# Construction of Nodal Bubbling Solutions for the Weighted Sinh-Poisson Equation

Authors: Yibin Zhang; Haitao Yang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider the weighted sinh-Poisson equationΔu+2ε2|x|2αsinhu=0 in B1(0), u=0 on ∂B1(0), where ε>0 is a small parameter, α∈(-1,+∞)∖{0}, and B1(0) is a unit ball in ℝ2. By a constructive way, we prove that for any positive integer m, there exists a nodal bubbling solution uε which concentrates at the origin and the other m-points q~l=(λcos(2π(l−1)/m),λsin(2π(l−1)/m)), l=2,…,m+1, such that as ε→0, 2ε2|x|2αsinhuε⇀8π(1+α)δ0+∑l=2m+18π(−1)l−1δq~l, where λ∈(0,1) and m is an odd integer with (1+α)(m+2)-1>0, or m is an even integer. The same techniques lead also to a more general result on general domains.


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## Body

## 1. Introduction

We are concerned with stationary Euler equations for an incompressible, homogeneous, and inviscid fluid on a bounded, smooth planar domainΩ, consider
(1)(w·∇)w+∇p=0,inΩ,divw≔∇·w=0,inΩ,w·ν=0,on∂Ω,
where w is the velocity field, p is the pressure, and ν is the unit outer normal vector to ∂Ω. Let us introduce the vorticity ω=curlw. By applying the curl operator to the first equation in (1), we have
(2)w·∇ω=0,inΩ.On the other hand, the second equation is equivalent to rewriting the velocity fieldw as
(3)w=(∇ψ)⊥:=(-∂x2ψ,∂x1ψ).
In return, the vorticity ω is expressed as ω=-Δψ in term of ψ, the so-called stream function. Now, the ansatz ω=ω(ψ) guarantees that (2) is also automatically satisfied, and then the Euler equations reduce to solving the Dirichlet elliptic problem as follows
(4)-Δψ=ω(ψ),inΩ,ψ=0,on∂Ω.Over the past decades, some vortex-type configurations for planar stationary turbulent Euler flows have aroused wide concern among the people (see [1–3]). Many functions ω(ψ) have been chosen in the physical perspective to describe turbulent Euler flows with vorticity ω concentrated in small “blobs”. For example, on the basis of the statistical mechanics approach, Joyce and Montgomery proposed the Stuart vortex pattern ω(ψ)=ε2eψ with a small positive parameter ε to describe positive vortices (see [4–8]). Meanwhile, they also proposed the Mallier-Maslowe vortex pattern ω(ψ)=2ε2sinhψ to describe coexisting positive and negative vortices (see [9, 10]). Recently, Tur and Yanovsky in [11] have used the singular ansatz ω(ψ)=ε2eψ-4πNδq to describe vortex patterns of necklace type with N+1-fold symmetry in rational shear flow, where N∈ℕ and δq denotes the Dirac mass at q∈Ω. Now, we adopt another new singular ansatz in [12] ω(ψ)=2ε2|x|2αsinhψ with α∈(-1,+∞)∖{0} to study the corresponding vortices with concentrated vorticity. To do it, we hope to investigate the effect of the presence of the weight |x|2α on the existence of nodal bubbling solutions for the weighted sinh-Poisson equation as follows:
(5)Δu+2ε2|x|2αsinhu=0,inB1(0),u=0,on∂B1(0),
where ε>0 is a small parameter, α∈(-1,+∞)∖{0}, and B1(0) is a unit ball in ℝ2.Let us first recall the two-dimensional sinh-Poisson equation as follows:(6)Δu+2ε2sinhu=0,
which relates to various dynamics of vorticity with respect to geophysical flows, rotating and stratified fluids, and fluid layers excited by electromagnetic forces (see [13–15] and the references therein) and the geometry of constant mean curvature surfaces studied by many works (see [16–20] and the references therein). Recently, the asymptotic behavior of solutions to (6) has been studied on a closed Riemann surface in [21, 22], and the authors applied the so-called “Pohozaev identity” and “symmetrization method,” respectively, to show that there possibly exist two different types of blow up for a family of solutions to (6). Furthermore, Grossi and Pistoia in [23] exhibited sign-changing multiple blow-up phenomena for the Dirichlet problem (6), more precisely, if 0∈Ω and Ω is symmetric with respect to the origin, for any integer k if ε is small enough, there exists a family of solutions to (6), which blows up at the origin, whose positive mass is 4πk(k-1) and negative mass is 4πk(k+1). This gives a complete answer to an open problem in [21]. Besides, for a similar equation, precisely the Neumann sinh-Gordon equation on a unit ball, Esposito and Wei in [24] also constructed a family of solutions with a multiple blow up at the origin. On the other hand, Bartolucci and Pistoia in [25] tried to construct blow-up solutions of (6) with Dirichlet boundary condition, and proved that for ε>0 small enough, there exist at least two pairs of solutions, which change sign exactly once, concentrate in the domain, and whose nodal lines intersect the boundary. Furthermore, Bartsch et al. in [26] obtained the existence of changing sign solutions for this equation on an arbitrary bounded domain Ω, which have three and four alternate-sign concentration points. In particular, when Ω has an axial symmetry they proved for each N∈ℕ there exists a nodal bubbling solution, which changes sign N-times and whose alternate-sign concentration points align on the symmetric axis of Ω. For (6) with Neumann boundary condition, Wei et al. in [27] constructed a family of solutions concentrating positively and negatively in the domain and its boundary. As for the presence of the weight |x|2α, the authors in [12] showed that there exists a family of nodal bubbling solutions uε to (5) only involving 0<α∉ℕ, such that 2ε2|x|2αsinhuε not only develops many positive and negative Dirac deltas with weight 8π and -8π, respectively, but also a Dirac data with weight 8π(1+α) at the origin.We mention that an analogous blow-up analysis can be applied to multiple blow-up solutions for the Liouville equation with or without singular data as follows:(7)-Δu=ε2eu-4π∑i=1M‍αiδqi,inΩ,u=0,on∂Ω,
where Ω is a smooth bounded domain in ℝ2, M≥0, αi>-1, qi∈Ω, δqi defines the Dirac mass at qi, and ε>0 is a small parameter. For the past decades, the asymptotic analysis for blow up solutions of problem (7) has been deeply studied in the vast literature (see [28–33] and the reference therein), which exhibits the quantiaztion properties of the weak limit of ε2eu as ε→0 if ε2∫Ω‍eu remains uniformly bounded, and characterizes the location of concentration points as critical points of a functional in terms of Green’s function. Reciprocally, an obvious problem is how to construct solutions of (7) with these properties. In [34, 35], the authors use the asymptotic analysis to construct solutions with multiple interior concentration points for (7) with M≥0 and 0<αi∉ℕ. More generally, by a constructive way, similar results related to -1<αi can also be obtained in [36–40] under some milder notions of stability of critical points. In particular, when M≥1 and αis are positive numbers, D’Aprile in [37] recently established a family of solutions to (7) consisting of N~:=∑i=1M‍Ni blow up points in Ω∖{q1,…,qM} as long as Ni<1+αi for any i, provided that the weights αi avoid the integers 1,2,…,N~-1, and so the result of del Pino et al. in [39] can be extended to the case of several singular sources.In this paper, we will continue the study of the existence of solutions to (5). We prove that there exists a family of solutions uε concentrating positively and negatively at the origin and outside the origin as long as α∈(-1,+∞)∖{0}. Concerning the sign-changing concentration at the origin and outside the origin, if we introduce the function f(λ):(0,1)→ℝ defined as
(8)f(λ)=-16π{∑l,j=2,l≠jm+1‍(-1)l+j×[12log((2πl-jm)λ4+1-2λ2=-16π{∑l,j=2,l≠jm+1‍(-1)l+j×v×cos(2πl-jm))=-16π{∑l,j=2,l≠jm+1‍(-1)l+j×v-log|2sin(πl-jm)|(λ4+1-2λ2cos(2πl-jm))]=v-16π+[(1+α)(m-2A)-A2]logλ=v-16π+mlog(1-λ2)∑l,j=2,l≠jm+1‍},
with A=∑l=2m+1(-1)l+1, our main result for problem (5) can be stated as follows.Theorem 1.
