New Classes of Spatial Central Configurations for N + N + 2-Body Problem

Abstract

Under arbitrary masses, in this paper, we discuss the existence of new families of spatial central configurations for the N + N + 2-body problem, 𝑁≥2. We study some necessary conditions and sufficient conditions for a families of spatial double pyramidical central configurations (d.p.c.c.), where 2N bodies are at the vertices of a nested regular N-gons Γ₁∪Γ₂, and the other two bodies are symmetrically located on the straight line that is perpendicular to the plane that contains Γ₁∪Γ₂ and passes through the geometric center of Γ₁∪Γ₂. We prove that if the bodies are in a d.p.c.c., then the masses on each N-gon are equal, and the other two are also equal. And also we prove the existence and uniqueness of the central configurations for any given ratios of masses.

1. Main Results

The Newtonian 𝑛-body problem (see [1–7]) concerns the motion of 𝑛 point particles with masses 𝑚_𝑗∈𝑅⁺ and positions 𝑞_𝑗∈𝑅³(𝑗=1,…,𝑛). The motion is governed by Newton’s law:𝑚_𝑗𝑞̈‌_𝑗=𝜕𝑈𝑞𝜕𝑞_𝑗, (1.1) where 𝑞=(𝑞₁,…,𝑞_𝑛) and 𝑈(𝑞) is the Newtonian potential:𝑈𝑞=_{1≤𝑘<𝑗≤𝑛}𝑚_𝑘𝑚_𝑗𝑞_𝑘−𝑞_𝑗. (1.2) Consider the space𝑋=𝑞=𝑞₁,…,𝑞_𝑛∈𝑅^3𝑛∶^𝑛_𝑘=1𝑚_𝑘𝑞_𝑘=0, (1.3) that is, suppose that the center of mass is fixed at the origin of the space. Because the potential is singular when two particles have the same position, it is natural to assume that the configuration avoids the set Δ={𝑞∶𝑞_𝑘=𝑞_𝑗 for some 𝑘≠𝑗}. The set 𝑋⧵Δ is called the configuration space.

Definition 1.1 (see [4]). A configuration 𝑞=(𝑞₁,…,𝑞_𝑛)∈𝑋⧵Δ is called a central configuration (c.c.) if there exists a constant 𝜆 such that 𝑞̈‌_𝑘=−𝜆𝑞_𝑘,𝑘=1,2,…,𝑛, or ^𝑛_{𝑗=1,𝑗≠𝑘}𝑚_𝑗𝑚_𝑘𝑞_𝑗−𝑞_𝑘³𝑞_𝑗−𝑞_𝑘=−𝜆𝑚_𝑘𝑞_𝑘,1≤𝑘≤𝑛, (1.4) and the value of the constant 𝜆 in (1.4) is uniquely determined by 𝑈𝜆=𝐼, (1.5) where 𝐼=^𝑛_𝑘=1𝑚_𝑘𝑞_𝑘². (1.6)

For any coordinate system, we have that, if the center of masses 𝑚₁,𝑚₂,…,𝑚_𝑛 with position vectors 𝑟₁,𝑟₂,…,𝑟_𝑛 is not at the origin, central configuration equations (1.4) are equivalent to the following:^𝑛_{𝑗=1,𝑗≠𝑘}𝑚_𝑗𝑚_𝑘𝑟_𝑗−𝑟_𝑘³𝑟_𝑗−𝑟_𝑘=−𝜆𝑚_𝑘𝑟_𝑘−𝑟₀,1≤𝑘≤𝑛, (1.7) where 𝑟₀=∑^𝑛_𝑘=1𝑚_𝑘𝑟_𝑘/∑^𝑛_𝑘=1𝑚_𝑘 is the center of masses 𝑚₁,𝑚₂,…,𝑚_𝑛.

The knowledge of central configurations allows us to compute homographic solutions (see [8]); there is a relation between central configurations and the bifurcations of the hypersurfaces of constant energy and angular momentum (see [9]); if the 𝑁 bodies are moving towards a simultaneous collision, then the bodies tend to a central configuration (see [10]). See also [11, 12].

Some examples of spatial central configurations are a regular tetrahedron with arbitrary positive masses at the vertices [13] and a regular octahedron with six equal masses at the vertices [12]. Double nested spatial central configurations for 2𝑁 bodies were studied for two nested regular polyhedra in [14]. More recently, the same authors studied central configurations of three regular polyhedra for the spatial 3𝑁-body problem in [15]. See also [16], where nested regular tetrahedrons are studied.

Recently, Hampton and Santoprete [17] provided new examples of stacked spatial central configurations—central configurations for the 𝑁-body problem where a proper subset of the 𝑁 bodies are already on a central configuration—for the 7-body problem where the bodies are arranged as concentric three- and two-dimensional simplex. New classes of stacked spatial central configurations for the 6-body problem which have four bodies at the vertices of a regular tetrahedron and the other two bodies on a straight line connecting one vertex of the tetrahedron with the center of the opposite face were studied in [18].

In this paper, we study new classes of spatial double pyramidical central configurations (d.p.c.c) for the 𝑁+𝑁+2-body that satisfy the following.(1)The position vectors 𝑟₁,𝑟₂,…,𝑟_𝑁 of masses 𝑚₁,𝑚₂,…,𝑚_𝑁 are at the vertices of a regular 𝑁-gon Γ₁, whose sides have length 𝜄₁. The position vectors 𝑟_𝑁+1,𝑟_𝑁+2,…,𝑟_2𝑁 of masses 𝑚_𝑁+1,𝑚_𝑁+2,…,𝑚_2𝑁 are at the vertices of another regular 𝑁-gon Γ₂, whose sides have length 𝜄₂. Two 𝑁-gons have a same geometric center and form a affine nested 𝑁-gons (base plane Π).(2)Let 𝐿 be the straight line perpendicular to the base plane Π that contains Γ₁ and Γ₂ and passes through the geometric center of Γ₁∪Γ₂. The position vectors 𝑟_2𝑁+1 and 𝑟_2𝑁+2 of masses 𝑚_2𝑁+1 and 𝑚_2𝑁+2 are on 𝐿 and on opposite sides with respect to the plane Π.

