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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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New solutions for critical Neumann problems in ℝ2

Shengbing Deng / Monica Musso
Published Online: 2017-07-28 | DOI: https://doi.org/10.1515/anona-2017-0092

Abstract

We consider the elliptic equation -Δu+u=0 in a bounded, smooth domain Ω in 2 subject to the nonlinear Neumann boundary condition uν=λueu2, where ν denotes the outer normal vector of Ω. Here λ>0 is a small parameter. For any λ small we construct positive solutions concentrating, as λ0, around points of the boundary of Ω.

Keywords: Elliptic partial differential equations; nonlinear Neumann boundary condition, Trudinger–Moser trace embedding

MSC 2010: 35J05; 35J08; 35J25; 35J60; 35J61

1 Introduction

Let Ω be a bounded domain in 2 with smooth boundary and λ>0. This paper is concerned with the existence of positive solutions to the boundary value problem

-Δu+u=0,u>0in Ω,uν=λueu2on Ω,(1.1)

where ν denotes the outer unitary normal vector of Ω. Elliptic equations with nonlinear Neumann boundary condition of exponential type arise in conformal geometry (prescribing Gaussian curvature of the domain and curvature of the boundary), see for instance [9, 10, 24] and references therein, and in corrosion modelling, see [7, 20, 26, 27].

Problem (1.1) is the Euler–Lagrange equation for the functional

Jλ(u)=12Ω[|u|2+u2]-λ2Ωeu2,uH1(Ω).

For functions uH1(Ω), the maximal growth of integrability on the boundary is of exponential type, due to the Trudinger trace embedding (see [29, 32])

H1(Ω)ueu2Lp(Ω)for all p1.

This optimal embedding is related to the critical Trudinger–Moser trace inequality

Cπ(Ω)=sup{Ωeπu2:uH1(Ω),Ω[|u|2+u2]=1}<+,

see [23]. It has been proven [33] that for any bounded domain Ω in 2, with smooth boundary, the supremum Cπ(Ω) is attained by a function uH1(Ω) with Ω[|u|2+u2]=1. Furthermore, for any α(0,π), the supremum Cα(Ω) is finite and it is attained, while Cα(Ω)= as soon as α>π. See also [11, 21, 22, 25, 30] for generalizations. Observe that critical points of the above constrained variational problem satisfy, after a simple scaling, an equation of the form (1.1).

The Trudinger–Moser trace embedding is critical, involving loss of compactness analogous to that related to the Trudinger–Moser embedding for functions u with zero boundary value,

H01(Ω)ueu2Lp(Ω)for all p1

for which the analogous problem to (1.1) is

Δu+λueu2=0in Ω,u=0on Ω,(1.2)

whose energy functional is given by Iλ:H01(Ω),

Iλ(u)=12Ω|u|2-λ2Ωeu2.

It is known that Iλ satisfies the compactness PS-condition for energy levels less that 2π (see [1]). Loss of compactness in H01(Ω) is described by the presence of families of blowing-up solutions for problem (1.2). It has been proven in [19] that if un solves problem (1.2) for λ=λn, with Iλn(un) bounded and λn0, then, passing to a subsequence, there is an integer k0 such that

Iλn(un)=2kπ+o(1),(1.3)

see [2, 19, 5]. This quantization property is not known for general Palais–Smale sequences associated to Iλ (see [3]). When k=1, a more precise description of the blowing-up behavior of these families of solutions is known [2]. On the other hand, a simple observation is that the functional Iλ has a mountain pass geometrical structure. In fact, in [1, 6] it is shown that there exists λ0 such that for all 0<λ<λ0, the mountain pass level stands below 2π where the PS-condition holds. Thus a solution to (1.2) always exists for this range of values of λ. As λ0, the family of mountain pass solutions satisfies (1.3) with k=1. In [31] it is proven that if Ω has a sufficiently small hole, a solution to (1.2), satisfying (1.3), exists. Further results were obtained in [15]: if Ω has a hole of any size, namely Ω is not simply connected, then a solution satisfying property (1.3) with k=2 exists. This solution happens to blow up exactly at two points in Ω. General conditions for the existence of solutions of problem (1.2) for small λ, which satisfy the bubbling condition (1.3), for any k1, are provided in [15], together with the precise characterization of their blow-up profile. In fact, blowing-up solutions satisfying (1.3) happen to blow up at exactly k points which are located in the interior of Ω. See also [4, 14, 16] for related results.

In this paper, we are concerned with the construction of solutions to (1.1), in the same spirit as the result described above in [15]. Assume that Ω is any bounded domain with smooth boundary. For any integer k we find existence of a pair of solutions uλ to problem (1.1) for small λ, whose energy satisfy the bubbling condition

Jλ(uλ)=kπ2+o(1)as λ0.

Furthermore, we give a precise description of their bubbling behavior.

To state our result, let us introduce the function φk:(Ω)k×(+)k with

φk(ξ,m)=φk(ξ1,,ξk,m1,,mk)

defined by

φk(ξ,m)=2(log2-1)j=1kmj2+2j=1kmj2log(mj2)-j=1kmj2H(ξj,ξj)-ijmimjG(ξi,ξj),(1.4)

where G is the Green function for the Neumann problem

{-ΔxG(x,y)+G(x,y)=0,xΩ,G(x,y)νx=2πδy(x),xΩ,(1.5)

and H its regular part defined as

H(x,y)=G(x,y)-2log1|x-y|.(1.6)

In this paper we establish the following:

Theorem 1.1.

Let Ω be a bounded domain in R2 with smooth boundary and let k1 be an integer. Then there exists λ0>0 such that, for all small λ with 0<λ<λ0, there exists a pair of solutions uλ1, uλ2 to problem (1.1) such that

12Ω[|uλi|2+(uλi)2]-λ2Ωe(uλi)2=k2π+o(1),i=1,2,

where o(1)0 as λ0. Moreover, for any i=1,2, passing to a subsequence, there exists

(ξi,mi)=(ξ1i,,ξki,m1i,,mki)(Ω)k×(+)k,

with ξ1ξ2, such that φk(ξi,mi)=0 and

uλi(x)=λ(j=1kmjiG(x,ξji)+o(1)),(1.7)

where o(1)0 on each compact subset of Ω¯{ξ1i,,ξki}.

These solutions blow up at points located near ξ1,,ξkΩ, while far away from these points the solutions look like a combination of Green function with positive weights m1,,mk. These points and parameters (ξ1,,ξk,m1,,mk) correspond to two distinct critical points of φk.

We can actually show a stronger version of this result. If Ω has more than one component, then pairs of families of solutions blowing up at k points on each component happen to exist. In reality, associated to each topologically nontrivial critical point situation associated to φk (for instance local maxima or saddle points possibly degenerate), a solution with concentration peaks at a corresponding critical point exists. We will not elaborate more on this point, and we refer the interested reader to [12].

It is important to remark the interesting analogy between these results and those known for other problems with exponential nonlinearity on the boundary, as

{-Δu+u=0in Ω,uν=λeuon Ω,(1.8)

see [7, 12, 13, 20, 26]. See also [8, 17, 18] for related problems.

In [12], a construction of solutions to (1.8) with λΩeuλ bounded is carried out: for any integer k1, there are at least two distinct families of solutions uλ which approach the sum of k Dirac masses at the boundary. The location of these possible points of concentration may be further characterized as critical points of the functional of k points ξ1,,ξk of the boundary defined as

Ψk(ξ1,,ξk)=-[j=1kH(ξj,ξj)+ijG(ξi,ξj)],

where G and H are defined in (1.5) and (1.6), respectively. Observe that the function Ψk only depends on the points on the boundary Ω and it does not depend on positive parameters m1,,mk. This is completely different from the case of the function φk which is defined in (1.4) and which determines the bubbling behavior of solutions to (1.1). Furthermore, it has been proven that, far from ξ1,,ξk, the solutions to problem (1.8) found in [12] look like

uλ(x)=j=1kG(x,ξj)+o(1)as λ0.

Thus, also the solutions to problem (1.8) found in [12] are combinations of Green function, far from the concentration points, but unlike the solutions obtained in Theorem 1.1 for problem (1.1), the weights in front of the Green functions are always equal to 1. Thus, to construct solutions to problem (1.1), not only we have to find the location of the bubbling points ξ1,,ξk on the boundary, but also the weights m1,,mk in front of the Green functions in (1.7).

The solutions predicted in Theorem 1.1 are constructed as a small additive perturbation of an appropriate initial approximation. A linearization procedure leads to a finite-dimensional reduction, where the reduced problem corresponds to that of adjusting variationally the location of the concentration points ξ1,,ξk and of the weights m1,,mk. A precise description of the approximation and a detailed outline of the proof and of the organization of the paper are given in Section 2.

Let us just mention that through out the paper, C will always denote an arbitrary positive constant, independent of λ, whose value changes from line to line.