For any positive integerm, there exists a nodal solution uε to problem (5) which concentrates at the origin and the other m-points q~l=(λ0cos(2π(l-1)/m),λ0sin(2π(l-1)/m)), l=2,…,m+1, such that as ε→0,
(9)2ε2|x|2αsinhuε⇀8π(1+α)δ0+∑l=2m+1‍8π(-1)l-1δq~l,
weakly in the sense of measures in B1(0)¯, where λ0 is an absolute minimum point of f(λ) in (0,1), m is an odd integer with (1+α)(m+2)-1>0, or m is an even integer. Moreover, for any δ>0, uε remains uniformly bounded on B1(0)∖(Bδ(0)∪⋃l=2m+1Bδ(q~l)), and as ε→0,
(10)supBδ(0)uε⟶+∞,supBδ(q~l)(-1)l-1uε⟶+∞∀l=2,…,m+1.Theorem1 is based on a constructive way which also works for the more generally weighted sinh-Poisson equation as follows:
(11)Δu+2ε2c(x)∏i=1M‍|x-qi|2αisinhu=0,inΩ,u=0,on∂Ω,
for ε>0 small, where Ω is a bounded smooth domain in ℝ2, M=n+k with M≥0, {q1,…,qM} are different singular sources in Ω, {α1,…,αn}⊂(-1,+∞)∖(ℕ∪{0}), {αn+1,…,αM}⊂ℕ, and c:Ω→ℝ is a continuous function such that c(qi)>0 for any i=1,…,M.To further state our results, we need to introduce some notations. LetG(x,y) be Green’s function of Δx such that for y∈Ω, -ΔxG(x,y)=δy(x) in Ω and G(x,y)=0 on ∂Ω, and let H(x,y) be its regular part defined as H(x,y)=G(x,y)+(1/2π)log|x-y|. Besides, let us denote S(x)=∏i=1M‍|x-qi|2αi, Ω′={x∈Ω:c(x)>0}, Γ={q1,…,qn}, J1={1,…,n}, J2={n+1,…,n+k}, J3={n+k+1,…,n+k+m}, and Δk+m={(qn+1,…,qn+k+m):qi=qjforsomei≠j}, where m≥0 is an integer. In what follows, we fix n different points qi,i∈J1, and define
(12)φk,mn(q)=-∑i∈J2∪J3‍di{∑j∈J1‍aiajdjG(qi,qj)+12diH(qi,qi)=-∑i∈J2∪J3‍divx+∑j∈J2∪J3,j≠i‍12aiajdjG(qi,qj)}-∑i∈⋃l=13J3‍dilogci(qi),
which is well defined on the following domain:
(13)Λk,m=(Ω′∖Γ)k+mΔk+m,
where q=(qn+1,…,qM+m), αi=0 for i∈J3, ci(x)=c(x)S(x)/|x-qi|2αi for i∈⋃l=13Jl, di=8π(1+αi) for i∈⋃l=13Jl, and ai=±1 for i∈⋃l=13Jl.Definition 2 (see [41]).
We say thatq* is a C0-stable critical point of φk,mn in Λk,m if for any sequence of functions ψj such that ψj→φk,mn uniformly on the compact subsets of Λk,m, ψj has a critical point ξj such that ψj(ξj)→φk,mn(q*).In particular, ifq* is a strict local maximum or minimum point of φk,mn, q* is a C0-stable critical point of φk,mn.Theorem 3.
Assume that{n,k,m}⊂ℕ∪{0} and q*=(qn+1*,…,qM+m*) is a C0-stable critical point of φk,mn in Λk,m with k+m≥1. Then, for any sufficiently small ε>0, there exists different points qε,l∈Ω′∖Γ, l∈J2∪J3, away from ∂Ω∪Γ, so that problem (11), for ql=qε,l,l∈J2, has a nodal solution uε such that as ε→0,
(14)2ε2c(x)∏i=1M‍|x-qi|2αisinhuε-∑J2∪J3‍aldlδqε,l⇀∑J1‍aldlδql,
weakly in the sense of measures in Ω¯. Moreover, up to a subsequence, there exists q~=(q~n+1,…,q~M+m)∈Λk,m such that
(15)φk,mn(q~)=φk,mn(q*),d(qε,l,q~l)⟶0,∀l∈J2∪J3,asε⟶0.