The central configurations studied in this paper are in some measure related to the double pyramidal central configurations (d.p.c.c) studied in [19] and the paper in [20]. The configuration in [19] consists of 𝑛 masses on a plane that are located at the vertices of a regular 𝑛-gon and two equal masses located on the line perpendicular to passing through the geometric center of the 𝑁-gon. And the authors in [20] assumed that the center of the 𝑁-gon is at the origin, that is, Σ^𝑛_𝑗=1𝑚_𝑗𝑟_𝑗=0, and more that the origin of the inertial system is the center of mass of the system, that is, Σ^𝑛+3_𝑗=1𝑚_𝑗𝑟_𝑗=0 and 𝑟_𝑛+3=0. In fact the origin is the geometric center of the 𝑁-gon. Hence the configuration in [20] is only to append a mass at the geometric center of the 𝑁-gon in [19].

As far as we know, the spatial central configurations studied here are very new. The number of bodies (masses) is increased to 2𝑁+2, it is not to suppose the origin of the inertial system, and the proofs are more difficult than those in [19, 20].

The main results of this paper are the following.

Theorem 1.2. Consider 𝑁+𝑁+2 bodies with masses 𝑚₁,𝑚₂,…,𝑚_2𝑁,𝑚_2𝑁+1,𝑚_2𝑁+2 located according to the following.(i)𝑟₁,𝑟₂,…,𝑟_𝑁 are at the vertices of a regular 𝑛-gon Γ₁ inscribed on a circle of radius 𝛼.(ii)𝑟_𝑁+1,𝑟_𝑁+2,…,𝑟_2𝑁 are at the vertices of another regular 𝑛-gon Γ₂ inscribed on a circle of radius 𝑎.(iii)𝑟_2𝑁+1 and 𝑟_2𝑁+2 are on the straight line 𝐿, on opposite sides with respect to the plane Π, where 𝐿 is the straight line that is perpendicular to Π and passes through the geometric center of Γ₁∪Γ₂. Let ℎ₁=distance(𝑟_2𝑁+1,Π) and ℎ₂=distance(𝑟_2𝑁+2,Π).
In order that the 𝑁+𝑁+2 bodies can be in a central configuration (c.c.), the following statements hold.
(1)If ℎ₁=ℎ₂=∶ℎ, then there is 𝑚_2𝑁+1=𝑚_2𝑁+2.(2)If ℎ₁=ℎ₂=∶ℎ, then not only 𝑚_2𝑁+1=𝑚_2𝑁+2, but also𝑚₁=𝑚₂=⋯=𝑚_𝑁𝑚=∶𝑚,_𝑁+1=𝑚_𝑁+2=⋯=𝑚_2𝑁=∶𝑚. (1.8)(3)The origin is the mass center of 𝑚₁,𝑚₂,…,𝑚_2𝑁 and also the mass center of 𝑚₁,𝑚₂,…,𝑚_2𝑁+2, that is, ^2𝑁_𝑗=1𝑚_𝑗𝑟_𝑗=0,_{2𝑁+2𝑗=1}𝑚_𝑗𝑟_𝑗=0. (1.9)(4)Get rid of masses 𝑚_2𝑁+1 and 𝑚_2𝑁+2, when ratio of masses 𝑏=𝑚_𝑁+1/𝑚₁ in some interval and the masses 𝑚₁,𝑚₂,…,𝑚_2𝑁 may form a central configuration.

Remark 1.3. Let 𝑚_2𝑁+1=𝑚_2𝑁+2, and the origin is the mass center of 𝑚₁,𝑚₂,…,𝑚_2𝑁, also it is the mass center of 𝑚₁,𝑚₂,…,𝑚_2𝑁+2, and then we have that ℎ₁=ℎ₂ and (1.8) hold. The conclusions are the oppose problem of some items in Theorem 1.2, which is similar to that in [20]. We may similarly prove ℎ₁=ℎ₂. The proof of (1.8) still see Theorem 1.2 in this paper.

Theorem 1.4. Under the suppositions of the positions for masses, and 𝑚₁=𝑚₂=⋯=𝑚_𝑁=∶𝑚, 𝑚_𝑁+1=𝑚_𝑁+2=⋯=𝑚_2𝑁=∶𝑚, 𝑚_2𝑁+1=𝑚_2𝑁+2 and ℎ₁=ℎ₂=∶ℎ, then 𝑚₁,…,𝑚_𝑁,𝑚_𝑁+1,…,𝑚_2𝑁,𝑚_2𝑁+1,𝑚_2𝑁+2 are in a c.c., if and only if the parameters 𝑏,𝑎,𝑐 and ℎ satisfy the following relationships: 𝜆𝑀=1(𝑁+𝑁𝑏+2𝑐)_𝑗≠𝑁1−𝜌_𝑗1−𝜌_𝑗³+_𝑗𝑏1−𝑎𝜌_𝑗1−𝑎𝜌_𝑗³+2𝑐1+ℎ^23/2,𝜆𝑀=1ℎ(𝑁+𝑁𝑏+2𝑐)𝑁ℎ1+ℎ^23/2+𝑁𝑏ℎ𝑎²+ℎ^23/2+𝑐4ℎ²,∑𝑏=^𝑗𝑎−𝜌_𝑗/𝑎−𝜌_𝑗³−𝑎∑_𝑗≠𝑁1−𝜌_𝑗/1−𝜌_𝑗³+2𝑐/𝑎²+ℎ^23/2−2𝑐𝑎/1+ℎ^23/2𝑎∑_𝑗1−𝑎𝜌_𝑗/1−𝑎𝜌_𝑗³−𝑎⁻²∑_𝑗≠𝑁1−𝜌_𝑗/1−𝜌_𝑗³, (1.10) where 𝑏=𝑚/𝑚,𝑐=𝑚_2𝑁+1/𝑚 and 𝜌_𝑗=exp(𝑖(2𝜋𝑗/𝑁)).

Theorem 1.5. Under the suppositions of the positions for masses, and 𝑚₁=𝑚₂=⋯=𝑚_𝑁=∶𝑚, 𝑚_𝑁+1=𝑚_𝑁+2=⋯=𝑚_2𝑁=𝑚, 𝑚_2𝑁+1=𝑚_2𝑁+2 and ℎ₁=ℎ₂=∶ℎ, then for any ratios of masses 𝑏=𝑚/𝑚 and 𝑐=𝑚_2𝑁+1/𝑚,𝑚₁,…,𝑚_𝑁,𝑚_𝑁+1,…,𝑚_2𝑁,𝑚_2𝑁+1,𝑚_2𝑁+2 may form a unique c.c. such that 𝑎∈(0,1) and ℎ∈(0,+∞).