2 A first approximation and outline of the argument

It is useful for our purpose to consider the change of variables u=λu~ so that problem (1.1) gets rewritten as

-Δu~+u~=0in Ω,u~>0in Ω,u~ν=λu~eλu~2on Ω.(2.1)

The first part of this section is devoted to constructing a good approximation for a solution to problem (2.1) and to estimate its error. To do so, let us introduce the following problem in the entire plane

Δv=0in +2,vν=evon +2,+2ev<.(2.2)

The positive solutions to problem (2.2) are the basic elements for our construction. So, let us recall that all positive solutions to (2.2) are given by

wt,μ(x)=wt,μ(x1,x2)=log2μ(x1-t)2+(x2+μ)2,

where t is any real number and μ>0 is any strictly positive number (see [24, 28, 34]). Set

wμ(x):=w0,μ(x)=log2μx12+(x2+μ)2.(2.3)

We next describe an approximate solution to (2.1) whose shape is given by the sum of functions wμ centered at points on the boundary of Ω and properly scaled. Let k be an integer, let ξ1,,ξk be points on the boundary of Ω and let m1,,mk be positive numbers. We assume there exists a positive, small number δ such that

|ξi-ξj|>δfor ij,δ<mj<1δ.(2.4)

We thus define the functions

uj(x)=log1|x-ξj-εjμjν(ξj)|2for any j=1,,k

and

U~(x)=j=1kmj[uj(x)+Hj(x)],(2.5)

where Hj is the unique solution to the problem

-ΔHj+Hj=-ujin Ω,Hjν=2εjμjeuj-ujνon Ω.(2.6)

In the above definitions, μj and εj are positive numbers. These numbers μj and εj will be defined later on in terms of λ,ξj and mj in order to ensure that U~ is a function very close to a solution for problem (2.1). Let us just mention that, a posteriori, the parameters εj will tend to zero, as λ0, namely

limλ0εj=0for any j=1,,k,(2.7)

while the numbers μj will remain bounded from above and strictly positive, as λ0. Taking this into account, we easily see that the shape of the function U~ change depending whether you evaluate it far from the fixed points ξj or in a region very close to one of the points ξj. Let us then describe carefully the shape of U~ in these two regions. For this purpose, we need the following:

Lemma 2.1.

Assume (2.4) and (2.7). For any 0<α<1, one has

Hj(x)=H(x,ξj)+εjαO(1)as λ0,

where O(1) denotes a function in Ω which is uniformly bounded as λ0, and H is the regular part of Green’s function defined in (1.5).

Proof.

We refer the reader to [12] for a proof of this lemma. ∎

A direct consequence of Lemma 2.1 is that, for a given δ>0 small and fixed, in the region |x-ξj|>δ for all j=1,,k, the function U~ looks like

U~(x)=j=1kmj[G(x,ξj)+O(εjα)]as λ0.(2.8)

Here and in what follows, with O(εjα) we denote a general function in Ω of the form εjαΘ(x), where Θ(x) is uniformly bounded in Ω as λ0.

Let us now examine U~ in a neighborhood of a given ξj. Assume that |x-ξj|<δ and set y=xεj, ξj=ξjεj. Explicit computations give that

U~(x)=mj[uj(x)+Hj(x)]+ijmi[ui(x)+Hi(x)]=mj[log2μj|x-ξj-εjμjν(ξj)|2-log(2μj)+H(x,ξj)+O(εjα)]+ijmi[log1|x-ξi-εiμiν(ξi)|2+H(x,ξi)+O(εiα)]=mj[log2μj|y-ξj-μjν(ξj)|2+2log1εj-log(2μj)+H(ξj,ξj)+O(|x-ξj|)+O(εjα)]+ijmi[log1|ξi-ξj|2+H(ξj,ξi)]+ijmi[log1|x-ξi-εiμiν(ξi)|2-log1|ξi-ξj|2+O(|x-ξj|)+O(εiα)]=mj[log2μj|y-ξj-μjν(ξj)|2+2log1εj-log(2μj)+H(ξj,ξj)+O(|x-ξj|)+O(εjα)]+ijmiG(ξj,ξi)+ijmi[log1|x-ξi-εiμiν(ξi)|2-log1|ξi-ξj|2+O(|x-ξj|)+O(εiα)]

as λ0. We set

wj(x)=wμj(x-ξjεj)=log2μj|y-ξj-μjν(ξj)|2,

and

βj=-log(2μj)+H(ξj,ξj)+ijmj-1miG(ξj,ξi),θ(x)=O(|x-ξj|)+j=1kO(εjα).

We thus write the above expansion in the following compact form: for |x-ξj|<δ,

U~(x)=mj(wj(x)+logεj-2+βj+θ(x))as λ0.(2.9)

Formulas (2.8) and (2.9) give a precise description of the function U~.

The solution to (2.1) we are looking for the form

u~=U~+ϕ,(2.10)

where U~ is defined as in (2.5), and ϕ represents a lower order correction. In fact, we aim at finding a solution u~ for a function ϕ small in some proper sense provided that the points ξj and the parameters mj are suitably chosen. Assuming for the moment that ϕ is small, we rewrite problem (2.1) as follows:

-Δϕ+ϕ=0in Ω,L(ϕ)=E+N(ϕ)on Ω,(2.11)

where

L(ϕ):=ϕν-[j=1kεj-1ewj]ϕ,E:=f(U~)-U~ν,(2.12)

and

N(ϕ):=f(U~+ϕ)-f(U~)-f(U~)ϕ+[f(U~)-j=1kεj-1ewj]ϕ.(2.13)

Here and in what follows f denotes the nonlinearity

f(u~)=λu~eλu~2.

It is not hard to believe that having a good approximation U~ to a solution of problem (2.1) is reflected into the fact that the function E is small, in some sense to be made precise. It is in this context that we will choose μj and εj in such a way that the error of approximation E for U~ is small around each point ξj under some appropriate norm.

Let us be more precise. The error E is clearly defined by (2.12). Assume that δ>0 is a small but fixed positive number and xΩ with |x-ξj|<δ. In this region, we have that

f(U~)=λ[mj(wj(x)+logεj-2+βj+θ(x))]eλ[mj(wj(x)+logεj-2+βj+θ(x))]2=(λmj(log1εj2+βj)+λmj(wj+O(1)))eλmj2(log1εj2+βj)2e2λmj2(log1εj2+βj)wje2λmj2(log1εj2+βj)θ(x)eλmj2(wj+θ(x))2=λmj(log1εj2+βj)(1+(log1εj2+βj)-1(wj+O(1)))×eλmj2(log1εj2+βj)2e2λmj2(log1εj2+βj)wje2λmj2(log1εj2+βj)θ(x)eλmj2(wj+θ(x))2

as λ0. We thus choose εj to be defined as

2λmj2(log1εj2+βj)=1.(2.14)

It is immediate to see that, with this definition, (2.7) holds true. Thanks to (2.14), one has

f(U~)=12mj(1+2λmj2(wj+O(1)))e12(log1εj2+βj)ewjeθ(x)eλmj2(wj+θ(x))2=12mjεj-1eβj2(1+2λmj2(wj+O(1)))ewjeθ(x)eλmj2wj2(1+O(λ)wj).

On the other hand, in the same region, we have

U~ν=ν[mj(wj(x)+logεj-2+βj+θ(x))]=mjεj-1ewj+j=1kO(εj2)as λ0.

Thus, in order to match at main order the two terms U~ν and f(U~) in a region near the point ξj, we fix the parameter μj such that the number βj satisfies

eβj2=2mj2.(2.15)

This condition defines the parameter μj as follows:

log(2μj)=-2log(2mj2)+H(ξj,ξj)+ijmimj-1G(ξi,ξj).(2.16)

With these choices of μj we get

f(U~)=mj(1+2λmj2(wj+O(1)))εj-1ewjeλmj2wj2(1+O(θ(x)))(1+O(λ)wj)=mj(1+2λmj2(wj+O(1)))εj-1ewjeλmj2wj2(1+O(λwj)).

As a conclusion, the choice we made of μj and of εj gives that in the region |x-ξj|<δ, the error of approximation can be described as follows:

E=mj{(1+2λmj2(wj+O(1)))eλmj2wj2(1+O(λwj))-1}εj-1ewj.

Let us mention now that a direct computation shows that for |x-ξj|>δγj, j=1,,k,

1λE=mjλ{(1+2λmj2(wj+O(1)))eλmj2wj2(1+O(λwj))-1}εj-1ewjCεj-1ewj=O(εj|logεj|2).(2.17)

Then we get

maxj=1k|x-ξj|>δγj|E|Cλmaxj=1,,kεj|logεj|2Cλ

for some positive constant C. Set γj=logεj-2. Let χBδγj(ξj) be the characteristic function on Bδγ(ξj)Ω and

ρj(x)=cγj{(1+1γj(wj+1))(1+1γj(1+|wj|))ewj22γj-1}εj-1ewj.

Taking into account (2.14), we get the following global bound on the error of approximation:

|E|Cλρ(x),where ρ(x):=j=1kρj(x)χBδγj(ξj)(x)+1.

We define the L-weight norm

h,Ω=supxΩρ(x)-1|h(x)|.

We thus have the validity of the following key estimate for the error term E:

E,ΩCλ.(2.18)

We conclude this section explaining the strategy to solve problem (2.11), which guarantees the existence of a solution to problem (2.1) of the form (2.10). In fact, we will solve problem (2.11) in two steps. The first step consists in solving problem (2.11) in a projected space. Let us be more precise.