Besides,uε remains uniformly bounded on Ω¯∖(⋃i∈J1Bδ(qi))∪(⋃i∈J2∪J3Bδ(q~i)) for any δ>0, and for any points ql,l∈J1, and q~l,l∈J2∪J3, as ε→0,
(16)supΩ¯∩Bδ(ql)aluε⟶+∞,supΩ¯∩Bδ(q~l)aluε⟶+∞.Note that for the casen=1 and k=0 (or n=k=0), Theorem 3 was partly proved in [12] (or [25]) only when c(x)=1 and αl∈(0,+∞)∖ℕ, l∈J1. In contrast with the results of [12, 25], this theorem provides a more complex concentration phenomenon involving the existence of changing-sign solutions for problem (11) with both positive and negative bubbles near the singular sources ql,l∈J1∪J2, and some other discrete points. Unlike the concentration set in [12] only contains singular sources ql with αl∈(0,+∞)∖ℕ and l∈J1, and no singular source points in the domain, our concentration set also contains some singular sources ql with αl∈(-1,0) and l∈J1, except for singular sources ql,l∈J2, where concentration points and singular sources coincide at the limit. As for the latter exception, which till now is a similar but very simple concentration phenomenon it appears only in [38] for the study of the Liouville equation with a singular source of integer multiplicity.In order to obtain multiple sign-changing blow up solutions of problem (11), we use a Lyapunov-Schmidt finite-dimensional reduction scheme and convert the problem into a finite-dimensional one, for a suitable asymptotic reduced energy, related to φk,mn in (12). Thus, a stable critical point of φk,mn leads to the existence of multiple sign-changing blow up solutions to (11). However, in view of different signs of Green’s functions in (12), it seems very difficult to find out a stable critical point of φk,mn for a general bounded domain Ω. A simple approach can help us to overcome this difficulty by imposing the very strong symmetry condition on the domain of the problem, namely, we use the symmetry of the unit ball B1(0) to reduce the problem of finding solutions of (5) to that of finding an absolute minimum point of f(λ) defined in (8), and so we get the existence of nodal bubbling solutions for (5) in Theorem 1. On the other hand, motivated by the obtained results in [23], we believe that Theorem 1 should be valid for a general domain than a unit ball. More precisely, we suspect that, if 0∈Ω and Ω is symmetric with respect to the origin, it is possible to construct a family of sign-changing blow up solutions whose maxima and minima are located alternately at the origin and the vertices of a regular polygon, and so Theorem 1 will be a consequence of this general result.It is important to point out that to prove the above results, we need to use classification solutions of the Liouville-type equation to construct approximate solutions of problem (5) (or (11)) as follows:
(17)Δu+|x|2αeu=0,inℝ2,∫ℝ2‍|x|2αeu<+∞,α∈(-1,+∞).In complex notations, a complete classification of the solutions of (17) takes the following form:
(18)u(z)=log8(1+α)2δ~2(δ~2+|zα+1-ξ|2)2,
where δ~>0, ξ∈ℂ if α∈ℕ∪{0} and ξ=0 if α∈(-1,+∞)∖(ℕ∪{0}) (see [33, 42–44]). Using classification solutions scaled up and projected to satisfy the Dirichlet boundary condition up to a right order, the approximate solutions can be built up as a summation of these initial approximations with some suitable signs. Thus, the nodal bubbling solutions can be constructed as a small additive perturbation of these approximations through the so-called “localized energy method,” which combined the Lyapunov-Schmidt finite dimensional reduction and variational techniques. Here, we follow [12, 25], but we will overcome some of the difficulties that the nodal concentration phenomenon brings by delicate analysis.


## 2. Construction of the Approximate Solution

In this section, we will provide a first approximation for the solutions of problem (11). Given a sufficiently small but fixed number δ>0, let us first fix n different points qi, i∈J1, and assume that points q=(qn+1,…,qn+k+m)∈Λk,m(δ), where(19)Λk,m(δ):={q∈Λk,m:mini∈J2∪J3d(qi,∂(Ω′∖Γ))≥2δ,mini,j∈J2∪J3,i≠jd(qi,qj)≥2δ(qi,∂(Ω′Γ))}.Suppose thatμi, i∈⋃l=13Jl, are positive numbers to be chosen later, we define
(20)ρi=ε(1/(αi+1)),vi=μi(1/(αi+1)),ui(x)=log8μi2(1+αi)2ci(qi)[μi2ε2+|x-qi|2(1+αi)]2.The ansatz is(21)Uq(x)=∑i∈⋃l=13Jl‍aiPui(x),
where ai=±1, P:H1(Ω)→H01(Ω) is a linear operator such that for any u∈H1(Ω), ΔPu=Δu in Ω, and Pu=0 on ∂Ω. By harmonicity, we easily get that
(22)Pui(x)=ui(x)+diH(x,qi)-log8μi2(1+αi)2ci(qi)+O(ρ),inC1(Ω¯),(23)Pui(x)=diG(x,qi)+O(ρ),inCloc(Ω¯∖{qi}),
where ρ:=max{ρi:i∈⋃l=13Jl}.Consider that the scaling of solution to problem (11) is as follows:
(24)v(y)=u(εy),∀y∈Ω-ε,
where Ωε=(1/ε)Ω, then v satisfies
(25)Δv+2ε4c(εy)S(εy)sinhv=0,inΩε,v=0,on∂Ωε.We will seek solutions of problem (25) of the form v=Vq+ϕ, where
(26)Vq(y)=Uq(εy)=∑i∈⋃l=13Jl‍aiPui(εy),∀y∈Ω-ε.In terms ofϕ, problem (11) (or (25)) becomes
(27)L(ϕ)∶=Δϕ+Wqϕ:=-[Rq+N(ϕ)],inΩε,ϕ=0,on∂Ωε,
where
(28)Wq=2ε4c(εy)S(εy)coshVq,Rq is the “error term”:
(29)Rq=ΔVq+2ε4c(εy)S(εy)sinhVq,
and N(ϕ) denotes the following “nonlinear term”:
(30)N(ϕ)=2ε4c(εy)S(εy)×[sinh(Vq+ϕ)-sinhVq-ϕcoshVq].Finally, in order to makeRq sufficiently small, we choose the parameters μi,i∈⋃l=13Jl, as
(31)log8μi2(1+αi)2ci(qi)=diH(qi,qi)+∑j≠i‍aiajdjG(qi,qj),
so that from Appendix, we have(32)Wq={(εviρi)2|zi|2αi{8(1+αi)2[1+O(ρi|zi|)+O(ρ)][1+|zi|2(1+αi)]2+O(ε4)},if|zi|≤δ(viρi)-1,O(ε4),otherwise,(33)Rq={(εviρi)2|zi|2αi{8ai(1+αi)2[O(ρi|zi|)+O(ρ)][1+|zi|2(1+αi)]2+O(ε4)},if|zi|≤δ(viρi)-1,O(ε4),otherwise,zi:=(1/viρi)(εy-qi).


