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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


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


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


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),


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


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


defined by


where G is the Green function for the Neumann problem


and H its regular part defined as


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


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


with ξ1ξ2, such that φk(ξi,mi)=0 and


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


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


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


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



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).


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


as λ0. We set




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


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)





Here and in what follows f denotes the nonlinearity


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


as λ0. We thus choose εj to be defined as


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


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


This condition defines the parameter μj as follows:


With these choices of μj we get


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:


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


Then we get


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


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


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


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},


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


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


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


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


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


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


and the finite-dimensional restriction


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,


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)



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



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


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




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


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)



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


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


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


This shows the validity of (3.7).

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


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


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


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


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


smooth, positive and bounded function such that


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


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


By the maximum principle and the Hopf Lemma we find that


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


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





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


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


holds. In fact, by Lemma 3.1, we have


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


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



η2{1in B(0,δ4εj),0in 2B(0,δ3εj),

so that


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


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


Observe first that, assuming R>R0, we have

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

Furthermore, we have


We claim that


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


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


Observe first that, assuming R>R0, we have

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



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


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


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


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


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


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


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


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


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


Summarizing all the above information, we get


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


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


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


for some constant C independent on λ.

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


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


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


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


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



|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


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


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


and on the boundary Ω,


with dij=ξslcij, and the orthogonality conditions now become


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




We then have

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





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


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


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


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


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


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


Thus we have


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


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




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


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


and the finite-dimensional restriction


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


We can rewrite




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

-Δvl+εl2(vl+12λml2)=0in Ωl



where Ωl=Ω-ξlεl. For any l, we define


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


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


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


Similarly, we get that for all s,l,


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


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


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:




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


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


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,


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


With the same argument, we get


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


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


We have the following result.

Lemma 4.2.

Let μj be given by (2.16). Then


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


is defined by



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


First, we write


Multiplying (2.6) by Hj, it yields


and multiplying (2.6) by uj again, we find


Then we get


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


We have


Using the definition of εj, we thus conclude that




On the other hand, we have


Multiplying (2.6) by Hi and integrating, we find




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


By (4.7) and (4.8) we find that


Then we conclude that


Finally, let us evaluate the third term in the energy


We write


We have


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




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


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


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


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


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


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


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


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




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


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


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


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,


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


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


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



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


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




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


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,


For the other terms we find






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


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,




Similarly, for δ4εj<r<δ3εj, we have


This shows that


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


and hence


First we recall that Δz0j=0 and, for 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


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


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


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


Since h is radial, this implies


Using (3.13), we see that


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


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


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


for δ4εj<r<δ3εj. From (3.13) we have


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


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


We write


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


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


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




Then we find


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


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


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




Thus we obtain


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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