Define in +2={(x1,x2):x2>0},

z0j(x1,x2)=1μj-2x2+μjx12+(x2+μj)2,z1j(x1,x2)=-2x1x12+(x2+μj)2.

It has been shown in [12] that these functions are all the bounded solutions to the linearized equation around wμj (2.3) associated to problem (2.2), that is they solve

Δψ=0in +2,-ψx2=ewμjψon +2.

For ξjΩ, we define Fj:Bδ(ξj)M to be a diffeomorphism, where M is an open neighborhood of the origin in +2 such that Fj(ΩBδ(ξj))=+2M, Fj(ΩBδ(ξj))=+2M. We can select Fj so that it preserves area. Define

Zij(x)=zij(εj-1Fj(x)),i=0,1,j=1,,k.(2.19)

Next, let us consider a large but fixed number R0>0 and a nonnegative radial and smooth cut-off function χ with χ(r)=1 if r<R0 and χ(r)=0 if r>R0+1, 0χ1. Then set

χj(x)=εj-1χ(εj-1Fj(x)).(2.20)

The problem we first solve is to find a function ϕ and numbers cij such that

{-Δϕ+ϕ=0in Ω,L(ϕ)=E+N(ϕ)+i=01j=1kcijZijχjon Ω,ΩϕZijχj=0for i=0,1,j=1,,k.(2.21)

Consider the norm

ϕ=supxΩ|ϕ(x)|.

We prove the following:

Proposition 2.1.

Let δ>0 be a small but fixed number. Assume that the points ξ1,,ξkΩ and the parameters m1,,mk satisfy

|ξi-ξj|δfor all ij,δ<mj<1δ.

Then there exist positive numbers λ0 and C, such that, for any 0<λ<λ0, problem (2.21) has a unique solution ϕ, cij which satisfies

ϕCλ,|cij|Cλ

for all λ<λ0. Moreover, if we consider the map (ξ,m)ϕ into the space C(Ω¯), the derivative Dξϕ and Dmϕ exists and defines a continuous function of (ξ,m). Besides, there is a constant C>0 such that

DξslϕCλ,DmsϕCλ

for all s,l.

The proof of this result is contained in Section 3.

At this stage of our argument, we have solved the nonlinear problem (2.21). In order to find a solution to the original problem we need to find ξ and m such that

cij(ξ,m)=0for all i=0,1,j=1,,k.(2.22)

This problem is indeed variational: it is equivalent to finding critical points of a function of ξ and m. Associated to (1.1), let us consider the energy functional Jλ given by

Jλ(u)=12Ω(|u|2+u2)-λ2Ωeu2,uH1(Ω),

and the finite-dimensional restriction

λ(ξ,m)=Jλ(λ(U~(ξ,m)+ϕ(ξ,m))),(2.23)

where ϕ is the unique solution to problem (2.21) given by Proposition 2.1. Critical points of λ correspond to solutions of (2.22) for a small λ, as the following result states.

Proposition 2.2.

Under the assumptions of Proposition 2.1, the functional Iλ(ξ,m) is of class C1. Moreover, for all λ>0 sufficiently small, if Dξ,mIλ(ξ,m)=0, then (ξ,m) satisfies (2.22).

The proof of the above proposition, together with the expansion of the functional 𝒥λ(ξ,m) is given in Section 4. Section 5 is devoted to concluding the proof of Theorem 1.1. The final Appendix, Section 6, contains the proofs of some estimates we have used through the paper.

3 Proof of Proposition 2.1

The proof of Proposition 2.1 is based on a fixed point argument and the invertibility property of the following linear problem: Given hL(Ω), find a function ϕ and constants cij such that

{-Δϕ+ϕ=0in Ω,L(ϕ)=h+i=01j=1kcijχjZijon Ω,ΩχjZijϕ=0for i=0,1,j=1,,k.(3.1)

We shall prove the validity of the following proposition:

Proposition 3.1.

Let δ>0 be a small but fixed number and assume that we have ξ1,,ξkΩ and m1,,mk with

|ξi-ξj|δfor all ij,δ<mj<1δ.(3.2)

Then there exist positive numbers λ0 and C such that, for any 0<λ<λ0 and any hL(Ω), there is a unique solution ϕTλ(h), and cijR to (3.1). Moreover,

ϕCh,Ω.

The proof of this result is based on the a-priori estimate for solutions to the following problem:

{-Δϕ+ϕ=finΩ,L(ϕ)=h+i=01j=1kcijχjZijon Ω,ΩχjZijϕ=0for i=0,1,j=1,,k.(3.3)

Define

f,Ω:=supxΩ(j=1kεjσ(1+|x-ξj-εjμjν(ξj)|)2+σ+1)-1|f(x)|,

where 0<σ<1. We have the validity of the following lemma.

Lemma 3.1.

Under the assumptions of Proposition 3.1, if ϕ is a solutions of (3.3) for some hL(Ω) and for some fL(Ω) with h,Ω,f,Ω< and cijR, then

ϕC[h,Ω+f,Ω].(3.4)

Proof.

We will carry out the proof of the a priori estimate (3.4) by contradiction. We assume then the existence of sequences λn0, points ξjnΩ and numbers mjn, μjn which satisfy relations (3.2) and (2.16), functions hn, fn with hn,Ω,fn,Ω0, ϕn with ϕn=1, constants cij,n,

-Δϕn+ϕn=fnin Ω,L(ϕn)=hn+i=02j=1kcij,nZijχjon Ω,ΩZijχjϕn=0for all i,j.

We will prove that in reality under the above assumption we must have that ϕn0 uniformly in Ω¯, which is a contradiction that concludes the result of the lemma. We will divide into the following several steps to prove this.

Passing to a subsequence we may assume that the points ξjn approach limiting, distinct points ξj* in Ω. Indeed, let us observe that fn0 locally uniformly in Ω¯, away from the points ξj. Away from the points ξj* we have then -Δϕn+ϕn0 uniformly on compact subsets on Ω¯{ξ1*,,ξk*}. Since ϕn is bounded, it follows also that passing to a further subsequence, ϕn approaches in C1 local sense on compacts of Ω¯{ξ1*,,ξk*} a limit ϕ* which is bounded and satisfies -Δϕ*+ϕ*=0 in Ω{ξ1*,,ξk*}. Furthermore, observe that far from {ξ1*,,ξk*}, hn0 locally uniformly on Ω{ξ1*,,ξk*} and so we also have ϕnν0 on Ω{ξ1*,,ξk*}. Hence ϕ* extends smoothly to a function which satisfies -Δϕ*+ϕ*=0 in Ω, and ϕ*ν=0 on Ω. We conclude that ϕ*=0, and the claim follows.

Indeed, for notational convenience, we shall omit the explicit dependence on n in the rest of the proof. Multiplying the first equation of (3.3) by Zij and integrating over B(ξj,δ), we find

l=01cljΩB(ξj,δ)χjZljZij=-ΩB(ξj,δ)hZij+ΩB(ξj,δ)L(Zij)ϕ-ΩB(ξj,δ)ϕνZij+ΩB(ξj,δ)(-ΔZij+Zij)ϕ-ΩB(ξj,δ)fZij.(3.6)

Having in mind that ϕn0 in C1 sense in ΩB(ξj,δ), we have

ΩB(ξj,δ)ϕνZij0as λ0.

Furthermore, a direct computation shows that

ΩB(ξj,δ)χjZljZij=Miδli+o(1)as λ0,

where Mi is some universal constant and δli=1 if i=l, and =0 if il. On the other hand, we have

ΩB(ξj,δ)(Zijν-[j=1kεj-1ewj]Zij)ϕ+ΩB(ξj,δ)(-ΔZij+Zij)ϕCϕ(3.7)

and

|ΩfZij|Cf**,Ω.(3.8)

In fact, estimate (3.8) is a direct consequence of the definition of the **,Ω-norm. Let us prove the validity of (3.7). Recall that in ΩB(ξj,δ), we have that Zij(x)=zij(εj-1Fj(x)), where Fj is chosen to preserve area (see (2.19)). Performing the change of variables y=εj-1Fj(x), we get that

ΩB(ξj,δ)(-ΔZij+Zij)ϕ=(1+o(1))+2B(0,δεj)(zij+εj2zij)ϕ~,(3.9)

where ϕ~(y)=ϕ(Fj-1(εjy)) and is a second order differential operator defined as follows:

=-Δ+O(εj|y|)2+O(εj)in +2B(0,δεj).(3.10)

Hence

|ΩB(ξj,δ)(-ΔZij+Zij)ϕ|Cϕ.

On the other hand, we observe that, after a possible rotation, we can assume that Fj(ξj)=I. Hence, using again the change of variables y=εj-1Fj(x), we get

ΩB(ξj,δ)L(Zij)ϕ=(1+o(1))+2B(0,δεj)(B(zij)-W~zij)b(y)ϕ~,(3.11)

where W~(y)=εjW(Fj-1(εjy)) with W(x)=j=1kεj-1ewj, and b(y) is a positive function, coming from the change of variables, which is uniformly positive and bounded as λ0. Furthermore, B is a differential operator of order one on +2. In fact, we have

B=-y2+O(εj|y|)on +2B(0,δεj)(3.12)

On the other hand, since

W(x)=εj-12μjεj2|x-ξj-εjμjν(ξj)|2(1+ljεlεjO(1)),

we get

W~(y)=2μjy12+μj2+lεlα(1+|y|)on +2B(0,δεj)(3.13)

for some 0<α<1. Thus we can conclude that

|ΩB(ξj,δ)L(Zij)ϕ|Cϕ.