## 3. The Linearized Problem and the Nonlinear Problem

In this section, we will first prove the bounded invertibility of the linearized operatorL under suitable orthogonality conditions. Let us define
(34)Li(ϕ)=Δϕ+8(1+αi)2|z|2αi[1+|z|2(1+αi)]2ϕ∀i∈⋃l=13Jl,zi0=1μi|z|2(1+αi)-1|z|2(1+αi)+1∀i∈⋃l=13Jl,zi1=4Re(z1+αi)|z|2(1+αi)+1∀i∈⋃l=23Jl,zi2=4Im(z1+αi)|z|2(1+αi)+1∀i∈⋃l=23Jl.The key fact to develop a satisfactory solvability theory for the operatorL is that L formally approaches Li under dilations and translations, and any bounded solution of Li(ϕ)=0 in ℝ2 is a linear combination of zi0, zi1, and zi2 for i∈J2∪J3 (see [34, 45, 46]), or proportional to zi0 for i∈J1 (see [35, 36, 41]). Let us denote
(35)Zi0(y)≔zi0(zi),∀i∈⋃l=13Jl,Zij(y)≔zij(zi),∀i∈⋃l=23Jl,j=1,2,θi(y)≔arg(zi),∀i∈J2,χi(y)≔χ(|zi|),∀i∈⋃l=13Jl,
where χ(r) is a smooth, nonincreasing cut-off function such that for a large but fixed number R0>0, χ(r)=1 if r≤R0, and χ(r)=0 if r≥R0+1.Additionally, we consider the following Banach space:(36)𝒞*={h:∥h∥*<+∞},
with the norm
(37)∥h∥*=supy∈Ω¯ε(1(1+|zi|)4+2α∑αi<0,i∈J1‍)-1|h(y)|×hg×(ε2+∑αi<0,i∈J1‍(εviρi)2×g×ng×|zi|2αi(1+|zi|)4+2α+2αi×g×ng+∑αi≥0,i∈J1∪J2∪J3‍(εviρi)2×g×ng×1(1+|zi|)4+2α∑αi<0,i∈J1‍)-1),
where α∈(-1,α0) and α0:=min{αi:i∈⋃i=13Ji}.Given thath∈𝒞* and q∈Λk,m(δ), we consider the linear problem of finding a function ϕ and scalars cij, i∈J2∪J3, j=1,2, such that
(38)L(ϕ)=h+∑j=12‍∑i=n+1n+k+m‍cijχicos2(12αiθi)Zij,inΩε,ϕ=0,on∂Ωε,∫Ωε‍χicos2(12αiθi)Zijϕdy=0,∀i∈J2∪J3,j=1,2.Our main interest in this problem is its bounded solvability for anyh∈𝒞*, uniform in small ε and points q∈Λk,m(δ), as the following result states.Proposition 4.
There exist positive numbersε0 and C such that for any h∈𝒞*, there is a unique solution ϕ∈L∞(Ωε), scalars cij∈ℝ, i∈J2∪J3, j=1,2, to problem (38) for all ε<ε0 and points q∈Λk,m(δ), which satisfies
(39)∥ϕ∥∞≤C(log1ε)∥h∥*.
Moreover, the mapq′↦ϕ is C1, precisely for l∈J2∪J3, s=1,2,
(40)∥∂ql,s′ϕ∥∞≤Cεvlρl(log1ε)2∥h∥*,
where q′:=(1/ε)q=((1/ε)qn+1,…,(1/ε)qn+k+m).We begin by stating a priori estimate for solutions of (38) satisfying orthogonality conditions with respect to Zi0, i∈J1, and Zij, i∈J2∪J3, j=0,1,2.Lemma 5.
There exist positive numbersε0 and C such that for any points q∈Λk,m(δ), h∈𝒞* and any solution ϕ to the following equation:
(41)L(ϕ)=h,inΩε,ϕ=0,on∂Ωε,
with the orthogonality conditions
(42)∫Ωε‍χiZi0ϕdy=0,∀i∈J1,∫Ωε‍χicos2(12αiθi)Zijϕdy=0,hhhlhhh∀i∈J2∪J3,j=0,1,2,
one has
(43)∥ϕ∥∞≤C∥h∥*,
for all ε<ε0.Proof.
We have the following steps.
Step  1. We first construct a suitable barrier. To realize it, we claim that for ε>0 small enough, there exist Rd>0 and
(44)ψ:Ωε∖⋃i=1n+k+mBi,Rd¯⟼ℝ
positive and uniformly bounded so that
(45)L(ψ)≤-∑i=1n+k+m‍(εviρi)21|zi|2α+4-ε2,
where -1<α<α0 and Bi,Rd={y∈Ωε:|zi|<Rd}. Take
(46)g0(z)=|z|2(1+α)-1|z|2(1+α)+1,Rd=1d3(1/2(1+α)),g1i(y)=g0(d|zi|),where|zi|≥Rd.
Thus, we have1/2≤g1i(y)≤1, and
(47)Δg1i(y)=-(εviρi)2d28(1+α)2|dzi|2α[1+|dzi|2(1+α)]2×g0(d|zi|)<-(εviρi)2d-2(1+α)(1+α)2|zi|2α+4.
By (32),
(48)Wq≤C1∑i=1n+k+m‍(εviρi)21|zi|2αi+4,hhhhhl∀y∈Ωε∖⋃i=1n+k+mBi,Rd¯.
So, ifd is taken small and fixed, by (47)-(48), it follows that for |zi|≥Rd,
(49)L(g1i)(y)<-∑i=1n+k+m‍(εviρi)21|zi|2α+4<0.
Letg~2 be a positive, bounded solution of -Δg~2=1 in Ω and g~2=0 on ∂Ω. Set g2(y)=g~2(εy). Then, g2 is a positive, uniformly bounded function in Ωε such that
(50)-Δg2(y)=ε2,inΩε,g2=0,on∂Ωε.
Consider(51)ψ(y)=C2∑i=1n+k+m‍g1i(y)+g2(y),hhhl∀y∈Ωε∖⋃i=1n+k+mBi,Rd¯,
where C2>0 is a sufficiently large constant. Obviously, ψ is a positive and uniformly bounded function. Moreover, in view of (48)–(50), it is easy to check that ψ satisfies the estimate (45), and the claim follows.
Step  2. Consider the following “inner norm”:
(52)∥ϕ∥l=sup⋃i=1n+k+mBi,Rd¯|ϕ|.
Let us claim that there exists a constantC3>0, such that
(53)∥ϕ∥∞≤C3(∥ϕ∥l+∥h∥*).
Let us take(54)ϕ~(y)=C4ψ(y)(∥ϕ∥l+∥h∥*),hl∀y∈Ωε∖⋃i=1n+k+mBi,Rd¯,
where C4>0 is chosen larger if necessary. Then, for y∈Ωε∖⋃i=1n+k+mBi,Rd,
(55)L(ϕ~±ϕ)(y)≤-C4(∥ϕ∥l+∥h∥*)×[∑i=1n+k+m‍(εviρi)21|zi|2α+4+ε2]±h(y)≤-|h(y)|±h(y)≤0,
for y∈∂Ωε,
(56)(ϕ~±ϕ)(y)=ϕ~(y)>0,
and for y∈⋃i=1n+k+m∂Bi,Rd,
(57)(ϕ~±ϕ)(y)>∥ϕ∥l±ϕ(y)≥0.
From the maximum principle, it follows that-ϕ(y)≤ϕ~(y)≤ϕ(y) on Ωε∖⋃i=1n+k+mBi,Rd¯, which provides the estimate (53).