2. Some Lemmas

Definition 2.1 (see [21]). If 𝑁×𝑁 matrix 𝐴=(𝑎_𝑖,𝑗) satisfies 𝑎_𝑖,𝑗=𝑎_{𝑖−1,𝑗−1},1≤𝑖,𝑗≤𝑁, (2.1) where we assume 𝑎_𝑖,0=𝑎_𝑖,𝑁 and 𝑎_0,𝑗=𝑎_𝑁,𝑗, then one calls that 𝐴 is a circular matrix.

Lemma 2.2 (see [21]). (i) If 𝐴 and 𝐵 are 𝑁×𝑁 circular matrices, for any numbers 𝛾 and 𝛽, then 𝐴+𝐵,𝐴−𝐵,𝐴𝐵 and 𝛾𝐴+𝛽𝐵 are also circular matrices, and 𝐴𝐵=𝐵𝐴.
(ii) Let 𝐴=(𝑎_𝑖,𝑗) be an 𝑁×𝑁 circular matrix; the eigenvalues 𝜆_𝑘 and the eigenvectors 𝑣_𝑘⃗ of 𝐴 are 𝜆_𝑘(𝐴)=_𝑗𝑎_1,𝑗𝜌^𝑗−1_𝑘−1,𝑣_𝑘⃗=1,𝜌_𝑘−1,𝜌²_𝑘−1,…,𝜌^𝑁−1_𝑘−1^𝑇. (2.2) (iii) Let 𝐴 and 𝐵 be circular matrices, and let 𝜆_𝑘(𝐴) and 𝜆_𝑘(𝐵) be eigenvalues of 𝐴 and 𝐵. Then the eigenvalues of 𝐴+𝐵,𝐴−𝐵 and 𝐴⋅𝐵 are 𝜆_𝑘(𝐴)+𝜆_𝑘(𝐵),𝜆_𝑘(𝐴)−𝜆_𝑘(𝐵) and 𝜆_𝑘(𝐴)⋅𝜆_𝑘(𝐵).

It is clear that.

Lemma 2.3. If 𝐴=(𝑎_𝑖,𝑗) is an 𝑁×𝑁 circular matrices, and 𝐴𝑋=0, where 𝑋=(𝑥₁,…,𝑥_𝑛)^𝑇, 𝑥_𝑖>0(𝑖=1,…,𝑁), then 𝑎_1,𝑗+⋯+𝑎_𝑁,𝑗𝑎=0,1≤𝑗≤𝑁,_𝑖,1+⋯+𝑎_𝑖,𝑁=0,1≤𝑖≤𝑁. (2.3)

Lemma 2.4. Let 𝐴 and 𝐵 be 𝑁×𝑁 Hermite circular matrices; then 𝐴+𝐵, 𝐴−𝐵, 𝐴𝐵 and 𝛾𝐴+𝛽𝐵(𝛾,𝛽∈𝑅) are also Hermite circular matrices.

Lemma 2.5. Let 𝐴 be a Hermite circular matrix; then the eigenvalues of 𝐴 are real number and(i) when 𝑛=2𝑚+1(𝑚≥1),𝐴 can be denoted by 𝐴=𝐴_2𝑚+1=cir(𝑎,𝑏₁,𝑏₂,…,𝑏_𝑚,𝑏_𝑚,…,𝑏₂,𝑏₁), where 𝑎∈𝑅 and 𝑏_𝑙 is a conjugate complex number of 𝑏_𝑙. It has eigenvalues 𝜆₀=𝑎+2^𝑚_𝑙=1Re𝑏_𝑙,𝜆_𝑘=𝑎+2^𝑚_𝑙=1Re𝑏_𝑙cos2𝑘𝜋𝑙2𝑚+1−Im𝑏_𝑙sin2𝑘𝜋𝑙2𝑚+1,1≤𝑘≤2𝑚, (2.4)(ii) when 𝑛=2𝑚(𝑚≥1), 𝐴 can be denoted by 𝐴=𝐴_2𝑚=cir(𝑎,𝑏₁,𝑏₂,…,𝑏_𝑚−1,𝑏_𝑚,𝑏_𝑚−1,…,𝑏₂,𝑏₁,). It has eigenvalues 𝜆₀=𝑎+2^𝑚−1_𝑙=1Re𝑏_𝑙+𝑏_𝑚,𝜆_𝑚=𝑎+2^𝑚−1_𝑙=1(−1)^𝑙Re𝑏_𝑙+(−1)^𝑚𝑏_𝑚,𝜆_𝑘=𝑎+2^𝑚−1_𝑙=1Re𝑏_𝑙cos2𝑘𝜋𝑙2𝑚−Im𝑏_𝑙sin2𝑘𝜋𝑙+()2𝑚−1^𝑘𝑏_𝑚,1≤𝑘≤2𝑚−1,𝑘≠𝑚. (2.5)

Lemma 2.6. The complex subspace 𝐿 of 𝐶^𝑁 generated by 𝑋₁=(1,1,…,1) and 𝑋₂=(1,𝜌,…,𝜌^𝑁−1), where 𝑁=2𝑘>2(𝜌=exp2𝜋𝐼/𝑁), and the complex subspace 𝐿 generated by 𝑋₁,𝑋₂, and 𝑋₃=(1,𝜌^𝑘+1,…,𝜌^{(𝑁−1)(𝑘+1)}), where 𝑁=2𝑘+1>3, all contain no real vectors other than the multiples of (1,1,…,1).

Remark 2.7. Lemmas 2.3–2.5 can be simply proved by properties of circular matrix and Hermite matrix, and after some algebraic computations, Lemma 2.6 can be also simply proved.

Lemma 2.8 (see [5]). Let 𝐴=(1/4)∑_𝑗≠𝑁csc(𝜋𝑗/𝑁); then 𝐴(𝑁) has the following asymptotic expansion for 𝑁 large: 𝐴(𝑁)∼𝑁2𝜋𝛾+log2𝑁𝜋+_𝑘≥0(−1)^𝑘2^2𝑘−1−1𝐵²_2𝑘𝜋^2𝑘−1()!12𝑘)(2𝑘𝑁^2𝑘−1, (2.6) where 𝛾 stands for the Euler-Mascheroni constant and 𝐵_2𝑘 stands for the Bernoulli numbers.