This shows the validity of (3.7).

We shall now estimate the term ΩhZij. Using the definition of the *,Ω-norm, we observe that

|ΩhZij|=h,ΩΩ(l=1kρlχB(ξl,δ)(x)+1)ZijCh,Ωl=1kΩB(ξl,δγl)γl{(1+wl+1γl)(1+1+|wl|γl)ewl22γl-1}εl-1ewl+Ch,ΩΩl=1kB(ξl,δγl)Zij.

Since Zij are uniformly bounded, as λ0, in Ωl=1kBδ(ξl), we just need to estimate

ΩBδγj(ξj)γj{(1+wj+1γj)(1+1+|wj|γj)ewj22γj-1}εj-1ewj.

Recall that the functions wj are defined as wj(x)=log2μj|y-ξj-μjν(ξj)|2, with y=xεj, ξj=ξjεj, and γj=-2logεj. Using the change of variables εjy=x-ξj, we have

ΩBδγj(ξj)γj{(1+wj+1γj)(1+1+|wj|γj)ewj22γj-1}εj-1ewj=ΩεjB(0,δγjεj)γj{(1+w¯j+1γj)(1+1+|w¯j|γj)ew¯j22γj-1}ew¯jCΩεjB(0,δγjεj)ew¯j=CΩεjB(0,δγjεj)1|y-μjν(0)|2Cμj-δγjεjμj1r2𝑑rC,

where Ωεj=Ω-ξjεj and w¯j=log2μj|y-μjν(0)|2. Therefore we get

|ΩhZij|Ch,Ω.

Collecting all estimates from (3.6), we find the validity of (3.5).

Set ε0=min{ε1,ε2,,εk}, and W(y)=j=1k2μj|y-ξj-μjν(ξj)|2. Arguing as in [12, Lemma 4.3], we have that for ε0>0 small enough, there exist R1>0, and

ψ:Ω\j=1kBR1ε0(ξj)

smooth, positive and bounded function such that

{-Δψ+ψj=1kε0σ|x-ξj|2+σ+1inΩ\j=1kBR1ε0(ξj),ψν-ε0-1W(xε0)ψj=1kε0σ|x-ξj|1+σ+1onΩ\j=1kBR1ε0(ξj).

Thus, by maximum principle in Ω\j=1kBR1ε0(ξj), the function |ϕ| can be bounded by

ϕ~=C1ψ(ϕi+f,Ω+h,Ω)

for some constant C1 independent of λ. Thus we get (3.14).

We now conclude our argument by contradiction to prove (3.4). From (3.5), we have that cij,n is bounded, thus we may assume that cij,ncij as n. By (3.14), we get that ϕni>c>0. That is

maxΩ¯\j=1kBRε0(ξj,n)|ϕn|c.

By the maximum principle and the Hopf Lemma we find that

maxΩ¯\j=1kBRε0(ξj,n)|ϕn|=maxΩ¯j=1kBRε0(ξj,n)|ϕn|.

Thus, we can find that there is some fixed s{1,2,,k} such that

maxΩ¯BRε0(ξs,n)|ϕn|c.

Set Ω0=Ω-ξs,nε0, and consider the change of variables

ϕ^n(z)=ϕn(ξs,n+ε0z),h^n(z)=hn(ξs,n+ε0z),f^n(z)=fn(ξs,n+ε0z),Z^ij(z)=Zij(ξs,n+ε0z).

Then

-Δϕ^n(z)+ε02ϕ^n(z)=ε02fn(z)inΩ0

and

ϕ^nν-ε0[j=1kεj-1ewj]ϕ^n=ε0h^n+i=01j=1kε0cij,nχjZ^ijon Ω0.

Then by elliptic estimate ϕ^n (up to subsequence) converges uniformly on compact sets to a nontrivial solution ϕ^0 of the problem

Δϕ=0in +2,ϕν-2μsx12+μs2ϕ=0on +2.

By the nondegeneracy result [12], we conclude that ϕ^ is a linear combination of z0s and z1s. On the other hand, we can take the limit in the orthogonality relation and we find that +2χϕ^zij=0 for i=0,1. This contradicts the fact that ϕ^0. This ends the proof of the lemma. ∎

Proof of Proposition 3.1.

In proving the solvability of (3.1), we may first solve the following problem: For given hL(Ω), with h*,Ω bounded, find ϕL(Ω) and dij, i=0,1, j=1,,k, such that

{-Δϕ+ϕ=i=01j=1kdijχjZijinΩ,ϕν-[j=1kεj-1ewj]ϕ=honΩ,ΩχjZijϕ=0fori=0,1,j=1,,k.(3.15)

First we prove that for any ϕ, dij solution to (3.15) the bound

ϕCh*,Ω(3.16)

holds. In fact, by Lemma 3.1, we have

ϕC(h*,Ω+i=01j=1kεj|dij|)(3.17)

and therefore it is enough to prove that εj|dij|Ch*,Ω.

Fix an integer j. To show that εj|dij|Ch*,Ω, we shall multiply equation (3.15) against a test function, properly chosen. Let us observe that the proper test function depends whether we are considering the case i=0 or i=1. We start with i=0. We define z^0j(y)=h(y)z0j(y), where

h(y)=log(δεj)-log|y|logδεj-logR.

In fact, we recognize that Δh=0 in B(0,δεj)B(0,R), h=1 on B(0,R) and h=0 on B(0,δεj). Let η1 and η2 be two smooth cut-off functions defined in 2 as

η1{1in B(0,R),0in 2B(0,R+1),

so that

0η11,|η1|C

and

η2{1in B(0,δ4εj),0in 2B(0,δ3εj),

so that

0η21,|η2|Cεjδ,|2η2|C(εjδ)2.

We assume that R>R0 (see (2.20)) and we define

Z~0j(x)=η1(εj-1Fj(x))Z0j(x)+(1-η1(εj-1Fj(x)))η2(εj-1Fj(x))z^0j(εj-1Fj(x))(3.18)

for xB(ξj,δ)Ω. We multiply equation (3.15) against Z~0j and we integrate by parts. We get

a=0,1dajΩχjZajZ~0j=Ω(-ΔZ~0j+Z~0j)ϕ+ΩhZ~0,j+ΩL(Z~0j)ϕ.

Observe first that, assuming R>R0, we have

dajΩχjZajZ~0j=dajΩχjZajZ0j=εjM0δa0daj(1+o(1))as λ0.(3.19)

Furthermore, we have

|ΩhZ~0j|Ch*,Ω.(3.20)

We claim that

-ΔZ~0j+Z~0j**,ΩC|logεj|,L(Z~0j)*,ΩC|logεj|.(3.21)

The proof of estimates (3.21) is postponed to the Appendix, Section 6. Assuming for the moment the validity of (3.21), from estimates (3.19)–(3.21) we conclude that

|εjd0j|C(h*,Ω+|logεj|-1ϕ).(3.22)

We shall now obtain an estimate similar to (3.22) for εjd1j. To do so, we use another test function. Indeed we multiply equation (3.15) against η2Z1j and we integrate by parts. We get

a=0,1dajΩχjZajη2Z1j=Ω(-Δ(η2Z1j)+η2Z1j)ϕ-Ωhη2Z1,j+ΩL(Z1j)η2ϕ+ΩZ1jη2νϕ.

Observe first that, assuming R>R0, we have

dajΩχjZajη2Z1j=dajΩχjZajZ1j=M1δa1εjd1j(1+o(1))as λ0,

and

|Ωhη2Z1j|Ch*,Ω.

Using the change of variables y=εj-1Fj(x), we get

ΩZ1jη2νϕ=Ωεjz1jη2νϕ~,

where Ωεj=Ωεj and ϕ~(y)=ϕ(Fj-1(εj-1y)). But z1j=O(11+r) and η2=O(εj) so

|ΩZ1jη2νϕ|Cεj|logεj|.

Using again the change of variables y=εj-1Fj(x), and proceeding similarly to (3.11), (3.12) and (3.13), one gets

ΩL(Zij)η2ϕ=(1+o(1))Ωεj[zijν-W~zij]η2ϕ~,

where ϕ~(y)=ϕ(Fj-1(εjy)) and b(y) is a positive function, coming from the change of variables, which is uniformly positive and bounded as λ0. Observe that

zijν-W~zij=O(εj1+r)+O(εjα1+r2)

for yΩεj and |y|<δεj-1, and this implies that

Ωεj|zijν-W~zij|Cεjα

for some 0<α<1. Thus we can conclude that

|ΩL(Zij)η2ϕ|Cεjαϕ.

Consider once again the change of variables y=εj-1Fj(x). Arguing as in (3.9) and (3.10), we get

Ω(-Δ(η2Zij)+η2Zij)ϕ=(1+o(1))Ωεj(-Δ(η2zij)+εj2η2zij)ϕ~,

where ϕ~(y)=ϕ(Fj-1(εjy)). We thus compute in yΩε1, with |y|<δεj-1,

Δ(η2z1j)=Δη2z1j+2η2z1j+η2Δz1j=O(ε121+r)+O(εj1+r)+η2Δz1j.

