Non-topological solutions in a generalized Chern-Simons model on torus

We consider a quasi-linear elliptic equation with Dirac source terms arising in a generalized self-dual Chern-Simons-Higgs gauge theory. In this paper, we study doubly periodic vortices with arbitrary vortex configuration. First of all, we show that under doubly periodic condition, there are only two types of solutions, topological and non-topological solutions as the coupling parameter goes to zero. Moreover, we succeed to construct non-topological solution with $k$ bubbles where $k\in\mathbb{N}$ is any given number. We believe that it is the first result for the existence of non-topological doubly periodic solution of the quasi-linear elliptic equation arising in a generalized self-dual Chern-Simons-Higgs gauge theory. To find a suitable approximate solution, it is important to understand the structure of quasi-linear elliptic equation.


Introduction
In this paper, we study a generalized self-dual Chern-Simons-Higgs gauge theory introduced by Burzlaff, Chakrabarti, Tchrakian in [1]. The Lagrangian density of the model in (2 + 1) dimensions is where A = (A 0 , A 1 , A 2 ) is a 3-vector gauge field, F αβ = ∂ ∂xα A β − ∂ ∂x β A α is the corresponding curvature, φ = φ 1 + iφ 2 is a complex scalar field called the Higgs field, D j = ∂ ∂x j − iA j , j = 0, 1, 2 is the gauge covariant derivative associated with A, α, β, µ, ν = 0, 1, 2, ε > 0 is a constant referred to as the Chern-Simons coupling parameter, ǫ αβγ is the Levi-Civita totally skew-symmetric tensor with ǫ 012 = 1, V is the Higgs potential function. The corresponding Bogomol'nyi equations for unknowns φ, A defined on R 2 are In view of Jaffe-Taubes' argument in [14], we introduce unknown v defined by where {p j } N j=1 are the zeros of φ(z), allowing their multiplicities. Then we obtain the following reduced equation: Here p j is called a vortex point. The equation (1.1) can be considered in R 2 or a two dimensional flat torus Ω due to the theory suggested by 't Hooft in [20].
We fix ε > 0 for a while. In R 2 , a solution v(x) is called a topological solution if lim |x|→+∞ v(x) = 0, and is called a non-topological solution if lim |x|→+∞ v(x) = −∞.
Yang in [24] found topological multi-vortex solutions of (1.1) by using the variational structure of the elliptic problem to produce an iteration scheme that yields the desired solution. After then, Chae and Imanuvilov in [4] constructed a non-topological multivortex solution v(x) of (1.1) satisfying v(x) = −(2N + 4 + σ) ln |x| + O(1) as |x| → +∞ for some σ > 0. To obtain the non-topological solution of (1.1), the authors in [4] observed that (1.1) is a perturbation of the Liouville equation and applied the arguments developed in [3]. In [3], Chae and Imanuvilov showed the existence of non-topological multi-vortex solutions of the relativistic Chern-Simons-Higgs model (see (1.6) below), using the implicit function theorem argument with Lyapunov-Schmidt reduction method. Now we consider the equation (1.1) on flat two torus Ω, where ε goes to 0. Since Moreover, from the maximum principle (see also [11,Lemma 3.1]), we note that any For the well known Chern-Simons-Higgs equation with ε → 0 (see (1.6) below), the corresponding properties (1.2) and (1.3) were important to classify the solutions according to their asymptotic behavior as ε → 0 (see [9,8] [11,Lemma 3.5], he also showed that the maximum solutions v ε,M of (1.1) are a monotone family in the sense that v ε 1 ,M > v ε 2 ,M whenever 0 < ε 1 < ε 2 < ε c . Therefore, in view of Theorem 1.1, the maximal solution obtained in [11] is a topological solution.
At this point, one might ask the existence of non-topological solution to (1.1) on Ω. In this paper, we obtain the affirmative answer for this question by constructing a bubbling non-topological solution solution v ε to (1.1) on Ω satisfying 4) in the sense of measure as ε → 0.
For the construction of bubbling solution solution v ε to (1.1) on Ω satisfying (1.4), we assume that N = 2k ∈ 2N. We note that since the equation (1.1) is quasi-linear, it is not easy to deal it directly. As in [11,23,24], we introduce a new dependent variable u defined by We have that F ′ (v) = 1 − e v and F ′′ (v) = −e v , which implies F is strictly increasing and invertible over (−∞, 0). Let G be the inverse function of F over (−∞, 0]. Then we see and only if u ε satisfies We remark that if lim ε→0 sup Ω u ε = −∞, then the equation (1.5) would be a perturbation of bubbling solutions W ε of the following Chern-Simons-Higgs equation: The relativistic Chern-Simons-Higgs model has been proposed in [12] and independently in [13] to describe vortices in high temperature superconductivity. The above equation was derived from the Euler-Lagrange equations of the CSH model via a vortex ansatz, see [12,13,22,25]. The equation (1.6) has been extensively studied not only in a flat torus Ω but also in the whole R 2 . We refer the readers to [2,3,5,6,7,8,15,16,17,18,19,21,22] and references therein. Among them, in a recent paper [16], Lin and Yan succeeded to construct bubbling non-topological solutions to (1.6) on Ω. Compared to (1.6), our equation has a difficulty caused by the nonlinear terms including implicit function G.
Therefore, to choose a suitable approximate solution, we should investigate the behavior of the function G near −∞ and carry out the analysis carefully. To state our result exactly, we introduce the following notations: Let G be the Green function satisfying −∆ x G(x, y) = δ y − 1 |Ω| for x, y ∈ Ω, and Ω G(x, y)dx = 0.
We let γ(x, y) = G(x, y) + 1 2π ln |x − y| be the regular part of the Green function G(x, y), and Then u 0 satisfies the following problem: We remind that N = 2k. We denote Ω (k) : ∈ Ω (k) be the critical point of the following function: We define where Ω i is any open set satisfying At this point, we introduce our main result.
Suppose that D (q) < 0. Then for ε > 0 small, there exists a non-topological solution To the best of our knowledge, Theorem 1.2 is the first result for the existence of nontopological solution solutions to (1.1) on Ω. We remark that in our paper, a limiting It would be an interesting problem to find other types of non-topological solution solution to (3.1), for The organization of this paper is as follows. In Section 2, we prove Theorem 1.1.
In Section 3, to prove Theorem 1.2, we present some preliminaries results and discuss about the invertibility of a linearized operator. Moreover, we find a suitable approximate solution and complete the proof of Theorem 1.2.
We claim that there exist C q > 0 such that ∇w ε L q (Ω) ≤ C q for any q ∈ (1, 2). Let Then (2.2) By lemma 7.16 in [10], if Ω φdx = 0, then there exist c, C > 0 such that Thus in view of (2.1), (2.3), and (1.2), we see that there exists constant C > 0, independent Now using (2.2), we complete the proof of our claim.