Lemma 2.9 (see [5]). Let Φ_𝜈(𝑥)=∑_𝑗1/𝑑^𝜈_𝑗, where 𝜈>0 and𝑑_𝑗=1+𝑥²−2𝑥cos(2𝜋𝑗/𝑁); then, for 0<𝑥<1, Φ_𝜈(𝑥) and all of its any order derivatives are positive. Moreover, the same is thus forΨ_𝜈(𝑥)=∑_𝑗cos(2𝜋𝑗/𝑁)/𝑑^𝜈_𝑗.

3. The Proof of Theorem

Proposition 3.1. The central configuration equations (1.4) or (1.5) is equivalent to the following:^𝑛_{𝑗=1,𝑗≠𝑘}𝑚_𝑗𝑟_𝑗−𝑟_𝑘𝑅_𝑗,𝑘−𝜆𝑀=0,1≤𝑘≤𝑛, (3.1) where 𝑀=∑^𝑛_𝑘=1𝑚_𝑘,𝑅_𝑗,𝑘=|𝑟_𝑗−𝑟_𝑘|⁻³.

Proof. From central configuration equation (1.4) or (1.5), we easily prove.

Denote 𝑟_𝑘=(𝑥_𝑘,𝑦_𝑘,𝑧_𝑘)∈𝑅³, and 𝑧⃗=(0,0,1). Observing (1.5), one could have a free choice of the origin for a configuration. Without loss of generality, consider that the origin is at the geometric center of Γ₁∪Γ₂, and let 𝑧_𝑘=0,1≤𝑘≤2𝑁.

Proposition 3.2. Under the hypotheses of Theorem 1.2, if ℎ₁=ℎ₂=∶ℎ, then the following equations are verified: 𝑚_2𝑁+1=𝑚_2𝑁+2,𝑅_2𝑁+1,𝑗=𝑅_2𝑁+1,𝑘=𝑅_2𝑁+2,𝑗=𝑅_2𝑁+2,𝑘𝑅,1≤𝑗≠𝑘≤𝑁,_{2𝑁+1,𝑁+𝑗}=𝑅_{2𝑁+1,𝑁+𝑘}=𝑅_{2𝑁+2,𝑁+𝑗}=𝑅_{2𝑁+2,𝑁+𝑘},1≤𝑗≠𝑘≤𝑁, (3.2)(3.3)(3.4) where 𝑅_𝑗,𝑘=|𝑟_𝑗−𝑟_𝑘|⁻³.

Proof. In (3.1), considering the equations along the direction 𝑧⃗, and 𝑘=1,𝑁+1, we have 𝑚_2𝑁+1𝑧_2𝑁+1−𝑧₁𝑅_1,2𝑁+1−𝜆𝑀+𝑚_2𝑁+2𝑧_2𝑁+2−𝑧₁𝑅_1,2𝑁+2−𝜆𝑀𝑚=0,_2𝑁+1𝑧_2𝑁+1−𝑧_𝑁+1𝑅_{𝑁+1,2𝑁+1}−𝜆𝑀+𝑚_2𝑁+2𝑧_2𝑁+2−𝑧_𝑁+1𝑅_{𝑁+1,2𝑁+2}−𝜆𝑀=0, (3.5) where 𝑅_1,2𝑁+1=𝑟₁²+𝑧_2𝑁+1^2−3/2,𝑅_1,2𝑁+2=𝑟₁²+𝑧_2𝑁+2^2−3/2,𝑅_{𝑁+1,2𝑁+1}=𝑟_𝑁+1²+𝑧_2𝑁+1^2−3/2,𝑅_{𝑁+1,2𝑁+2}=𝑟_𝑁+1²+𝑧_2𝑁+2^2−3/2,𝑧₁=𝑧_𝑁+1=0. (3.6)
By the ℎ₁=ℎ₂, we have 𝑧_2𝑁+1=−𝑧_2𝑁+2,𝑅_1,2𝑁+1=𝑅_1,2𝑁+2,𝑅_{𝑁+1,2𝑁+1}=𝑅_{𝑁+1,2𝑁+2}. (3.7) Hence 𝑚_2𝑁+1−𝑚_2𝑁+2𝑅_1,2𝑁+1−𝜆𝑀𝑚=0,_2𝑁+1−𝑚_2𝑁+2𝑅_{𝑁+1,2𝑁+1}−𝜆𝑀=0. (3.8) Obviously 𝑅_1,2𝑁+1≠𝑅_{𝑁+1,2𝑁+1}, so 𝑚_2𝑁+1=𝑚_2𝑁+2. (3.9) By our assumptions for the origin, (3.3), (3.4) clearly holds.

Equation (3.9) of Proposition 3.2 proves item (1) of Theorem 1.2.

Proposition 3.3. Under the hypotheses of Theorem 1.2, if 𝑚_2𝑁+1=𝑚_2𝑁+2, the origin is the mass center of 𝑚₁,𝑚₂,…,𝑚_2𝑁, and it also is the mass center of 𝑚₁,𝑚₂,…,𝑚_2𝑁+2, then ℎ₁=ℎ₂=∶ℎ (3.10) and (3.3), (3.4) hold.

Proof. The proof is very similar to that in [20].

Proposition 3.4. Under the hypotheses of Theorem 1.2, if ℎ₁=ℎ₂=∶ℎ, then the masses at the vertices of circle Γ₁ are equal, and also the masses at the vertices of circle Γ₂ are equal, that is, 𝑚₁=𝑚₂=⋯=𝑚_𝑁,𝑚_𝑁+1=𝑚_𝑁+2=⋯=𝑚_2𝑁. (3.11)

