_i(u_i)=

u∈H₀¹(Ω_i,ε^δ ) u=u_i onΩ_i,ε .

We regardu∈H₀¹(Ω_i,ε^δ )as an element inH¹(Rⁿ)by definingu(x)=0 forx /∈Ω_i,ε^δ .

Proposition 1.For eachu_i∈H¹(Ω_i,ε), i∈ {0, . . . , k+m}, there exists a unique minimizerP_i(u_i)ofΓ_εonX^ε_i(u_i), which satisfies the following:

(i) w=P_i(u_i)∈H₀¹(Ω_i,ε^δ )solves

⎧⎪

⎨

⎪⎩

w−V_εw+f_λ(w)=0 inΩ_i,ε^δ \Ω_i,ε,

w=ui on∂Ωi,ε,

w=0 on∂Ω_i,ε^δ ,

(ii) P_i:H¹(Ω_i,ε)→H₀¹(Ω_i,ε^δ )is of classC¹, and for allh∈H¹(Ω_i,ε),w=P_i(u_i)hsolves

⎧⎪

⎨

⎪⎩

w−V_εw+f_λ(P_i(u_i))w=0 inΩ_i,ε^δ \Ω_i,ε,

w=h inΩ_i,ε,

w=0 on∂Ω_i,ε^δ ,

(iii) there exists a positive constantC, independent of smallε >0such that P_i(u_i)

εCu_iε,Ω_i,ε for allu_i∈H¹(Ω_i,ε), i∈ {0, . . . , k+m}. This proposition can be proved in a similar way as in [36], so we omit the proof.

Foru=(u₀, . . . , u_k+m), u_i∈H¹(Ω_i,ε), let X_∗^ε(u) =

u∈H_ε u=u_ionΩ_i,ε, i=0, . . . , k+m .

By the same procedure as in the proof of Proposition 1, we can prove the following proposition.

Proposition 2.For eachu=(u₀, . . . , u_k+m)∈H¹(Ω_0,ε)× · · · ×H¹(Ω_k+m,ε), there exists a unique minimizerϕ(u) ofΓ_εonX_∗^ε(u), which satisfies the following:

(i) w=ϕ(u) solves

w−V_εw+f_λ(w)=0 in(Ω_0,ε∪ · · · ∪Ω_k,ε)^c, w=u_i on∂Ω_i,ε(i=0, . . . , k),

(ii) ϕ:H¹(Ω_0,ε)× · · · ×H¹(Ω_k,ε)→H_εis of classC¹, and for allh∈H¹(Ω_i,ε),v=^∂ϕ(_∂u^u)_i hsolves

⎧⎨

⎩

v−V_εv+f_λ(ϕ(u))v =0 in(Ω_0,ε∪ · · · ∪Ω_k,ε)^c,

v=h inΩ_i,ε,

v=0 inΩ_j,ε, j =i,

(iii) there exists a positive constantC, independent of smallε >0such that ϕ(u)

εCuε. Let ϕ(˜ u) =ϕ(u) −k+m

i=0 Pi(ui). Then it follows that ϕ(˜ u) ∈X^ε_∗(0). Now we obtain the following estimates forϕ(˜ u).

Proposition 3.For anyR >0andε0>0, there exist constantsC, c >0such that ϕ(˜ u)

εCe^−c/ε forε∈(0, ε0)anduεR.

Proof. Forui∈H¹(Ωi,ε),i=0, . . . , k+m,wi=Pi(ui)∈H₀¹(Ω_i,ε^δ )solves

⎧⎪

⎨

⎪⎩

w−V_εw+f_λ(w)=0 inΩ_i,ε^δ \Ω_i,ε,

w=u_i on∂Ω_i,ε,

w=0 on∂Ω_i,ε^δ .

By Proposition 1, there exists a constantC >0, independent ofuwithuεR, such thatw_iH¹_(Ω^δ

i,ε\Ωi,ε)C. Hence by elliptic estimates (refer to [23]), we deduce that for eachs∈(0,1),there exists a constantc >0 such that w_iL∞(Ω_i,ε^sδ\Ω_i,ε)C. By comparison principle, we conclude that for someC >0,independent of smallε >0,

w_iCe^−c/ε onΩ_i,ε^δ \Ω_i,ε^δ/2.

Then, it follows from boundary Schauder estimates [23, Corollary 8.36] that|w_i|_C1,θ(∂Ω_i,ε^δ )Ce^−c/ε. Denotingϕ= ϕ(u) =_k+m

i=0 P_i(u_i)+ ˜ϕ(u), we see that ϕsatisfies

ϕ−V_εϕ+f_λ(ϕ)=0 in(Ω_0,ε∪ · · · ∪Ω_k+m,ε)^c,

ϕ=u_i on∂Ω_i,ε.

IfuεR, it follows from (iii) of Proposition 2 thatϕ(u) εC. Hence by elliptic estimates, we see that for each s∈(0,1),

ϕ(u)

L^∞(Ω_0,ε^sδ∪···∪Ω_k^sδ_+m,ε)^cC.

Then by comparison principle, we deduce that ϕCe^−c/ε in

k+m i=0

Ω_i,ε^3δ/2\Ω_i,ε^δ/2.