Step  3. We prove the priori estimate (43) by contradiction. Let us assume the existence of a sequence εj→0, and points qj=(qn+1j,…,qn+k+mj)∈Λk,m(δ), functions hj with ∥hj∥*→0, solutions ϕj with ∥ϕj∥∞=1, such that (41)-(42) hold. From the estimate (53), ∥ϕj∥l≥κ for some κ>0, namely, supBi,Rd|ϕj|≥κ for some i. To simplify the notation, let us denote ε=εj and qi=qij. Set ϕ^j(z)=ϕj((viρiz+qi)/ε) and h^j(z)=hj((viρiz+qi)/ε). By (32), ϕ^j satisfies
(58)Δϕ^j+8(1+αi)2|z|2αi[1+|z|2(1+αi)]2[1+O(ρi|z|)Δϕ^j+O(ρ)]ϕ^j=(viρiε)2h^j,
for z∈BRd(0). Obviously, for β∈[1,-(1/α)], (viρi/ε)2h^j→0 in Lβ(BRd(0)). Since 8(1+αi)2|z|2αi/[1+|z|2(1+αi)]2 is bounded in Lβ(BRd(0)) and ∥ϕ^j∥∞=1, the elliptic regularity theory implies that ϕ^j converges uniformly over compact sets near the origin to a bounded nontrivial solution of Li(ϕ^)=0 in ℝ2. Then, ϕ^ is proportional to zi0 for i∈J1 (see [35, 36, 41]), or a linear combination of zi0, zi1, and zi2 for i∈J2∪J3 (see [34, 45, 46]). However, our assumed conditions (42) on i and ϕj pass to the limit and yield ∫‍χzi0ϕ^dz=0 for i∈J1, or ∫‍χcos2((1/2)αiθ)zilϕ^dz=0 for i∈J2∪J3, l=0,1,2, which implies that ϕ^≡0. This is absurd because ϕ^ is nontrivial.We will give next a priori estimate for the solutions of (41) satisfying orthogonality conditions with respect to Zij, i∈J2∪J3, j=1,2.Lemma 6.
There exist positive numbersε0 and C such that for any points q∈Λk,m(δ), h∈𝒞* and any solution ϕ to (41) with the following orthogonality conditions:
(59)∫Ωε‍χicos2(12αiθi)Zijϕdy=0,∀i∈J2∪J3,j=1,2,
one has
(60)∥ϕ∥∞≤C(log1ε)∥h∥*,
for all ε<ε0.Proof.
Consider the radial solutionψi,i∈⋃l=13Jl, for the following equation:
(61)Δψi(r)=0,inR<r<δ(viρi)-1,ψi(r)=1,onr=R,ψi(r)=0,onr=δ(viρi)-1,
where R>R0+1 is a large number. Then, ψi(r) is explicitly given by
(62)ψi(r)=log[δ(viρi)-1]-logrlog[δ(viρi)-1]-logR.
Letη1 and η2 be radial smooth cut-off functions on ℝ2 so that
(63)0≤η1≤1;|∇η1|≤Cinℝ2;η1≡1inBR(0);η1≡0inℝ2∖BR+1(0);0≤η2≤1;|∇η2|≤Cinℝ2;η2≡1inBδ/4(0);η2≡0inℝ2∖Bδ(0).
Set(64)ψ~i(y)=ψi(zi),η1i(y)=η1(zi),η2i(y)=η2(viρizi).
Besides, let us define(65)ϕ~=ϕ+∑i=1n+k+m‍diZ~i0,
where
(66)Z~i0(y)=η1iZi0+(1-η1i)η2iψ~iZi0,
and di is chosen such that for i∈J1,
(67)di∫Ωε‍χi|Zi0|2+∫Ωε‍χiZi0ϕ=0,
and for i∈J2∪J3,
(68)di∫Ωε‍χicos2(12αiθi)Zi02+∫Ωε‍χicos2(12αiθi)Zi0ϕ=0.
Thus,(69)L(ϕ~)=h+∑i=1n+k+m‍diLZ~i0,inΩε,ϕ~=0,on∂Ωε,
and ϕ~ satisfies the orthogonality conditions (42). By (43),
(70)∥ϕ~∥∞≤C{∥h∥*+∑i=1n+k+m‍|di|·∥LZ~i0∥*}.
Multiplying (69) by Z~i0,i∈⋃l=13Jl, and integrating by parts, it follows that
(71)〈L(Z~i0),ϕ~〉=〈Z~i0,h〉+di〈L(Z~i0),Z~i0〉,
where 〈f,g〉=∫Ωε‍fg. Then, for i∈⋃l=13Jl, by (70) and (71),
(72)|di〈L(Z~i0),Z~i0〉|≤C∥h∥*{1+∥L(Z~i0)∥*}≤+C∥L(Z~i0)∥*∑l=1n+k+m‍|dl|·∥L(Z~l0)∥*.
Let us claim that fori∈⋃l=13Jl, there exists some constant C>0 independent of ε, such that
(73)∥L(Z~i0)∥*=O(1|logε|),(74)〈L(Z~i0),Z~i0〉<-C|logε|{1+O(1|logε|)}.
Once these estimates (73)-(74) are proven, it easily follows that
(75)|di|≤C(log1ε)∥h∥*.
This, together with (65), (70), and (73), easily gives the estimate (60) of ϕ.Proofs of (73) and (74).
Let us first define that(76)Wi0=8(1+αi)2|zi|2αi[1+|zi|2(1+αi)]2,z~i0(zi)=η1zi0+(1-η1)η2(viρizi)ψizi0.
Forri:=|zi|≤R, by (32),
(77)L(Z~i0)=(εviρi)2Wi0zi0{O(ρi|zi|)+O(ρ)},I1=∫ri≤R‍Z~i0·L(Z~i0)=∫|z|≤R‍Wi0zi02[O(ρi|z|)+O(ρ)]=O(ρ).
ForR<ri≤R+1, Z~i0(y)=η1zi0+(1-η1)ψizi0, 1-ψi=O(1/|logε|), and |∇ψi|=O(1/|logε|). Then,
(78)L(Z~i0)=(εviρi)2{2(1-ψi)∇η1∇zi0=(εviρi)2jjj+2(1-η1)∇ψi∇zi0-2zi0∇η1∇ψi=(εviρi)2jjj+(1-ψi)zi0Δη1=(εviρi)2jjj+Wi0z~i0[O(ρi|zi|)+O(ρ)]}(zi)=(εviρi)2O(1|logε|).
Furthermore,(79)I2=∫R≤ri≤R+1‍Z~i0·L(Z~i0)=∫R≤|z|≤R+1‍z~i0{(1-ψi)zi0Δη1=∫R≤|z|≤R+1‍z~i0vv+2(1-ψi)∇η1∇zi0-2zi0∇η1∇ψi=∫R≤|z|≤R+1‍z~i0vv+2(1-η1)∇ψi∇zi0}dz+O(ρ).
Integrating by parts the first term ofI2, it follows that
(80)I2=∫R≤|z|≤R+1‍2(1-η1)z~i0∇ψi∇zi0-zi02∇ψi∇η1-zi02(1-ψi)2|∇η1|2+O(ρ).
Integrating by parts the first term again, it also follows that(81)I2=∫|z|=R+1‍ψi′(|z|)ψizi02-∫R≤|z|≤R+1‍zi02|(1-η1)∇ψi-∫R≤|z|≤R+1‍dzi02+(1-ψi)∇η1|2+O(ρ).