Proof. Because if 𝑟→𝜀𝑟 is a transformation in a central configuration, then 𝜆→(1/𝜀²)𝜆 can be a new parameter of a central configuration. We say that the old and the new are equivalent. Hence without loss of generality, we may let 𝛼=1,0<𝑎<1. Then the vectors of positions based on the previous assumptions can be interpreted by the following: 𝑟_𝑘=𝜌_𝑘(),𝑟,01≤𝑘≤𝑁_𝑁+𝑘=𝜌_𝑘),𝑟,0(1≤𝑘≤𝑁_2𝑁+1=0⋅𝜌₀),𝑟,ℎ(ℎ>0_2𝑁+2=0⋅𝜌₀,−ℎ(ℎ>0), (3.12) where 𝜌_𝑘=exp((2𝜋𝑘/𝑁)𝑖),𝜌_𝑘=𝑎𝜌_𝑘,𝑎>0, and 𝑖=√−1,𝜌_𝑘 denote the 𝑁 complex 𝑘th roots of unity, that is, that 𝑚_𝑘(1≤𝑘≤𝑁) each locates at the vertices 𝑟_𝑘 of the one regular 𝑁-gon Γ₁,𝑚_𝑁+𝑘(1≤𝑘≤𝑁) each locates at the vertices 𝑟_𝑁+𝑘 of the other regular 𝑁-gon Γ₂, and 𝑚_2𝑁+1 and 𝑚_2𝑁+2 lie on the vertices of 𝑟_2𝑁+1,𝑟_2𝑁+2. Then the center of masses is 𝑟₀=∑_𝑗𝑚_𝑗𝑟_𝑗+𝑚_𝑁+𝑗𝑟_𝑁+𝑗+𝑚_2𝑁+1𝑟_2𝑁+1+𝑚_2𝑁+2𝑟_2𝑁+2𝑀, (3.13) where 𝑀=_𝑗𝑚_𝑗+𝑚_𝑁+𝑗+𝑚_2𝑁+1+𝑚_2𝑁+2. (3.14)
In (3.13)-(3.14) and throughout this paper, unless other restricted, all indices and summations will range from 1 to 𝑁.
Let ℎ₁=ℎ₂=∶ℎ; then 𝑚_2𝑁+1=𝑚_2𝑁+2. Now we discuss all equations for the 2𝑁 masses on the base plane Π. According to (3.1), then _𝑗≠𝑘𝑚_𝑗1𝜌_𝑘−𝜌_𝑗³−𝜆𝑀𝜌_𝑘−𝜌_𝑗+_𝑗𝑚_𝑁+𝑗1𝜌_𝑘−𝜌_𝑗³−𝜆𝑀𝜌_𝑘−𝜌_𝑗+2𝑚_2𝑁+111+ℎ^23/2−𝜆𝑀𝜌_𝑘=0,_𝑗𝑚_𝑗1𝜌_𝑘−𝜌_𝑗³−𝜆𝑀𝜌_𝑘−𝜌_𝑗+_𝑗≠𝑘𝑚_𝑁+𝑗1𝜌_𝑘−𝜌_𝑗³−𝜆𝑀𝜌_𝑘−𝜌_𝑗+2𝑚_2𝑁+11𝑎²+ℎ^23/2−𝜆𝑀𝑎𝜌_𝑘=0. (3.15)
Multiplying both sides by 𝜌_𝑁−𝑘, noting that |𝜌_𝑘−𝜌_𝑗|=|𝜌_𝑘||1−𝜌_𝑗−𝑘|=|1−𝜌_𝑗−𝑘| and 𝜌_𝑘=𝑎𝜌_𝑘, we see that (3.15) may be written as _𝑗≠𝑘𝑚_𝑗11−𝜌_𝑗−𝑘³−𝜆𝑀1−𝜌_𝑗−𝑘+_𝑗𝑚_𝑁+𝑗11−𝑎𝜌_𝑗−𝑘³−𝜆𝑀1−𝑎𝜌_𝑗−𝑘+2𝑚_2𝑁+111+ℎ^23/2−𝜆𝑀=0,_𝑗𝑚_𝑗1𝑎−𝜌_𝑗−𝑘³−𝜆𝑀𝑎−𝜌_𝑗−𝑘+_𝑗≠𝑘𝑚_𝑁+𝑗1𝑎−𝑎𝜌_𝑗−𝑘³−𝜆𝑀𝑎−𝑎𝜌_𝑗−𝑘+2𝑚_2𝑁+11𝑎²+ℎ^23/2−𝜆𝑀𝑎=0, (3.16) where 𝑘=1,2,…,𝑁.
Now we define the 𝑁×𝑁 circular matrices 𝐶=[𝑐_𝑘,𝑗], 𝐴=[𝑎_𝑘,𝑗], 𝐵=[𝑏_𝑘,𝑗], and 𝐷=[𝑑_𝑘,𝑗] as follows: 𝑐_𝑘,𝑗𝑐=0,for𝑘=𝑗,_𝑘,𝑗=11−𝜌_𝑗−𝑘³−𝜆𝑀1−𝜌_𝑗−𝑘𝑎,for𝑘≠𝑗,_𝑘,𝑗=11−𝑎𝜌_𝑗−𝑘³−𝜆𝑀1−𝑎𝜌_𝑗−𝑘,𝑏_𝑘,𝑗=1𝑎−𝜌_𝑗−𝑘³−𝜆𝑀𝑎−𝜌_𝑗−𝑘,𝑑_𝑘,𝑗𝑑=0,for𝑘=𝑗,_𝑘,𝑗=1𝑎−𝑎𝜌_𝑗−𝑘³−𝜆𝑀𝑎−𝑎𝜌_𝑗−𝑘for𝑘≠𝑗. (3.17) Also define 1⃗=(1,…,1)^𝑇,1𝐸=21+ℎ^23/2−𝜆𝑀⋅⃗11,𝐹=2𝑎𝑎²+ℎ^23/2−𝜆𝑀⋅1⃗. (3.18) We see that (3.16) holds if and only if the matrix equation 𝑚𝐶𝐴𝐸𝐵𝐷𝐹¹⋮𝑚_𝑁𝑚_𝑁+1⋮𝑚_2𝑁𝑚_2𝑁+1=0 (3.19) has a positive solution. Let 𝑚⃗=𝑚₁,…,𝑚_𝑁^𝑇,𝑚⃗=𝑚_𝑁+1,…,𝑚_2𝑁^𝑇. (3.20) Then (3.19) is equivalent to 𝐶𝑚⃗+𝐴𝑚⃗+𝑚_2𝑁+1⃗⃗𝐸=0,𝐵𝑚⃗+𝐷𝑚+𝑚_2𝑁+1𝐹=0. (3.21)(3.22) Noticing that 𝐴,𝐵,𝐶, and 𝐷 are 𝑁×𝑁 circular matrix, using the properties of circular matrix, we know that they must have positive real eigenvector 1⃗. Each of (3.21) and (3.22) left multiplies 1^𝑇⃗=(1,1,…,1); there are _𝑘𝑚_{𝑘𝑗≠𝑁}11−𝜌_𝑗³−𝜆𝑀1−𝜌_𝑗+_𝑘𝑚_{𝑁+𝑘𝑗}11−𝑎𝜌_𝑗³−𝜆𝑀1−𝑎𝜌_𝑗+2𝑚_2𝑁+11⋅𝑁1+ℎ^23/2−𝜆𝑀=0,_𝑘𝑚_𝑘𝑗1𝑎−𝜌_𝑗³−𝜆𝑀𝑎−𝜌_𝑗+_𝑘𝑚_{𝑁+𝑘𝑗≠𝑁}1𝑎−𝑎𝜌_𝑗³−𝜆𝑀𝑎−𝑎𝜌_𝑗+2𝑚_2𝑁+11⋅𝑁𝑎²+ℎ^23/2−𝜆𝑀𝑎=0. (3.23) By (3.21) and (3.22) we have (𝐶𝐷−𝐴𝐵)𝑚⃗+2𝑚_2𝑁+111+ℎ^23/2−𝜆𝑀1𝐷−𝑎𝑎²+ℎ^23/2−𝜆𝑀𝐴⃗⃗()⃗1=0,𝐴𝐵−𝐶𝐷𝑚+2𝑚_2𝑁+111+ℎ^23/2−𝜆𝑀1𝐵−𝑎𝑎²+ℎ^23/2−𝜆𝑀𝐶1⃗=0⃗. (3.24) From Lemma 2.2 we see that ((1/(1+ℎ²)^3/2)−(𝜆/𝑀))𝐷−𝑎((1/(𝑎²+ℎ²)^3/2)−(𝜆/𝑀))𝐴, ((1/(1+ℎ²)^3/2)−(𝜆/𝑀))𝐵−𝑎((1/(𝑎²+ℎ²)^3/2)−(𝜆/𝑀))𝐶, and 