On the other hand, in this region we have -Δz1j+εj2z1j=O(εj1+r2)+O(εj21+r). Thus

Ωεj|-Δ(η2zij)+εj2η2zij|Cεj|logεj|.

Summarizing all the above information, we get

|εjd1j|C(h,Ω+εjϕ).(3.23)

Estimates (3.22), (3.23) combined with (3.17) yield |εjdij|Ch,Ω, which gives the validity of (3.16). Now consider the Hilbert space

={ϕH1(Ω):ΩχjZijϕ=0 for all i=0,1,j=1,,k},

endowed the norm ϕH12=Ω(|ϕ|2+ϕ2). Problem (3.15), expressed in a weak form, is equivalent to find ϕ such that

Ω(ϕψ+ϕψ)-Ω[j=1kεj-1ewj]ψ=Ωhψforallψ.

With the aid of Fredholm’s alternative we obtain unique solvability of (3.15), which is guaranteed by (3.16).

In order to solve (3.1), let YlsL(Ωε), dijls, be the solution of (3.15) with h=χsZls, that is

{-ΔYls+Yls=i=01j=1kdijlsχjZijinΩ,Ylsν-[j=1kεj-1ewj]Yls=χsZlsonΩ,ΩχjZijYls=0forl=0,1,s=1,,k,(3.24)

Then there is a unique solution YlsL(Ω) of (3.24), and

YlsC,εj|dijls|C

for some constant C independent on λ.

Multiplying (3.24) by Zij, and integrating by parts, we have

i=01j=1kB(ξj,δ)dij,lsχj(Zij)2=δilδjsB(ξj,δ)χj(Zij)2+o(1),

where δil,δjs are Kronecker’s delta. Then we get

d0j,0s=aδjs+o(1),d1j,1s=bδjs+o(1)

with a,b>0 independent of εj. Hence the matrix D1 (or D2) with entries d0j,0s (or d1j,1s) in invertible for small εj and Di-1C (i=1,2) uniformly in εj.

Now, given hL(Ω) we find ϕ1, dij, solution to (3.15). Define constants cls as

l=01s=1kclsdijls=-dij,for all i=0,1,j=1,,k.

The above linear system is almost diagonal, since arguing as before one can show that

dijls=εj-1Miδjsδil(1+o(1))as λ0,

where Mi is a positive universal constant. Then define

ϕ=ϕ1+l=01s=1kclsYls.

A direct computation shows that ϕ satisfies (3.1) and furthermore

ϕϕ1+l=01s=1k|cls|Ch*,Ω+i=01j=1kεj|dij|Ch*,Ω

by (3.16). This finishes the proof of Proposition 3.1. ∎

Remark 3.1.

A slight modification of the proof above also shows that for any hL(Ω) and fL(Ω), with h*,Ω, f**,Ω<, the problem

{-Δϕ+ϕ=finΩ,L(ϕ)=h+i=01j=1kcijχjZijon Ω,ΩχjZijϕ=0fori=0,1,j=1,,k,

has a unique solution ϕ, cij, i=0,1, j=1,,k, satisfying

ϕC(h*,Ω+f**,Ω)

and

|cij|C(h*,Ω+f**,Ω)for all i=0,1,j=1,,k,

with C independent of λ.

The result of Proposition 3.1 implies that the unique solution ϕ=Tλ(h) of (3.1) defines a continuous linear map from the Banach space 𝒞 of all functions h in L(Ω) for which h,Ω< into L, with norm bounded uniformly in λ.

Lemma 3.2.

The operator Tλ is differentiable with respect to the variable ξ=(ξ1,,ξk) on Ω satisfying (3.2), and for m=(m1,,mk), one has the estimate

DξTλ(h)Ch,Ω,DmTλ(h)Ch,Ω(3.25)

for a given positive C, independent of λ, and for all λ small enough.

Proof.

Differentiating equation (3.1), formally Z:=ξslϕ for all s,l, should satisfy in Ω the equation

-ΔZ+Z=0inΩ,

and on the boundary Ω,

L(Z)=-ξsl(j=1kεj-1ewj)ϕ+i=01j=1kcijξsl(χjZij)+i=01j=1kdijZijχj

with dij=ξslcij, and the orthogonality conditions now become

ΩZijχjZ=0ifsj,ΩZisχsZ=-Ωξsl(Zisχs)ϕ.

We consider the constants αab, a=0,1, b=1,,k, defined as

αabΩχb2|Zab|2=Ωξsl(Zabχb)ϕfora=0,1,b=1,,k.

Define

Z~=Z+a=0,1b=1kαabχbZab.

We then have

{-ΔZ~+Z~=f1in Ω,L(Z~)=h1+i=01j=1kdijZijχjon Ω,ΩχjZijZ~=0fori=0,1,j=1,,k,

where

f1=a=0,1b=1kαab(-Δ(χbZab)+χbZab)

and

h1=-ξsl(j=1kεj-1ewj)ϕ+i=01j=1kcijξls(Zijχj)+a=0,1b=1kαabL(χbZab).

Hence, using the result of Proposition 3.1 we have Z~C(h1,Ω+f1,Ω). By the definition of αab, we get |αab|Cϕ. Since ϕCh,Ω and |cij|Ch,Ω, we obtain Z~Ch,Ω. Hence we get

ξslTλ(h)Ch,Ωforalls,l.

An analogous computation holds true if we differentiate with respect to mj. ∎

We are now in a position to prove Proposition 2.1.

Proof of Proposition 2.1.

In terms of the operator Tλ defined in Proposition 3.1, problem (2.21) becomes

ϕ=Tλ(E+N(ϕ)):=A(ϕ).(3.26)

For a given number γ>0, let us consider the region

γ:={ϕC(Ω¯):ϕγλ}.

From Proposition 3.1, we get A(ϕ)C[E,Ω+N(ϕ),Ω]. We claim that

f(U~)-j=1kεj-1ewj,ΩCλ,f′′(U~),ΩC.(3.27)

We postpone the proofs of (3.27) to the Appendix, Section 6. From (2.18), (3.27), from the definition of N(ϕ) in (2.13), it follows that

A(ϕ)C(λ+ϕ2+λϕ).

We then get that A(γ)γ for a sufficiently large but fixed γ and all small λ. Moreover, for any ϕ1,ϕ2γ, a straightforward computation gives

N(ϕ1)-N(ϕ2),ΩC[(maxi=1,2ϕi)+λ]ϕ1-ϕ2.

Thus we have

A(ϕ1)-A(ϕ2)CN(ϕ1)-N(ϕ2),ΩC[maxi=1,2ϕi+λ]ϕ1-ϕ2.

Hence the operator A has a small Lipschitz constant in γ for all small λ, and therefore a unique fixed point of A exists in this region.

We shall next analyze the differentiability of the map (ξ,m)=(ξ1,,ξk,m1,,mk)ϕ. Assume for instance that the partial derivative ξslϕ exists for s=1,,k, l=1,2. Since ϕ=Tλ(N(ϕ)+E), formally we have that

ξslϕ=(ξslTλ)(N(ϕ)+E)+Tλ(ξslN(ϕ)+ξslE).

From (3.25), we have ξslTλ(N(ϕ)+E)CN(ϕ)+E,ΩCλ. On the other hand,

ξslN(ϕ)=[f(U~+ϕ)-f(U~)-f′′(U~)ϕ]ξslU~+ξsl(Zijν-[j=1kεj-1ewj])ϕ+[f(U~+ϕ)-f(U~)]ξslϕ+(f(U~)-[j=1kεj-1ewμj])ξslϕ.

Then

ξslN(ϕ),ΩC{ϕ2+λϕ+ϕξslϕ+λξslϕ}.

Since ξslE,Ωλ, Proposition 3.1 guarantees that ξslϕCλ for all s,l. An analogous computation holds true if we differentiate with respect to mj. Then the regularity of the map (ξ,m)ϕ can be proved by standard arguments involving the Implicit Function Theorem and the fixed point representation (3.26). This concludes proof of the proposition. ∎

4 Proof of Proposition 2.2 and expansion of the energy

Up to now we have solved the nonlinear problem (2.21). In order to find a solution to the original problem, we need to find ξ and m such that

cij(ξ,m)=0foralli=0,1,j=1,,k.(4.1)

We recall the following definitions: the energy functional associated to problem (1.1) is

Jλ(u)=12Ω(|u|2+u2)-λ2Ωeu2,uH1(Ω),

and the finite-dimensional restriction

λ(ξ,m)=Jλ(λ(U~(ξ,m)+ϕ(ξ,m))),

where ϕ is the unique solution to problem (2.21) given by Proposition 2.1. Critical points of λ correspond to solutions of (4.1) for a small λ, as the result of Proposition 2.2 states. We give the proof of this result.

Proof of Proposition 2.2.

A direct consequence of the results obtained in Proposition 2.1 and the definition of function U~ is the fact that the map (ξ,m)λ(ξ,m) is of class C1.

From Proposition 2.1, we observe that

Dξ,mλ(ξ,m)=Dξ,mJλ(λ(U~(ξ,m)+ϕ(ξ,m)))[λDξ,mU~(ξ,m)](1+o(1)).