Then,(82)I2=∫|z|=R+1‍ψi′(|z|)ψizi02+O(1|logε|2).
ForR+1<ri≤δ(4viρi)-1, Z~ij(y)=ψizij(zi). Then,
(83)L(Z~i0)=(εviρi)2{2∇ψi∇zi0+ψiΔzi0=(εviρi)2vv+Wi0ψizi0[1+O(ρi|zi|)=(εviρi)2vv+Wi0ψizi0v+O(ρ)]}(zi).
Note that|∇zi0|≤C/|zi|3+2αi, |∇ψi|≤C/(|zi|log(1/ε)), and |Wi0|≤C/|zi|4+2αi. Then,
(84)L(Z~i0)=(εviρi)21|zi|4+2αi×{O(1|logε|)+O(ρi|zi|)+O(ρ)}.
Besides,(85)I3=∫R+1≤ri≤δ(4viρi)-1Z~i0·L(Z~i0)=∫R+1≤|z|≤δ(4viρi)-1ψizi0×{[O(ρi|z|)+O(ρ)]2∇ψi∇zi0+Wi0ψizi0×[O(ρi|z|)+O(ρ)]}=(∫|z|=δ(4viρi)-1-∫|z|=R+1)ψi′(|z|)×ψizi02-∫R+1≤|z|≤δ(4viρi)-1zi02|∇ψi|2+O(ρ).
Some simple computations show that(86)∫|z|=δ(4viρi)-1ψi′(|z|)ψizi02=O(1|logε|2),(87)∫R+1≤|z|≤δ(4viρi)-1zi02|∇ψi|2≥π2log(δ/4viρi)-log(R+1)(log(δ/viρi)-logR)2.
Then, there exists some constantC>0 independent of ε and R such that
(88)I3<-∫|z|=R+1‍ψi′(|z|)ψizi02<-C|logε|{1+O(1|logε|)}.
Forδ(4viρi)-1<ri≤δ(viρi)-1, Z~i0(y)=η2(viρizi)ψizi0. Then,
(89)L(Z~i0)=(εviρi)2{ψizi0Δη2(viρizi)=(εviρi)2vv+2∇η2(viρizi)∇ψizi0=(εviρi)2vv+2ψi∇η2(viρizi)∇zi0=(εviρi)2vv+2η2(viρizi)∇ψi∇zi0=(εviρi)2vv+Wi0η2(viρizi)ψizi0=(εviρi)2vv×[O(ρi|zi|)+O(ρ)]}.
Note thatψi=O(1/|logε|), |∇ψi|=O(ρi/|logε|), and |∇zi0|=O(ρi3+2αi). Then,
(90)L(Z~i0)=(εviρi)2O(ρi2|logε|).
Furthermore,(91)I4=∫δ(4viρi)-1<ri≤δ(viρi)-1‍Z~i0·L(Z~i0)=O(1|logε|2).
As a result, the estimates (73)-(74) can be easily derived from (77)–(91).Proof of Proposition4.
We first establish the a priori estimate (39). Testing (38) against η2iZij, i∈J2∪J3, j=1,2, and integrating by parts, it follows that
(92)|cij|≤C(εviρi)2(|〈ϕ,L(η2iZij)〉|+|〈h,η2iZij〉|).
Observe that(93)L(η2iZij)=(εviρi)2{O(ρi(1+|zi|)2+αi)=(εviρi)2vb+O(ρi2(1+|zi|)1+αi)}+η2(viρizi)zij(Wq-Wi0).
Then,(94)〈ϕ,L(η2iZij)〉=O(ρ∥ϕ∥∞).
Note that〈h,η2iZij〉=O(∥h∥*). This, together with (92)–(94), implies that
(95)|cij|≤C(εviρi)2(∥h∥*+ρ∥ϕ∥∞).
By (60),
(96)∥ϕ∥∞≤C(log1ε){∑i=n+1n+k+m‍∥h∥*≤C(log1ε)b+∑j=12‍∑i=n+1n+k+m‍(viρiε)2|cij|}.
Combining this with (95), we find that the a priori estimate (39) holds. Furthermore,
(97)|cij|≤C(εviρi)2∥h∥*,
which implies that there exists a unique trivial solution to problem (38) with h≡0.
Next, we prove the solvability of problem (38). Consider the following Hilbert space:
(98)ℍ={ϕ∈H01(Ωε):nnnϕsatisfiestheorthogonalityconditions(59)H01},
endowed with the usual norm ∥ψ∥H01=∫Ωε‍|∇ψ|2. Problem (38) is equivalent to that of finding ϕ∈ℍ such that
(99)∫Ωε‍∇ϕ∇ψ-Wqϕψ+hψ=0,∀ψ∈ℍ.
By Fredholm’s alternative, this is equivalent to the uniqueness of solutions to this problem, which is guaranteed by (39). As a consequence, there exists a unique solution ϕ=T(h), scalars cij, i∈J2∪J3, j=1,2, for problem (38) with h∈𝒞*, where T:𝒞*↦L∞(Ωε) is a continuous linear map satisfying ∥T(h)∥∞≤C(log(1/ε))∥h∥*.
Finally, we give the a priori estimate (40). Let us Differentiate (38) with respect to the parameters ql,s′, l∈J2∪J3, s=1,2. Formally, Z=∂ql,s′ϕ should satisfy
(100)L(Z)=-ϕ∂ql,s′Wq=+∑j=12‍∑i=n+1n+k+m‍{(12αiθi)cij∂ql,s′+∑j=12‍hnn∑i=n+1n+k+m‍×[χicos2(12αiθi)Zij]+∑j=12hnn‍∑i=n+1n+k+m‍+dijχicos2(12αiθi)Zij},
where (still formally) dij=∂ql,s′cij. The orthogonality conditions now become
(101)∫Ωε‍χicos2(12αiθi)ZijZ=-∫Ωε‍ϕ∂ql,s′[χicos2(12αiθi)Zij],=-∫Ωε‍ϕ∂q′l,sv∀i∈J2∪J3,j=1,2.
We consider the constantsbij, i∈J2∪J3, j=1,2, defined as
(102)bij∫Ωε‍χicos2(12αiθi)|Zij|2=-∫Ωε‍ϕ∂ql,s′[χicos2(12αiθi)Zij].
By (31), it follows that |∂ql,s′logμi|=O(ε) uniformly holds on Λk,m(δ), which implies that for i∈J2∪J3, j=1,2,
(103)|bij|≤{Cεvlρl∥ϕ∥∞,ifi=l,Cε∥ϕ∥∞,ifi≠l.
Define(104)Z~=Z-∑j=12‍∑i=n+1n+k+m‍bijη2iZij,f=∑j=12‍∑i=n+1n+k+m‍{(η2iZij)cij∂ql,s′∑j=12‍∑i=n+1n+k+m‍vv×[χicos2(12αiθi)Zij]∑j=12‍∑i=n+1n+k+m‍vv-bijL(η2iZij)}-ϕ∂ql,s′Wq.