Existence of bubbling non-topological solution solution
In this section, we want to construct a bubbling non-topological solution solution v ε to Without loss of generality, from now on, we assume that |Ω| = 1.
As we mentioned in the introduction, if lim ε→0 sup Ω u ε = −∞, then u ε would be related to the following Chern-Simons-Higgs equation: In [16], bubbling solutions for the above Chern-Simons-Higgs equation have been constructed as following: We denoteW We want to find solution u ε to (3.1) in the following form: where η x,µ is a perturbation term. To find η x,µ which makes that u ε in the form (3.2) is a solution to (3.1), we consider the following linearized operator We see that u ε is a solution to (3.1) if η x,µ satisfies where To show the invertibility of the linear operator L x,µ , we need to introduce suitable function spaces. For fixed a small constant α ∈ (0, 1 2 ), we define Let Let χ i (|y|) be a smooth function satisfying We use the following notations .., k, j = 1, 2, where x i = (x i1 , x i2 ) and y = (y 1 , y 2 ). The estimations for Y x,µ,0 , Y x i ,µ i ,j has been known: Proof. See the estimation (3.8) in [16].
We define two subspace of X α,x,µ , Y α,x,µ as where c 0 , c i,j are chosen so that Q x,µ ξ ∈ F x,µ . For the projection operator Q x,µ , we have the following result. The following lemma will be useful for our arguments.
The functiong x,µ (η) was introduced in [16], and the following estimations were obtained:
We remind that q is a non-degenerate critical point of G * (q) with D (q) < 0. Now we have the following proposition.
Step 2. Now we claim that B x,µ is a contraction map.
Step 3. In view of Step 1, Step 2, and contraction mapping theorem, there exists a unique solution η x,µ ∈ S x,µ of (3.6). Moreover, from Theorem 3.4, Lemma 3.2, and (3.11), we obtain that where C > 0 is independent of ε > 0. Now we complete the proof of Proposition 3.6.
By Proposition 3.6, we get that for any µ ∈ β 0 √ ε , β 1 √ ε , and any x close to q, where q is a non-degenerate critical point of G * (q) with D (q) < 0, there is η x,µ ∈ S x,µ such that where c 0 , c ij are constants satisfying In the following, we will choose x, µ suitably ( depending on ε ) such that the corresponding c 0 , c ij are zero and hence the solution η x,µ is exactly the solution to (3.3) which implies that u ε = 1 + u 0 +W x,µ + η x,µ is a solution to (3.1). It is standard to prove the following lemma.