We can rewrite

(U~+ϕ)(ξ,m)(x)=mlvl(x-ξlεl)+12λml

with

vl(y):=wμl(y)+j=1k(O(|εly+ξl-ξj|)+O(εj2))for|y|δεl.

Since U~+ϕ is the solution of (2.21), it follows that vl satisfies

-Δvl+εl2(vl+12λml2)=0in Ωl

and

vlν-(1+2λml2vl)evleλml2vl2=ml-1εli=01j=1kcijεj-1χ(Fj(εly+ξl-ξj)εj)zij(Fj(εly+ξl-ξj)εj)onΩl,

where Ωl=Ω-ξlεl. For any l, we define

Il(vl)=12Ωl[|vl|2+εl2(vl+12λml2)2]-Ωlevleλml2vl2.

We note that λ(ξ,m)=λml2Il(vl). We compute the differential Dmsλ(ξ,m), s=1,,k; thus we have

Dmsλ(ξ,m)=λmlεli=01j=1k(Ωlεj-1χ(Fj(εly+ξl-ξj)εj)zij(Fj(εly+ξl-ξj)εj)Dmsvl(y)𝑑y)cij.

Now, fix i and j. We compute the coefficient in front of cij. To this end, we choose l=j and obtain

Ωlεj-1χ(Fj(εly+ξl-ξj)εj)zij(Fj(εly+ξl-ξj)εj)Dmsvl(y)𝑑y=μjms+2z0j2(y)𝑑y(1+o(1)).

Thus we concludes that for any s=1,2,,k, we have

Dmsλ(ξ,m)=λmlεlj=1kμjms+2z0j2(y)𝑑yc0j(1+o(1)).

Similarly, we get that for all s,l,

Dξs1λ(ξ,m)=λmlεl[j=1k(μjξs1+2z0j2(y)𝑑y)c0j+(+2z1s2(y)𝑑y)c1s](1+o(1)).

Thus, we can conclude that Dξ,mλ(ξ,m)=0 is equivalent to the following system:

[j=1kμjmsc0j](1+o(1))=0for s=1,2,,k,(4.2)[Aj=1kμjξs1c0j+c1s](1+o(1))=0for all s(4.3)

for some fixed constant A, with o(1) small in the sense of the L-norm as λ0. The conclusion of the lemma follows if we show that the matrix μjms of dimension k×k is invertible in the range of the points ξj and parameters mj we are considering. Indeed, this fact implies unique solvability of (4.2). Inserting this in (4.3), we get unique solvability of (4.3).

Consider the definition of the μj, in terms of the parameters mj and points ξj given in (3.2). These relations correspond to the gradient DmF(m,ξ) of the function F(m,ξ) defined as

F(m,ξ)=12j=1kmj2[-2log(2mj2)-log(2μj)+2+H(ξj,ξj)]+ijmimjG(ξi,ξj).

We set sj=mj2. Then the above function can be written as

F(s,ξ)=12j=1ksj[-2log(2sj)-log(2μj)+2+H(ξj,ξj)]+ijG(ξi,ξj)sisj.

This function is a strictly convex function of the parameters sj, for parameters sj uniformly bounded and uniformly bounded away from 0 and for points ξj in Ω uniformly far away from each other and from the boundary. For this reason, the matrix (2Fsisj) is invertible in the range of parameters and points we are considering. Thus, by the Implicit Function Theorem, relation (2.16) defines a diffeomorphism between μj and mj. This fact gives the invertibility of (μjms). This concludes the proof of the lemma. ∎

In order to solve for critical points of the functional λ, a key step is its expected closeness to the functional Jλ(λU~). This fact is contained in the following lemma.

Lemma 4.1.

The following expansion holds:

λ(ξ,m)=Jλ(λU~)+ϑλ(ξ,m),

where

|ϑλ(ξ,m)|+|ϑλ(ξ,m)|=O(λ3)

uniformly on points ξ1,,ξk and parameters m1,,mk satisfying the constraints in Proposition 3.1.

Proof.

Taking into account DJλ(λ(U~+ϕ))[ϕ]=0, a Taylor expansion gives

Jλ(λ(U~+ϕ))-Jλ(λU~)=λ01D2Jλ(λ(U~+tϕ))[ϕ]2(1-t)𝑑t=λ01(Ω[N(ϕ)+E]ϕ+Ω[f(U~)-f(U~+tϕ)]ϕ2)(1-t)𝑑t.(4.4)

Since ϕCλ, we have Jλ(λ(U~+ϕ))-Jλ(λU~)=ϑλ(ξ,m)=O(λ3). Let us differentiate with respect to ξ. We use the representation (4.4) and differentiate directly under the integral sign; we get that, for all j,l,

ξjl[Jλ(λ(U~+ϕ))-Jλ(λU~)]=λ01(Ωξjl[(N(ϕ)+E)ϕ]+Ωξjl[(f(U~)-f(U~+tϕ))ϕ2])(1-t)𝑑t.

Since ξjlϕCλ and by the computations in the proof of Lemma 2.1, we get

ξjl[Jλ(λ(U~+ϕ))-Jλ(λU~)]=ξjlϑλ(ξ,m)=O(λ3).

With the same argument, we get

mj[Jλ(λ(U~+ϕ))-Jλ(λU~)]=O(λ3).

The continuity in ξ and m of all these expressions is inherited from that of ϕ and its derivatives in ξ and m in the L-norm. This concludes the proof. ∎

We end this section with the asymptotic estimate of Jλ(U), where

U(x)=λU~(x)=λj=1kmj[log1|x-ξj-εjμjν(ξj)|2+Hj(x)]

and Jλ is the energy functional associated to (1.1), whose definition is as follows:

Jλ(u)=12Ω(|u|2+u2)-λ2Ωeu2.

We have the following result.

Lemma 4.2.

Let μj be given by (2.16). Then

Jλ(U)=kπ2-|Ω|2λ+πφk(ξ,m)λ+λ2Θλ(ξ,m),(4.5)

where |Ω| denotes the measure of Ω, and Θλ(ξ,m) is a function, uniformly bounded with its derivatives, as λ0, for points ξ and parameters m satisfying (3.2). Furthermore, the function

φk(ξ,m)=φk(ξ1,,ξk,m1,,mk)

is defined by

φk(ξ,m)=2(log2-1)j=1kmj2+2j=1kmj2log(mj2)-j=1kmj2H(ξj,ξj)-ijmimjG(ξi,ξj).

Proof.

Let us set U(x)=j=1kUj(x) with Uj(x)=λmj[uj(x)+Hj(x)], where uj(x)=log1|x-ξj-εjμjν(ξj)|2 and Hj is defined in (2.6). Then

Jλ(U)=j=1k12Ω(|Uj|2+Uj2)+12ijΩ(UiUj+UiUj)-λ2ΩeU2=I1+I2+I3.

First, we write

Ω(|Uj|2+Uj2)=λmj2[Ω|uj|2+Ωuj2+Ω|Hj|2+Ω(Hj)2+2ΩujHj+2ΩujHj].

Multiplying (2.6) by Hj, it yields

Ω|Hj|2+Ω(Hj)2=-ΩujHj+ΩHjνHj=-ΩujHj+2εjμjΩeujHj-ΩujνHj,

and multiplying (2.6) by uj again, we find

Ωuj2+ΩHjuj=-ΩujHj+2εjμjΩeujuj-Ωujνuj,

Then we get

Ω(|Uj|2+Uj2)=λmj2[Ω|uj|2-Ωujνuj+ΩujHj-ΩujνHj+2εjμjΩeuj(uj+Hj)]=2λmj2εjμjΩeuj(uj+Hj)=2λmj2Ωεjμj|x-ξj-εjμjν(ξj)|2(log1|x-ξj-εμjν(ξj)|2+H(x,ξj)+O(εjα)).

Taking the change of variables y=x-ξjεjμj, we have

Ω(|Uj|2+Uj2)=2λmj2[Ωεjμj1|y-ν(0)|2(log1|y-ν(0)|2+H(ξj+εjμjy,ξj)-2log(μjεj))+O(εjα)]=2λmj2[Ωεjμj1|y-ν(0)|2(log1|y-ν(0)|2+H(ξj,ξj)-2log(εj)-2log(2μj)+2log2)]+2λmj2[Ωεjμj1|y-ν(0)|2(H(ξj+εjμjy,ξj)-H(ξj,ξj))+O(εjα)].

We have

Ωεjμj1|y-ν(0)|2=π+O(εjα),Ωεjμj1|y-ν(0)|2log1|y-ν(0)|2=-2πlog(2)+O(εjα),Ωεjμj1|y-ν(0)|2(H(ξj+εμjy,ξj)-H(ξj,ξj))=Ωεjμj1|y-ν(0)|2O(εjα|y|α)=O(εjα).

Using the definition of εj, we thus conclude that

Ω(|Uj|2+Uj2)=π+2λmj2[πH(ξj,ξj)-2πlog(2mj2)-2πlog(2μj)+O(εjα)].

Therefore

I1=kπ2+j=1kλmj2[πH(ξj,ξj)-2πlog(2mj2)-2πlog(2μj)+O(εjα)].(4.6)

On the other hand, we have

ijΩ(UiUj+UiUj)=ijλmimj[Ωuiuj+ΩuiHj+ΩujHi+ΩHiHj+Ωuiuj+ΩuiHj+ΩujHi+ΩHiHj].