We then have(105)L(Z~)=f+∑j=12‍∑i=n+1n+k+m‍dijχicos2(12αiθi),inΩε,ϕ=0,on∂Ωε,∫Ωε‍χicos2(12αiθi)ZijZ~=0,∀i∈J2∪J3,j=1,2.
Thus,Z=∂ql,s′ϕ can be uniquely expressed as
(106)Z=T(f)+∑j=12‍∑i=n+1n+k+m‍bijη2iZij.
Furthermore, elliptic regularity theory implies thatϕ=T(h) is differentiable with respect to ql,s′, l∈J2∪J3, s=1,2. Note that ∥L(η2iZij)∥*=O(ρ), ∥∂ql,s′Wq∥*=O(ρlαl), ∥∂ql,s′[χlcos2((1/2)αlθl)Zlj]∥*=O(ρl-αl), and ∥∂ql,s′[χicos2((1/2)αiθi)Zij]∥*=O(ρi1-αi) for i≠l. This, together with (39) and (97)–(106), implies that
(107)∥Z∥∞≤C(log1ε)∥f∥*+C∑j=12‍∑i=n+1n+k+m‍|bij|≤C(log1ε)×∑j=12‍∑i=n+1n+k+m‍{(εviρi)2∥h∥*×∑j=12‍∑i=n+1n+k+m‍ff×∥∂ql,s′[χicos2(12αiθi)Zij]∥*×∑j=12‍∑i=n+1n+k+mff‍+ρ|bij|(εviρi)2}+C(log1ε)∥ϕ∥∞∥∂ql,s′Wq∥*+Cεvlρl∥ϕ∥∞≤Cεvlρl(log1ε)2∥h∥*,
which implies that the estimate (40) holds.Now, we solve the auxiliary nonlinear problem: forq∈Λk,m(δ), we find a function ϕ and scalars cij, i∈J2∪J3, j=1,2, such that(108)L(ϕ)=-[Rq+N(ϕ)]=+∑j=12‍∑i=n+1n+k+m‍cijχicos2(12αiθi)Zij,∑j=12‍∑i=n+1n+k+m‍=cijχicos2(12αiθi)inΩε,ϕ=0,on∂Ωε,∫Ωε‍χicos2(12αiθi)Zijϕdy=0,hhhhhhhl∀i∈J2∪J3,j=1,2.The following result can be proved through arguments similar to these of [39].Proposition 7.
There exist positive numbersε0 and C such that for any ε<ε0 and points q∈Λk,m(δ), there is a unique solution ϕ∈L∞(Ωε), scalars cij∈ℝ, i∈J2∪J3, j=1,2, of problem (108), which satisfies
(109)∥ϕ∥∞≤Cρlog1ε.
Moreover, the mapq′↦ϕ is C1, precisely for l∈J2∪J3, s=1,2,
(110)∥∂ql,s′ϕ∥∞≤Cρεvlρl(log1ε)2,
where ρ=max{ρi:i∈⋃l=13Jl}.


## 4. Variational Reduction

In what follows, we only need to find a solution of problem (27) (or (25)) with k+m≥1, and hence to problem (108) if points q∈Λk,m(δ) satisfy
(111)cij(q′)=0,∀i∈J2∪J3,j=1,2.To realize it, we consider the energy functionalJε associated with problem (25), namely,
(112)Jε(v)=12∫Ωε‍|∇v|2-2ε4∫Ωε‍c(εy)S(εy)coshv,hhhhhhhhhhhhhhhhhhhhl∀u∈H01(Ωε).We define(113)Fε(q)=Jε(Vq+ϕ),
where Vq is defined in (26) and ϕ is the unique solution of problem (108). Critical points of Fε correspond to solutions of (111) for small ε, as the following result states.Lemma 8.
For any pointsq∈Λk,m(δ), the functional Fε(q) is of class C1. Moreover, for ε>0 small enough, if DqFε(q)=0, then points q∈Λk,m(δ) satisfy (112).Proof.
Observe that forl∈J2∪J3, s=1,2,
(114)∂ql,sFε(q)=ε-1DJε(Vq+ϕ)×[∂ql,s′(Vq+ϕ)]=ε-1∑j=12‍∑i=n+1n+k+m‍cijKil(j,s),
where
(115)Kil(j,s)=∫Ωε‍χicos2(12αiθi)Zij(∂ql,s′Vq+∂ql,s′ϕ)dy.
Then, ifDqFε(q)=0, we have
(116)∑j=12‍∑i=n+1n+k+m‍cijKil(j,s)=0,∀l∈J2∪J3,s=1,2.
Set(117)Fl=∫0R0+1‍16(1+αl)r3(1+αl)[1+r2(1+αl)]2χ(r)dr,hhhhhhhhlhhhhhhhhl∀l∈J2∪J3.
A simple computation shows that(118)∂ql,s′Vq=εvlρl4(1+αl)al|zl|2αlzl,s1+|zl|2(1+αl)+O(ε),hhhhhhhhhhhhl∀l∈J2∪J3,s=1,2.
This, together with the estimate (110) of ∂ql,s′ϕ, implies that
(119)Kil(j,s)={(viρiε)2O(ρεvlρl|logε|2),∀i≠lorj≠s,vlρlε[Al+O(ρ|logε|2)],∀i=landj=s,
where
(120)Al={πalFl,∀l∈J3,14πalFl,∀l∈J2.
Set(121)c~ij=(viρiε)2cij,∀i∈J2∪J3,j=1,2.
Thus, (116) can be rewritten as: for any l∈J2∪J3, s=1,2,(122)c~lsalFl=∑j=12‍∑i=n+1n+k+m‍c~ijaiO(ρ|logε|2),hhhhhhl∀l∈J2∪J3,s=1,2.
This is a diagonal dominant system and we thus getcij=0 for all i,j.Next, we give a precise asymptotic expansion ofFε(q) defined in (113).Lemma 9.
The following precise asymptotic expansion holds(123)Fε(q)=-∑i∈J1‍di{∑j∈J1,i≠j‍12diH(qi,qi)=-∑i∈J1‍div+∑j∈J1,i≠j‍12aiajdjG(qj,qi)}+∑i∈⋃l=13Jl‍di{log8(1+αi)2+∑i∈∪l=13Jl‍divv-2(1+logε)}+φk,mn(q)+O(ρ),
uniformly for any points q∈Λk,m(δ), where φk,mn(q) is defined in (12).Proof.
Let us first give a priori estimate ofθε(q), where θε(q)=Fε(q)-Jε(Vq). Using DJε(Vq+ϕ)[ϕ]=0, a Taylor expansion and an integration by parts, it follows that
(124)θε(q)=-∫01‍D2Jε(Vq+sϕ)[ϕ]2sds=-∫01‍(∫Ωε‍[N(ϕ)+Rq]ϕ-=∫01‍vv+2ε4c(εy)S(εy)-∫01‍=vv×[coshVq-cosh-∫01‍vvv=ffff×(Vq+sϕ)]ϕ2∫Ωε‍)sds.