𝐶𝐷−𝐴𝐵 are circular matrix, and we know that they must have positive real eigenvector 1⃗. By the properties of circular matrix, (3.24) can be written as (𝐶𝐷−𝐴𝐵)⋅𝑚⃗+𝛾₁⋅⃗⃗()⋅⃗1=0,𝐴𝐵−𝐶𝐷𝑚+𝛾₂⋅1⃗=0⃗, (3.25) where 𝛾₁⋅1⃗=2𝑚_2𝑁+111+ℎ^23/2−𝜆𝑀1𝐷−𝑎𝑎²+ℎ^23/2−𝜆𝑀𝐴⃗𝛾1,₂⋅1⃗=2𝑚_2𝑁+111+ℎ^23/2−𝜆𝑀1𝐵−𝑎𝑎²+ℎ^23/2−𝜆𝑀𝐶⃗𝛾1,₁=2𝑚_2𝑁+111+ℎ^23/2−𝜆𝑀_𝑗≠𝑁1𝑎−𝑎𝜌_𝑗³−𝜆𝑀𝑎−𝑎𝜌_𝑗1−𝑎𝑎²+ℎ^23/2−𝜆𝑀_𝑗11−𝑎𝜌_𝑗³−𝜆𝑀1−𝑎𝜌_𝑗,𝛾₂=2𝑚_2𝑁+111+ℎ^23/2−𝜆𝑀_𝑗1𝑎−𝜌_𝑗³−𝜆𝑀𝑎−𝜌_𝑗1−𝑎𝑎²+ℎ^23/2−𝜆𝑀_𝑗≠𝑁11−𝜌_𝑗³−𝜆𝑀1−𝜌_𝑗. (3.26) We easily prove 𝛾₁,𝛾₂∈𝑅, and from (3.23), we have _𝑘𝑚_𝑁+𝑘11+ℎ^23/2−𝜆𝑀_𝑗≠𝑁1𝑎−𝑎𝜌_𝑗³−𝜆𝑀𝑎−𝑎𝜌_𝑗1−𝑎𝑎²+ℎ^23/2−𝜆𝑀_𝑗11−𝑎𝜌_𝑗³−𝜆𝑀1−𝑎𝜌_𝑗+_𝑘𝑚_𝑘11+ℎ^23/2−𝜆𝑀_𝑗1𝑎−𝜌_𝑗³−𝜆𝑀𝑎−𝜌_𝑗1−𝑎𝑎²+ℎ^23/2−𝜆𝑀_𝑗≠𝑁11−𝜌_𝑗³−𝜆𝑀1−𝜌_𝑗=0, (3.27) that is, 𝛾_1𝑘𝑚_𝑁+𝑘+𝛾_2𝑘𝑚_𝑘=0,𝑚_𝑗>0,𝑚_𝑗>0. (3.28) Hence one has the following.
(1) If 𝛾₁=0, then 𝛾₂=0. By (3.25), there are ()⃗()⋅⃗⃗𝐶𝐷−𝐴𝐵⋅𝑚⃗=0,𝐶𝐷−𝐴𝐵𝑚=0. (3.29)(3.30) We notice that (3.29) or (3.30) must have positive real solutions, which is equivalent to that 𝐶𝐷−𝐴𝐵 has positive real eigenvectors corresponding to eigenvalue 0.
But we notice that (3.29) and (3.30) must hold, and for 𝑘=1, we have an eigenvalue 𝜆₁=0, and an matching eigenvector 𝑣₁⃗=(1,1,…,1)^𝑇 of 𝐶𝐷−𝐴𝐵. Noticing that 𝐴,𝐵,𝐶,and 𝐷 are Hermite circular matrices, from the properties of circular and Hermite matrix in Lemmas 2.2 and 2.4, then 𝐶𝐷−𝐴𝐵=𝐺 is also a Hermite circular matrix. We may denote 𝐺 by cir(𝑎₀,𝑔₁,𝑔₂,…,𝑔_𝑚−1,𝑔_𝑚,𝑔_𝑚−1,…,𝑔₂,𝑔₁) when 𝑁=2𝑚, where 𝑎₀,𝑔_𝑚∈𝑅. We also denote 𝐺 by cir(𝑎₀,𝑔₁,𝑔₂,…,𝑔_𝑚−1,𝑔_𝑚,𝑔_𝑚,…,𝑔₂,𝑔₁) when 𝑁=2𝑚+1, where 𝑎₀∈𝑅. Using Lemmas 2.4 and 2.5, after complex computation, we may prove that the kernel of 𝐺 is at most a subspace in 𝐿 when 𝑁=2𝑚>2, and a subspace in 𝐿 when 𝑁=2𝑚+1>3 (see [22]), where the meanings of 𝐿 and 𝐿 are in Lemma 2.6. Hence the kernel of 𝐺 does not contain any positive real vectors other than multiplication of 1⃗=(1,1,…,1).
When 𝑁=2,3, we may easily prove the conclusion.
(2) If 𝛾₁≠0, then 𝛾₂≠0; from (3.25) and (3.28),we get (𝐴𝐵−𝐶𝐷)_𝑘𝑚_𝑘𝑚⃗−_𝑘𝑚_𝑁+𝑘𝑚⃗=0⃗. (3.31) If _𝑘𝑚_𝑘𝑚⃗−_𝑘𝑚_𝑁+𝑘𝑚⃗≠0, (3.32) let 𝐺=𝐴𝐵−𝐶𝐷=(𝑔_𝑖𝑗), which is not zero circular matrix; by Lemmas 2.2 and 2.3 we see that ∑_𝑗𝑔_𝑖𝑗=0 and 𝐺 has an eigenvalue 𝜆₁=0. Using the properties of circular matrix, we have (𝐴𝐵−𝐶𝐷)1⃗=0⃗ or 1⃗^𝑇(𝐴𝐵−𝐶𝐷)=0⃗^𝑇. Let 1⃗^𝑇 left multiplies (3.25); we get 𝛾₁=𝛾₂=0, which is a contradiction to the supposition. So _𝑘𝑚_𝑘𝑚⃗−_𝑘𝑚_𝑘𝑚⃗=0, (3.33) and then 𝑚_𝑁+𝑗=𝑏𝑚_𝑗, where 𝑏=∑_𝑘𝑚_𝑘/∑_𝑘𝑚_𝑘, that is, 𝑚⃗=𝑏𝑚⃗. Substituting it into (3.21)-(3.22), we have (𝐶+𝑏𝐴)𝑚⃗+2𝑚_2𝑁+111+ℎ^23/2−𝜆𝑀⃗()1=0,𝐵+𝑏𝐷𝑚⃗+2𝑎⋅𝑚_2𝑁+11𝑎²+ℎ^23/2−𝜆𝑀1⃗=0. (3.34) Then 1𝑎⋅𝑎²+ℎ^23/2−𝜆𝑀()−1𝐶+𝑏𝐴1+ℎ^23/2−𝜆𝑀(𝐵+𝑏𝐷)𝑚⃗=0. (3.35) Noticing that 𝑎⋅((1/(𝑎²+ℎ²)^3/2)−(𝜆/𝑀))(𝐶+𝑏𝐴)−((1/(1+ℎ²)^3/2)−(𝜆/𝑀))(𝐵+𝑏𝐷) is a Hermite circular matrix, similarly we also have 𝑚⃗=(𝑚₁,𝑚₁,…,𝑚₁)^𝑇 and 𝑚⃗=(𝑚_𝑁+1,…,𝑚_𝑁+1), where 𝑚₁,𝑚_𝑁+1>0.