Multiplying (2.6) by Hi and integrating, we find

ΩHjHi+ΩHjHi=-ΩujHi+2εjμjΩeujHi-ΩujνHi.

Hence

ijΩ(UiUj+UiUj)=ijλmimj[Ωuiuj+ΩuiHj+ΩujHi+Ωuiuj+ΩuiHj+2εjμjΩeujHi-ΩujνHi].(4.7)

Multiplying (2.6) by ui again and integrating, we find

ΩHjui+Ωujui=-ΩHjui+2εjμjΩeujui-Ωujνui.(4.8)

By (4.7) and (4.8) we find that

ijΩ(UiUj+UiUj)=ijλmimj[Ωuiuj-Ωujνui+ΩujHi-ΩujνHi+2εjμjΩeuj(ui+Hi)].

Then we conclude that

I2=ijλπmimj[G(ξi,ξj)+O(εi2log1εi+εj2log1εj)+O(εiα+εjα)].(4.9)

Finally, let us evaluate the third term in the energy

λ2ΩeU2(x)=λ2j=1kΩB(ξj,δεj)eU2(x)=I+λ2Ω\j=1kB(ξj,δεj)eU2(x)=II.(4.10)

We write

ΩB(ξj,δεj)eU2(x)=ΩB(ξj,δεj|logεj|)eU2(x)=:IA+Ω(B(ξj,δεj)\B(ξj,δεj|logεj|))eU2(x)=:IB.

We have

IA=ΩB(ξj,δεj|logεj|)eU2(x)=ΩB(ξj,δεj|logεj|)e[λmj(-2logεj+βj+wj+θ(x))]2=εj-1eβj2ΩB(ξj,δεj|logεj|)ewjeθ(x)eλmj2[wj2+2wjθ(x)+θ2(x)]=2mj2εj-1ΩB(ξj,δεj|logεj|)2μj|x-ξjεj-μjν(ξj)|2(1+O(λ))=2mj2Ω-ξjεjμjB(0,δ|logεj|μj)2|y-ν(0)|2(1+O(λ))=4πmj2(1+λΘλ(m,ξ)),(4.11)

with Θλ(m,ξ) a function, uniformly bounded with its derivatives, as λ0. And

|IB|Cδ|logεj|δεj-121r2elog2rγj2rdr(sett=logr)=CR1+logγj2R2+γj24e-2t+4t2γj2𝑑tCR1+logγj2R2+γj24e-t𝑑t=O(λ).(4.12)

Moreover,

II=λ2[|Ω|+j=1kλ2Θλ(m,ξ)],(4.13)

where |Ω| denotes the measure of Ω, and Θλ(m,ξ) is a function, uniformly bounded with its derivatives, as λ0. Then from (4.10)–(4.13), we get

I3=-2λπj=1kmj2(1+λΘλ(m,ξ)).(4.14)

Hence from (4.6), (4.9) and (4.14) we obtain

Jλ(U)=π[j=1kmj2H(ξj,ξj)+ijmimjG(ξi,ξj)-2j=1kmj2-2j=1kmj2log(2mj2)]λ+kπ2-|Ω|2λ-2πλj=1kmj2log(2μj)+o(λ).

By the choice of μj in (2.16), we get that the function Θ(ξ,m) in the expansion (4.5) is uniformly bounded, as λ0, for points ξ and parameters m satisfying (3.2). In order to prove that also the derivatives, in ξ and in m, of this function Θ(ξ,m) are uniformly bounded, as λ0, in the same region, one argues similarly as for the C0 expansion of Jλ(U). We leave the details to the reader. Thus the proof of this lemma is complete. ∎

5 Proof of Theorem 1.1

In this section, we will prove the main result.

Proof of Theorem 1.1.

Let 𝒟 be the open set such that

𝒟¯{(ξ,m)(Ω)k×+k:ξiξj for all ij)}.

From Proposition 2.2, the function

uλ(x)=λ(U~(ξ,m)+ϕ(ξ,m)),

where U~(ξ,m) defined by (2.5) and ϕ is the unique solution to problem (2.21) given by Proposition 2.1, is a solution of problem (1.1) if we adjust (ξ,m) so that it is a critical point of λ(ξ,m) defined by (2.23). This is equivalent to finding a critical point of

~λ(ξ,m)=1πλ[λ(ξ,m)-kπ2+|Ω|2λ].

On the other hand, from Lemmas 4.1 and 4.2, for (ξ,m)𝒟 satisfying (2.4), we have

~λ(ξ,m)=φk(ξ,m)+o(1)Θλ(m,ξ),(5.1)

where Θλ(m,ξ) and Θλ(m,ξ) are uniformly bounded in consider region as λ0. Thus we need to find a critical point of

φk(ξ,m)=2(log2-1)j=1kmj2+2j=1kmj2log(mj2)-j=1kmj2H(ξj,ξj)-ijmimjG(ξi,ξj).

We make the change of variables sj=mj2, and set b=2(log2-1). And we next find a critical point of

φk(ξ,s)=bj=1ksj+2j=1ksjlog(sj)-[j=1ksjH(ξj,ξj)+ijksisjG(ξi,ξj)],

which is well defined on 𝒟¯. For j{1,2,,k}, we have

sjφk(ξ,s)=b+2+2log(sj)-H(ξj,ξj)-12ijksisjG(ξi,ξj)

and

sjsj2φk(ξ,s)=2sj+14ijksisj1sjG(ξi,ξj),sjsi2φk(ξ,s)=14ijk1sisjG(ξi,ξj).

We have that φk(ξ,s) is strictly convex as a function of s, and it is bounded below. Hence it has a unique minimum point, which we denote by s¯=(s¯1,,s¯k), each component of s¯ is a function of points ξ1,,ξk, namely s¯j=s¯j(ξ1,,ξk), satisfying

b+2+2log(s¯j)-H(ξj,ξj)-12ijks¯is¯jG(ξi,ξj)=0.(5.2)

We have:

  • (1)

    s¯j is a C1 function with respect to ξ defined in (Ω)k.

  • (2)

    There is a positive constant c0 such that s¯jc0 for each j=1,,k.

  • (3)

    s¯j+ as |ξi-ξj|0 for some ij.

In fact, (1) directly holds by the Implicit Function Theorem. Moreover, since G(ξi,ξj) is positive and H(ξj,ξj) is bounded, from (5.2) we have

s¯j2>e-(b+2)+H(ξj,ξj)

Then we get (2). Furthermore, for some ij, we have G(ξi,ξj)+ as |ξi-ξj|0, so (3) holds by (5.2).

A direct computation shows that

Φk(ξ):=φk(ξ,s¯)=-2j=1ks¯j(ξ)

for ξΩ^k={(ξ1,,ξk)(Ω)k:ξiξj if ij}.

Given one component 𝒞0 of Ω, let Λ:S1𝒞0 be a continuous bijective function that parameterizes 𝒞0. Set

Ω~k={(ξ1,,ξk)𝒞0k:|ξi-ξj|>δ for ij}.

Next, we find a critical point of Φk. The function Φk is C1, bounded from above in Ω~k, and from (3) we have

Φk(ξ)=Φk(ξ1,,ξk)-as|ξi-ξj|0 for someij.

Hence, since δ is arbitrarily small, Φk has an absolute maximum M in Ω~k.

On the other hand, using Ljusternik–Schnirelmann theory as in the proof in [12], we get that Φk has at least two distinct points in Ω~k. Let cat(Ω~k) be the Ljusternik–Schnirelmann category of Ω~k relative to Ω~k, which is the minimum number of closed and contractible in Ω~k sets whose union covers Ω~k. We will estimate the number of critical points for Φk below by cat(Ω~k).

Indeed, by contradiction, suppose that cat(Ω~k)=1. This means that Ω~k is contractible in itself, namely there exist a point ξ0Ω~k and a continuous function Γ:[0,1]×Ω~kΩ~k such that, for all ξΩ~k,

Γ(0,ξ)=ξ,Γ(1,ξ)=ξ0.

Define f:S1Ω~k to be the continuous function given by

f(ξ¯)=(Λ(ξ¯),Λ(e2πi1kξ¯),,Λ(e2πik-1kξ¯)).

Let η:[0,1]×S1S1 be the well-defined continuous map given by

η(t,ξ¯)=Λ-1π1Γ(t,f(ξ¯)),

where π1 is the projection on the first component. The function η is a contraction of S1 to a point and this gives a contradiction; then the claim follows.

Therefore we have cat(Ω~k)2 for any k1. Define

c=supCΞinfξCΦk(ξ),

where

Ξ={CΩ~k:C is closed and cat(C)2}.

Then by the Ljusternik–Schnirelmann theory we obtain that c is a critical level.

If cM, we conclude that Φk has at least two distinct critical points in Ω~k. If c=M, there is at least one set C such that cat(C)2, where the function Φk reaches its absolute maximum. In this case we conclude that there are infinitely many critical points for Φk in Ω~k.