Note that∥N(ϕ)∥*=O(∥ϕ∥∞2), ∥Rq∥*=O(ρ), and ∥2ε4c(εy)S(εy)[coshVq-cosh(Vq+sϕ)]∥* = O(∥ϕ∥∞). These, together with (109), imply that θε(q)=O(ρ2|logε|), and so Fε(q)=Jε(Vq)+O(ρ2|logε|).
Next, we only need to give an asymptotic expansion ofJε(Vq). Observe that
(125)Jε(Vq)=-12∑i,j∈⋃l=13Jl‍aiaj∫Ω‍PuiΔujdx-∫Ωε‍Wqdy.
By (22),
(126)-∫Ω‍PuiΔuj=∫Ω‍ε2cj(qj)|x-qj|2αjeuj×{8μi2(1+αi)2ci(qi)ui+diH(x,qi)×vv-log8μi2(1+αi)2ci(qi)+O(ρ)}=Iij+Jij,
where
(127)Iij=∫Ωvjρj‍8(1+αj)2|zj|2αj[1+|zj|2(1+αj)]2×log1[μi2ε2+|vjρjzj+qj-qi|2(1+αi)]2dzj,Jij=∫Ωvjρj‍8(1+αj)2|zj|2αj[1+|zj|2(1+αj)]2×{diH(qj,qi)+O(ρj|zj|)+O(ρ)}dzj.
Note that(128)∫Ωvjρj‍8(1+αj)2|z|2αj[1+|z|2(1+αj)]2dz=dj+O(ε2),(129)∫Ωvjρj‍8(1+αj)2|z|2αj[1+|z|2(1+αj)]2O(ρj|z|)dz=O(ρj)+O(ε2),
and for i≠j,(130)log[μi2ε2+|vjρjzj+qj-qi|2(1+αi)]=2(1+αi)log|qj-qi|+O(ρj|zj|)+O(ε2).
Then, for anyi≠j, by (128)–(130),
(131)Iij=-4(1+αi)djlog|qj-qi|+O(ρ).
By (128),(132)Ijj=-2∫Ωvjρj‍8(1+αj)2|z|2αj[1+|z|2(1+αj)]2×log[1+|z|2(1+αj)]dz-4[dj+O(ε2)]log(μjε).
Making the complex changes of variablesω=z1+αj with ω=|ω|eiθ~, we get
(133)Ijj=-2∫02π(1+αj)‍dθ~∫0(μjε)-1‍8|ω|[1+|ω|2]2hhhhhhhhhhhhhhhhhl×log[1+|ω|2]d|ω|=-4djlog(μjε)+O(ρ)=-4djlog(μjε)-2dj+O(ρ).
On the other hand, by (128)-(129), we have
(134)Jij=didjH(qi,qj)+O(ρ).
Thus, by (131)–(134),
(135)-∫Ω‍PuiΔuj={didjG(qi,qj)+O(ρ),∀i≠j,dj{djH(qj,qj)-4log(μjε)-2}+O(ρ),∀i=j.
Besides, by (32),
(136)∫Ωε‍Wq=∑⋃l=13Jl‍di+O(ρ).
Using the choice forμi’s by (31), together with (125), (135), and (136), it follows that
(137)Jε(Vq)=∑i∈⋃l=13Jl‍di{∑j≠i,j∈∪l=13Jl‍log8(1+αi)2ci(qi)∑i∈∪l=13Jl‍diVvbV-2(1+logε)-12diH(qi,qi)∑i∈∪l=13Jl‍divVbV-12∑j≠i,j∈⋃l=13Jl‍aiajdjG(qj,qi)}+O(ρ).
This, together with (12), easily gives the asymptotic expansion (123) of Fε(q).


## 5. Proofs of Theorems

Proof of Theorem3.
According to Lemma8, we only need to find a critical point of the function, consider
(138)F~ε(q)=Fε(q)-∑i∈⋃l=13Jl‍di{log8(1+αi)2-2(1+logε)}+∑i∈J1‍di{∑j∈J1,i≠j‍12diH(qi,qi)+∑i∈J1‍dibhhA+∑j∈J1,i≠j‍12aiajdjG(qj,qi)}.
By (123), F~ε=φk,mn+O(ρ)→φk,mn uniformly holds on Λk,m(δ). By Definition 2, there exists a critical point qε of Fε such that φk,mn(qε)→φk,mn(q*). Moreover, up to a subsequence, there exists points q~=(q~n+1,…,q~M+m)∈Λk,m such that qε→q~∈Λk,m as ε→0, and φk,mn(q*)=φk,mn(q~). Thus, vε=Vqε+ϕ(qε) is a family of solutions of problem (25) (or (27)). Set uε(x)=Vqε((1/ε)x)+ϕ(qε)((1/ε)x) for any x∈Ω. As a result, uε is a family of solutions of problem (11) with the qualitative properties predicted by the theorem, as it can be easily shown.Proof of Theorem1.
Setq1=(0,0), c(x)=1, and al=(-1)l-1 for l=1,…,m+1. If α∈ℕ,
(139)φ1,m0(q)=-32π2×{∑l,j=2,l≠jm+1‍α2π∑l=2m+1‍log|ql|+(1+α)2H(0,0)JJJJJ,kJJ+∑l=2m+1‍H(ql,ql)+2(1+α)JJJJJ,kJJ×∑l=2m+1‍(-1)l+1G(0,ql)JJJJJ,kJJ+∑l,j=2,l≠jm+1‍(-1)l+jG(ql,qj)},
and if α∈(-1,+∞)∖(ℕ∪{0}),
(140)φ0,m1(q)=-32π2×{α2π∑l=2m+1‍log|ql|+2(1+α)×vm×∑l=2m+1‍(-1)l+1G(0,ql)+∑l=2m+1‍H(ql,ql)×vm+∑l,j=2,l≠jm+1‍(-1)l+jG(ql,qj)}.
We will seek a nodal solution for problem (5) with the concentration points 0 and q~l=(λcos(2π(l-1)/m),λsin(2π(l-1)/m)), l=2,…,m+1. Note that
(141)G(x,y)=-log|x-y|2π=+log(|x|2|y|2+1-2〈x,y〉)4π,H(x,x)=log(1-|x|2)2π.
By Theorem3, we can reduce the problem of finding solutions of (5) to that of finding a C0-stable critical point of the function f(λ):(0,1)→ℝ defined in (8). Obviously, limλ→0+f(λ)=limλ→1-f(λ)=+∞, which implies that f(λ) has an absolute minimum point λ0 in (0,1). Then, λ0 is a C0-stable critical point of f(λ).



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*Source: 873948-2013-11-27.xml*