Remark 3.5. Proposition 3.4 proves item (2) of Theorem 1.2.

Proposition 3.6. Under the hypothesis of Theorem 1.2, if ℎ₁=ℎ₂, then the origin is the mass center of 𝑚₁,𝑚₂,…,𝑚_2𝑁 and also is the mass center of 𝑚₁,𝑚₂,…,𝑚_2𝑁+2.

Proof. We have already 𝑚_2𝑁+1=𝑚_2𝑁+2,𝑚₁=𝑚₂=⋯=𝑚_𝑁𝑚_𝑁+1,=𝑚_𝑁+2=⋯=𝑚_2𝑁. (3.36)(3.37)(3.38) So there are ^2𝑁_𝑗=1𝑚_𝑗𝑟_𝑗=0,_{2𝑁+2𝑗=1}𝑚_𝑗𝑟_𝑗=0, (3.39) by the positions of 𝑟_𝑗,𝑗=1,2,…,2𝑁+2.

Remark 3.7. Proposition 3.6 proves item (3) of Theorem 1.2. By (3.37) and (3.38), item (4) of Theorem 1.2 may be proved.

Proposition 3.8. Under the hypothesis of Theorem 1.4, the conclusion of Theorem 1.4 holds.

Proof (The Proof of the Necessary). By the hypothesis of Theorem 1.4, let 𝑚=𝑏𝑚,𝑚_2𝑁+1=𝑐𝑚; from (3.23), there are _𝑗≠𝑁11−𝜌_𝑗³−𝜆𝑀1−𝜌_𝑗+𝑏_𝑗11−𝑎𝜌_𝑗³−𝜆𝑀1−𝑎𝜌_𝑗1+2𝑐1+ℎ^23/2−𝜆𝑀=0,_𝑗1𝑎−𝜌_𝑗³−𝜆𝑀𝑎−𝜌_𝑗+𝑏_𝑗≠𝑁1𝑎−𝑎𝜌_𝑗³−𝜆𝑀𝑎−𝑎𝜌_𝑗1+2𝑐𝑎²+ℎ^23/2−𝜆𝑀𝑎=0. (3.40) We know that _𝑗1−𝜌_𝑗=𝑁,_𝑗𝑏1−𝑎𝜌_𝑗=𝑏𝑁,_𝑗𝑎−𝜌_𝑗=𝑎𝑁,_𝑗𝑏𝑎−𝑎𝜌_𝑗=𝑎𝑏𝑁. (3.41) From (3.40) we have 𝜆𝑀=1𝑁+𝑁𝑏+2𝑐_𝑗≠𝑁1−𝜌_𝑗1−𝜌_𝑗³+𝑏_𝑗1−𝑎𝜌_𝑗1−𝑎𝜌_𝑗³+2𝑐1+ℎ^23/2,𝜆𝑀=1𝑎(𝑁+𝑁𝑏+2𝑐)_𝑗𝑎−𝜌_𝑗𝑎−𝜌_𝑗³+𝑏_𝑗≠𝑁𝑎−𝑎𝜌_𝑗𝑎−𝑎𝜌_𝑗³+2𝑐𝑎²+ℎ^23/2, (3.42) and similarly we also have 𝜆𝑀=1ℎ(𝑁+𝑁𝑏+2𝑐)𝑁ℎ1+ℎ^23/2+𝑁𝑏ℎ𝑎²+ℎ^23/2+𝑐ℎ4ℎ³ (3.43) by (3.1). Then 𝑎_𝑗≠𝑁1−𝜌_𝑗1−𝜌_𝑗³+𝑏_𝑗1−𝑎𝜌_𝑗1−𝑎𝜌_𝑗³+2𝑐1+ℎ^23/2=_𝑗𝑎−𝜌_𝑗𝑎−𝜌_𝑗³+𝑏_𝑗≠𝑁𝑎−𝑎𝜌_𝑗𝑎−𝑎𝜌_𝑗³+2𝑐𝑎²+ℎ^23/2, (3.44)∑𝑏=^𝑗𝑎−𝜌_𝑗/𝑎−𝜌_𝑗³−𝑎∑_𝑗≠𝑁1−𝜌_𝑗/1−𝜌_𝑗³+2𝑐/𝑎²+ℎ^23/2−2𝑐𝑎/1+ℎ^23/2𝑎∑_𝑗1−𝑎𝜌_𝑗/1−𝑎𝜌_𝑗³−𝑎⁻²∑_𝑗≠𝑁1−𝜌_𝑗/1−𝜌_𝑗³, (3.45) where (3.42), (3.43), (3.45) imply (1.10).