Thus we obtain that the function Φk has at least two distinct critical points in Ω~k, denoted by ξ1,ξ2. Hence (ξ1,s¯(ξ1)) and (ξ2,s¯(ξ2)) are two distinct critical points for the function φk(ξ,s). From (5.1) we then have that ~λ(ξ,m) has at least two critical points. This ends the proof of Theorem 1.1. ∎

6 Appendix

Proof of the first estimate in (3.21).

We shall prove

-ΔZ~0j+Z~0j**,ΩC|logεj|,

where Z~0j is defined in (3.18). Perform the change of variables y=εj-1Fj(x) and denote

z~0j(y)=Z~0j(Fj-1(εjy)).

Then

-ΔZ~0j+Z~0j=(z~0j+εj2z~0j),

where is defined in (3.10). We shall show that

|(z~0j+εj2z~0j)|C|logεj|[εj2+j=1m(1+|y-ξj|)-2-σ],yΩεj.

This fact implies the first estimate in (3.21).

Let us first consider the region where |y|<R. In this region, z~0j=z0j. Since Δz0j=0 and since (3.10) holds, we have

(z~0j+εj2z~0j)=O(εj)for |y|<R.(6.1)

In the region R+1<|y|<δ4εj, we have z~0j=hz0j. Therefore, in this region,

|Δz~0j|=2|hz0j|Cr3logδεj,r=|y|.

For the other terms we find

|2z~0j||2h|z0j+2|hz0j|+h|2z0j|=O(1r2logδεj)+O(1r3logδεj)+O(1r3)

and

|z~0j||h|z0j+h|z0j|=O(1rlogδεj)+O(1r2).

Hence

(z~0j+εj2z~0j)=O(1r3logδεj)+O(εjrlogδεj)+O(εjr2)+εj2z~0j,R+1<r<δ4εj.(6.2)

In the region δ4ε<r<δ3ε the definition of z~0j is

z~0j=η2hz0j.

We will estimate each term of (3.10) using the facts that η2=O(εjδ), |2η2|=O(εj2δ2) and that in the considered region h=O(1logδεj) which implies also z~0j=O(1logδεj). We obtain, for δ4εj<r<δ3εj,

Δz~0j=O(εj2δ2logδεj)

and

2z~0j=O(εj2δ2logδεj)+η2(2hz0j+2hz0j+h2z0j)=O(εj2δ2logδεj).

Similarly, for δ4εj<r<δ3εj, we have

z~0j=η2hz0j+η2hz0j+η2hz0j=O(εjδlogδεj).

This shows that

(z~0j+εj2z~0j)=O(ε2δ2logδεj),δ4εj<r<δ3εj.(6.3)

Thus we only need to estimate the size of z~0j+εj2z~0j in the region R<r<R+1. In this region we have

z~0j=η1z0j+(1-η1j)hz0j

and hence

Δz~0j=O(1logδεj)++η1Δz0j+(1-η1)Δ(hz0j),R<r<R+1.

First we recall that Δz0j=0 and, for R<r<R+1,

Δ(hZ0j)=2hz0j+O(εj)=O(1logδεj)+O(εj).

Thus

z~0j+εj2z~0j=O(1logδε),R<r<R+1.

This bound and (6.1), (6.2) and (6.3) imply (3.21). ∎

Proof of the second estimate in (3.21).

We shall prove

L(Z~0j)*,ΩC|logεj|.

We perform the change of variables y=εj-1Fj(x). We already observed that we can assume that Fj(ξj)=I. Hence,

L(Z~0j)=(1+o(1))[B(z~0j)-W~z~0j],

where z~0j=Z~0j(Fj-1(εjy)) and W~(y)=W(Fj-1(εjy)). Recall that B is the differential operator of order one on +2 defined in (3.12), and W~ is described in (3.13). Thus in the region y(Ωεj), with |y|<R, we get

B(z~0j)-W~z~0j=O(εj).(6.4)

Next, in the region R<|x|<R+1 we have

z~0j=O(1logδεj)+η1(1-h)z0j+hz0j.

Since h is radial, this implies

B(z~0j)=-hz0jx2+O(1R2logδεj)+O(Rεjlogδεj),R<|y|<R+1,y+2.

Using (3.13), we see that

B(z~0j)-W~z~0j=O(1R2logδεj)+O(Rεjlogδεj),R<|y|<R+1,y+2.(6.5)

Using the fact that h has zero normal derivative on +2, we deduce

B(h~z0j)=-hz0jx2+O(εjlogδεj)+O(εjr),R+1<r<δεj.(6.6)

On the other hand, using (3.13), we have in R+1<r<δεj that

B(z~0j)-W~z~0j=O(εjlogδεj)+O(εjαr)(6.7)

for some 0<α<1. Finally, we consider δ4εj<r<δ3εj. In this region we have z~0j=η2hz0j and h,z0j=O(1logδεj), η¯2=O(εjδ). Using these facts, estimate (6.6) and that η2 has zero normal derivative, we find

B(z~0j)=B(η2)hz0j+η2B(hz0j)=O(εj2rδlogδεj)+O(1r2)+O(εjlogδεj)+O(εjr)

for δ4εj<r<δ3εj. From (3.13) we have

W~=O(εjαr),δ4εj<r<δεj.

Thus we conclude that for yΩεj, δ4εj<r<δ3εj,

B(z~0j)-W~z~0j=O(εj2rδlogδεj)+O(1r2)+O(εjlogδεj)+O(εjr).(6.8)

Estimates (6.4), (6.5), (6.7) and (6.8) give the validity of the second estimate in (3.21). ∎

Proof of the first estimate in (3.27).

We shall prove

f(U~)-j=1kεj-1ewj,ΩCλ.

We write

f(U~)=λeλU~2+2λ2U~2eλU~2=:Ia+Ib.

For xΩ, far away from the points ξj, namely for |x-ξj|>δ, i.e. |y-ξj|>δεj for all j=1,2,,k, a consequence of (2.8) is that Ia=λO(1), Ib=λ2O(1). Then we have

f(U~)1outer=λO(1),(6.9)

where 1outer is the characteristic function of the set {y:|y-ξj|>δεj,j=1,,k}. Moreover, for |x-ξj|>δ, we have

j=1kεj-1ewj1outer=O(1)j=1kεj=O(1)j=1k2mj2e-14mj21λ=λO(1).(6.10)

On the other hand, fix the index j in {1,2,,k}, for |x-ξj|<δ, from (2.9), (2.14) and (2.15) we have

Ia=2λmj2eλmj2wj2(1+O(λwj))εj-1ewj,

and

Ib=1mj2(1+2λmj2(wj+O(1)))eλmj2wj2(1+O(λwj))εj-1ewj.

Then we find

Ia1inter=λj=1kρj(x)O(1),Ib1inter-j=1kεj-1ewj=λj=1kρj(x)O(1),(6.11)

where 1inter is the characteristic function of the set j=1k{y:|y-ξj|<δεj}. Then from (6.9), (6.10), (6.11) and the definition of the -norm, we obtain the first estimate in (3.27). ∎

Proof of the second estimate in (3.27).

We write

f′′(U~)=6λ2U~eλU~2+4λ3U~3eλU~2:=Ic+Id.

For xΩ, far away from the points ξj, namely for |x-ξj|>δγj, i.e. |y-ξj|>δγjεj for all j=1,2,,k, a consequence of (2.8) and (2.17) is that Ic=λ2O(1), Id=λ3O(1). Then we have

f′′(U~)1outer=λ2O(1).

On the other hand, fix the index j in {1,2,,k}, for |x-ξj|<δγj, from (2.9), (2.14) and (2.15) we have

Ic=6λ(1+2λmj2(wj+O(1)))eλmj2wj2(1+O(λwj))εj-1ewj=6λ{(1+2λmj2(wj+O(1)))eλmj2wj2(1+O(λwj))-1}εj-1ewj+6λεj-1ewj=12λ2mj212λmj2{(1+2λmj2(wj+O(1)))eλmj2wj2(1+O(λwj))-1}εj-1ewj=ρj(x)+6λεj-1ewj=λρj(x)O(1)=λρj(x)O(1)

and

Id=12mj6(1+2λmj2(wj+O(1)))3e12(log1εj2+βj)ewjeθ(x)eλmj2(wj+θ(x))2=12mj6(1+2λmj2(wj+O(1)))3eβj2εj-1ewjeθ(x)eλmj2wj2(1+O(λ)wj)=1mj4(1+2λmj2(wj+O(1)))eλmj2wj2(1+O(λwj))εj-1ewj=2λmj212λmj2{(1+2λmj2(wj+O(1)))eλmj2wj2(1+O(λwj))-1}εj-1ewj=ρj(x)+1mj4εj-1ewj=ρj(x)O(1)=ρj(x)O(1).

Thus we obtain

f′′(U~)1inter=O(1)j=1kρj(x).

Collecting all this information and using the definition of the -norm, we obtain the second estimate in (3.27). ∎

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About the article

Received: 2017-04-10

Accepted: 2017-05-22

Published Online: 2017-07-28


The research of the first author has partly been supported by NSFC No. 11501469 and Chongqing Research Program of Basic Research and Frontier Technology cstc2016jcyjA0032. The research of the second author has partly been supported by Fondecyt Grant 1160135 and Millennium Nucleus Center for Analysis of PDE, NC130017.


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 615–644, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0092.

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