Then it follows from interior Schauder estimates [23, Theorem 8.32] that ϕ(u)

C^1,θ(k+m

i=0 ∂Ω_i,ε^δ )Ce^−c/ε. Now a functionw_i=P_i(u_i)satisfies

(w_i−ϕ)−V_ε(w_i−ϕ)+f_λ(w_i)−f_λ(ϕ)

w_i−ϕ (w_i−ϕ)=0 inΩ_i,ε^δ \Ω_i,ε. Multiplying byw_i−ϕand integrating by parts, we have

∂Ω_i,ε^δ \Ωi,ε

(w_i−ϕ)∂(w_i−ϕ)

∂n dS−

Ω_i,ε^δ \Ωi,ε

∇(w_i−ϕ) ²dx

−

Ω_i,ε^δ \Ω_i,ε

V_ε(w_i−ϕ)²dx+

Ω_i,ε^δ \Ω_i,ε

fλ(wi)−fλ(ϕ)

wi−ϕ (w_i−ϕ)²dx=0.

Sincewi−ϕ=0 on∂Ωi,εandwi−ϕ_C¹_(∂Ω^δ

i,ε)=O(e^−c/ε),we see that

Ω_i,ε^δ \Ω_i,ε

∇(w_i−ϕ) ²+

Ω_i,ε^δ \Ω_i,ε

(w_i−ϕ)²=O e^−c/ε

;

hence

˜ϕ_H1(Ω^δ_i,ε\Ω_i,ε)= ϕ−w_iH1(Ω^δ_i,ε\Ω_i,ε)=O e^−c/ε

. (2.1)

We note that ϕ(˜ u) = ϕ(u) on Rⁿ \ (Ω_0,ε^δ ∪ · · · ∪Ω_k^δ_+m,ε). From a decay property of ϕ, we see that ϕ_H1(Rⁿ\Ω_0,ε^δ ∪···∪Ω_k^δ_+m,ε)=O(e^−c/ε).Thus combining this with (2.1), we get the required estimate. 2

LetIε(u) =Γε(ϕ(u)). Then from Proposition 3, we conclude that for any R, ε0>0,there exist constantsC, c >0 such that

Iε(u) −

k+m i=0

Γε

Pi(ui)

Ce^−c/ε foruεRandε∈(0, ε0). (2.2)

Moreover, we have the following properties forI_ε(u).

Proposition 4.The following hold.

(i) A vector functionu=(u₀, . . . , u_k+m)∈H¹(Ω_0,ε)× · · · ×H¹(Ω_k+m,ε)is a critical point ofI_εif and only ifϕ(u) is a critical point ofΓ_ε.

(ii) The functionalu→I_ε(u) satisfies(PS)condition ifΓ_εdoes.

(iii) For anyR >0,i=0, . . . , k+mandε₀>0,there exist constantsC, c >0such that ∂I_ε

∂u_i(u0, . . . , uk+m)−dΓ_ε(P_i(u_i)) du_i

Ce^−c/ε forε∈(0, ε0)anduεR.

Proof. (i) For allζ=(ζ₀, . . . , ζ_k+m)∈H¹(Ω_0,ε)× · · · ×H¹(Ω_k+m,ε), we haveϕ˜(u) ζ ∈X_∗^ε(0).Then, it follows from the definition ofϕthat for allh∈X^ε_∗(0),

I_ε(u) ζ=Γ_ε ϕ(u)

ϕ(u) ζ

=Γ_ε

ϕ(u) _k+m

i=0

P_i(u_i)ζ_i+ ˜ϕ(u) ζ

=Γ_ε

ϕ(u) _k+m

i=0

P_i(u_i)ζ_i

=Γ_ε

ϕ(u) _k+m

i=0

P_i(u_i)ζ_i+h

. Then, from the factH_ε= {_k+m

i=0 P_i(u_i)ζ_i+h|ζ_i∈H¹(Ω_i,ε), h∈X_∗^ε(0)},the equivalence of (i) follows.

(ii) We see from Propositions 1 and 2 that forζ=(ζ₀, . . . , ζ_k+m)∈H¹(Ω_0,ε)× · · · ×H¹(Ω_k+m,ε), Γ_ε

ϕ(u)_k+m

i=0

P_i(ui)ζi+h

I_ε(u)ζ

I_ε(u)ζε

I_ε(u)

k+m i=0

P_i(u_i)ζ_i+h ε

.

Thus it follows that Γ_ε(ϕ(u)) I_ε(u) . Note that Γε(ϕ(u)) =Iε(u) and for some C >0, uεϕ(u)ε Cuε. Thus, we conclude thatIεsatisfies (PS) condition ifΓεdoes.

(iii) Forh=(h₀, . . . , h_k+m),u=(u₀, . . . , u_k+m)withh_i, u_i∈H¹(Ω_i,ε), a functionv=_k+m

i=0

∂ϕ(u)

∂ui h_i solves v−V_εv+f_λ(ϕ(u))v =0 in(Ω0,ε∪ · · · ∪Ω_k+m,ε)^c,

v=hi inΩi,ε.

By the minimization characterization of v, we have vε ChεC for a constant C independent of h with hε1.

Similarly, a functionwi=P_i(ui)hi solves

⎧⎨

⎩

w−Vεw+f_λ(Pi(ui))w=0 inΩ_i,ε^δ \Ωi,ε,

w=h_i inΩ_i,ε,

w=0 on∂Ω_i,ε^δ .

By the minimization characterization ofwi, we havewiεChiεC for a constantC independent ofhwith hε1.

Since ^k+m

i=0

Γ_ε ϕ(u)

w_i−

k+m i=0

Γ_ε P_i(u_i)

w_i

^k+m

i=0

Ω_i,ε^δ

∇

ϕ(u) −P_i(u_i)

· ∇w_i+V_ε

ϕ(u) −P_i(u_i) w_idx