Proof of Sufficient. For 𝑁≥2, under the suppositions of Theorem 1.4, then 𝑚₁,𝑚₂,…,𝑚_2𝑁+2 in a c.c. if and only if (3.42)–(3.43) hold, which are equivalent to (1.10). Hence the proof of the sufficient was finished.

Proposition 3.9. Under the hypothesis of Theorem 1.5, for any ratios of masses 𝑏=𝑚/𝑚 and 𝑐=𝑚_2𝑁+1/𝑚,𝑚₁,𝑚₂,…,𝑚_2𝑁+2 may form a unique c.c. such that 𝑎∈(0,1) and ℎ∈(0,+∞).

Here is Theorem 1.5.

Proof. Under the suppositions of the positions for masses, and ℎ₁=ℎ₂=∶ℎ,𝑚₁=𝑚₂=⋯=𝑚_𝑁=∶𝑚,𝑚_𝑁+1=𝑚₂=⋯=𝑚_2𝑁=∶𝑚, and 𝑚_2𝑁+1=𝑚_2𝑁+2, then for any ratios of masses 𝑏=𝑚/𝑚 and 𝑐=𝑚_2𝑁+1/𝑚,𝑚₁,𝑚₂,…,𝑚_2𝑁+2 are in a unique c.c. such that 𝑎∈(0,1),ℎ>0, if and only if that (1.10)–(1.14) or (3.42)–(3.43) have a unique positive solution on 0<𝑎<1 and ℎ>0 for any given numbers 𝑏,𝑐(>0).
Let 𝑎=𝑥,ℎ=𝑦, and 𝐾𝑥,𝑦=𝑥_𝑗≠𝑁1−𝜌_𝑗1−𝜌_𝑗³+𝑏_𝑗1−𝑥𝜌_𝑗1−𝑥𝜌_𝑗³+2𝑐1+𝑦^23/2−_𝑗𝑥−𝜌_𝑗𝑥−𝜌_𝑗³−𝑏_𝑗≠𝑁𝑥−𝑥𝜌_𝑗𝑥−𝑥𝜌_𝑗³−2𝑐𝑥²+𝑦^23/2,𝐿𝑥,𝑦=_𝑗≠𝑁1−𝜌_𝑗1−𝜌_𝑗³+𝑏_𝑗1−𝑥𝜌_𝑗1−𝑥𝜌_𝑗³+2𝑐1+𝑦^23/2−𝑁1+𝑦^23/2−𝑁𝑏𝑥²+𝑦^23/2−𝑐4𝑦³. (3.46) It suffices to prove that 𝐾(𝑥,𝑦)=0,and 𝐿(𝑥,𝑦)=0 have a unique positive solution (𝑥,𝑦) for any given ratios 𝑏,𝑐(>0) in 𝑥∈(0,1), and 𝑦∈(0,+∞).
Now let 𝑑²_𝑗=1+𝑥²−2𝑥cos2𝜋𝑗𝑁,𝑃(𝑥)=_𝑗1𝑑³_𝑗,𝑄(𝑥)=_𝑗cos2𝜋𝑗/𝑁𝑑³_𝑗,𝐴=_𝑗≠𝑁1−𝜌_𝑗1−𝜌_𝑗³=142^(𝑁/2)−1_𝑗=1csc𝜋𝑗𝑁1+1when𝑁iseven,2^(𝑁−1)/2_𝑗=1csc𝜋𝑗𝑁Φ(𝑥)=when𝑁isodd,_𝑗1𝑑_𝑗. (3.47) It follows from the definitions that Φ(𝑥)=1+𝑥²𝑃(𝑥)−2𝑥𝑄(𝑥), (3.48) and it implies 𝑃(𝑥)−𝑥𝑄(𝑥)=𝑄(𝑥)+𝑥(𝑄(𝑥)−𝑥𝑃(𝑥)). (3.49) Since 𝑑Φ(𝑥)(𝑥)𝑑𝑥=𝑄−𝑥𝑃, (3.50) then now 𝐾 and 𝐿 in (3.46) can be written as follows: 𝐾=𝑏𝑥,𝑦𝑥−𝑥²𝐴+𝑏𝑥(𝑃(𝑥)−𝑥𝑄(𝑥))+(𝑄(𝑥)−𝑥𝑃(𝑥))+2𝑐𝑥1+𝑦^23/2−2𝑐𝑥²+𝑦^23/2=𝑏𝑥−𝑥²𝐴+𝑏𝑥Φ(𝑥)+1+𝑏𝑥²𝑑Φ+𝑑𝑥2𝑐𝑥1+𝑦^23/2−2𝑐𝑥²+𝑦^23/2,𝐿𝑥,𝑦=_𝑗≠𝑁1−𝜌_𝑗1−𝜌_𝑗³+𝑏_𝑗1−𝑥𝜌_𝑗1−𝑥𝜌_𝑗³+2𝑐1+𝑦^23/2−𝑁1+𝑦^23/2−𝑁𝑏𝑥²+𝑦^23/2−𝑐4𝑦³=𝐴+𝑏𝑄(𝑥)+𝑥𝑑Φ+𝑑𝑥2𝑐1+𝑦^23/2−𝑁1+𝑦^23/2−𝑁𝑏𝑥²+𝑦^23/2−𝑐4𝑦³, (3.51) where 𝑥=𝑎∈(0,1). From Lemmas 2.8 and 2.9 and their proofs, and with implicit function theory, after some complex calculation (some ideas partially see [23]), we can prove that 𝐾=0, and 𝐿=0 have only one solution for any given ratios of masses 𝑏(>0) and 𝑐(>0) such that 0<𝑥<1, and 0<ℎ<+∞.

Acknowledgments

The authors are grateful to the reviewer(s) for their thoughtful comments. This work is partially supported by the Natural Science Foundation (NSF) of China, supported by NSF of Chongqing, NSF of Chongqing Education Committee, and SXXY-11ZD-19.