Doubly periodic mixed type solution of Non-Abelian Chern–Simons

Abstract

We are concerned with the bosonic sector of an $N = 2$ supersymmetric Chern–Simons–Higgs theory in $2 + 1$ dimensions. Here the gauge group is $SU (N) \times U (1)$ and has $N_{f}$ flavors of fundamental matter fields. Recently, Chen, Han, Lozano, Schaposnik showed the existence of two gauge-distinct solutions carrying the same physical energy on a two dimensional flat torus. In this paper, we find a solution which has a different asymptotic behavior from Chen, Han, Lozano, Schaposnik’s result.

Keywords

Non-Abelian Chern–Simons–Higgs vortices nonlinear elliptic PDE system doubly periodic solutions

1. Introduction

During the last few decades, the theories for magnetic vortex configurations have been extensively investigated in condensed-matter and particle physics (see [1,3,16,26,41,44,45,50]). In $(2 + 1)$ space-time dimensions, the usual Maxwell term of the Abelian Higgs model can be replaced by a Chern–Simons (CS) term leading to the Abelian Chern–Simons–Higgs theory (see [2,15,19,20,22,23,27,29,39]). We refer the readers to [1,4,5,25,30,38,46–48,51–53] for the study related to the existence of self-dual vortex solutions of the Abelian Chern–Simons model.

In [17 ,18], it was shown that with the scalar fields breaking the symmetry to $Z_{N}$ and gauge fields restricted to the Cartan subalgebra, there are topologically stable vortex solutions in $SU (N)$ gauge theories with both Yang–Mills and Chern–Simons terms (see also [37,54]). By adding flavor to the Yang–Mills–Higgs theory, one can arrange the symmetry breaking so that some global diagonal combination of color and flavor groups survives. This pattern of symmetry breaking is known as color-flavor locking procedure (see [33,40,45]). Within the color-flavor locking symmetry breaking pattern and a suitable cylindrically symmetric ansatz, the existence of genuine non-Abelian vortex solutions were discussed in [9,10,37].

In [10 ,37], the authors are concerned with the truncated bosonic sector of the $N = 2 SUSY SU (N) \times U (1)$ Chern–Simons–Higgs action in $2 + 1$ dimensions (see [24,42] for complete $SUSY$ Lagrangian) $\begin{array}{l} S & = \int d^{3} x {\frac{κ_{1}}{2} ϵ^{μ ν ρ} F_{μ ν}^{0} A_{ρ}^{0} + \frac{κ_{2}}{2} ϵ^{μ ν ρ} (F_{μ ν}^{I} A_{ρ}^{I} - \frac{1}{3} f^{I J K} A_{μ}^{I} A_{ν}^{J} A_{ρ}^{K}) \\ (1.1) & + {(D_{μ} ϕ^{f})}^{†} (D_{μ} ϕ^{f}) - V [ϕ, ϕ^{†}]}, \end{array}$ where $ϵ^{012} = 1$ , $g^{00} = 1$ , $f^{I J K}$ are the structure constants of the non-Abelian group, $κ_{2} = \frac{m}{8 π}$ , the potential $V [ϕ, ϕ^{†}]$ is a sixth order polynomial of the form $\begin{array}{l} V [ϕ, ϕ^{†}] & = \frac{1}{16 κ_{1}^{2} N^{2}} ϕ_{f}^{†} ϕ^{f} {(ϕ_{g}^{†} ϕ^{g} - N ξ)}^{2} \\ + \frac{1}{4 κ_{2}^{2}} ϕ_{f}^{†} τ^{I} τ^{J} ϕ^{f} (ϕ_{g}^{†} τ^{I} ϕ^{g}) (ϕ_{h}^{†} τ^{J} ϕ^{h}) \\ (1.2) & - \frac{1}{4 κ_{1} κ_{2} N} {(ϕ_{f}^{†} τ^{I} ϕ^{f})}^{2} (ϕ_{g}^{†} ϕ^{g} - N ξ), \end{array}$ $μ, ν, ρ = 0, 1, 2$ are Lorentz indices, $I, J, K = 1, \dots, N^{2} - 1$ are $SU (N)$ “color” gruop indices, and $τ_{I}$ are the anti-Hermitian generators of $SU (N)$ . The complex scalar multiplets $ϕ_{i}^{f}$ , besides the color index $i, j, k = 1, \dots, N$ , have additional flavor index $f, g, h = 1, \dots, N_{f}$ . The field strengths and covariant derivatives are defined by $\begin{matrix} (1.3) & \{\begin{matrix} D_{μ} ϕ_{i}^{f} = \partial_{μ} ϕ_{i}^{f} + {(A_{μ}^{SU (N)})}_{i}^{j} ϕ_{j}^{f} + {(A_{μ}^{U (1)})}_{i}^{j} ϕ_{j}^{f}, \\ A_{μ}^{SU (N)} = A_{μ}^{I} τ_{I}, A_{μ}^{U (1)} = A_{μ}^{0} τ_{0}, \\ F_{μ ν}^{0} = \partial_{[μ} A_{ν]}^{0}, F_{μ ν}^{I} = \partial_{[μ} A_{ν]}^{I} + f^{I J K} A_{μ}^{J} A_{ν}^{K} . \end{matrix} \end{matrix}$ The Euler–Lagrange equations of this model is written in the form $\begin{matrix} (1.4) & \{\begin{matrix} κ_{1} ϵ_{μ}^{α β} F_{α β}^{0} = ϕ_{f}^{†} τ^{0} D_{μ} ϕ^{f} - {(D_{μ} ϕ^{f})}^{†} τ^{0} ϕ^{f}, \\ κ_{2} ϵ_{μ}^{α β} F_{α β}^{I} = ϕ_{f}^{†} τ^{I} D_{μ} ϕ^{f} - {(D_{μ} ϕ^{f})}^{†} τ^{I} ϕ^{f}, \\ D_{μ} D^{μ} ϕ^{f} = \frac{\partial V}{\partial ϕ_{f}^{†}} . \end{matrix} \end{matrix}$ After a BPS reduction [3,44], one can show that the energy minimizer satisfies $\begin{matrix} (1.5) & \{\begin{matrix} D_{0} ϕ^{f} \mp i (\frac{1}{4 κ_{1} N} (ϕ_{g}^{†} ϕ^{g}) ϕ^{f} - \frac{1}{2 κ_{2}} ϕ_{g}^{†} τ^{I} ϕ^{f}) = 0, \\ D_{\mp} ϕ^{f} = 0 . \end{matrix} \end{matrix}$ Although it is difficult to deal (1.5) directly, if we use a suitable ansatz as in [10,37] and the Gauss law, then (1.5) can be reduce to the following nonlinear elliptic system: $\begin{matrix} (1.6) & \{\begin{matrix} Δ u_{1} = \frac{1}{ε^{2}} {- \frac{[N - 1 + κ]}{N} e^{u_{1}} (1 - \frac{[N - 1 + κ]}{N} e^{u_{1}}) \\ + \frac{[κ - 1]}{N} e^{u_{2}} (1 - \frac{(1 + [N - 1] κ)}{N} e^{u_{2}}) - \frac{[κ - 1] (N - [N - 2] [κ - 1])}{N^{2}} e^{u_{1} + u_{2}}} \\ + 4 π \sum_{i = 1}^{d_{1}} m_{1, i} δ_{p_{1, i}}, \\ Δ u_{2} = \frac{1}{ε^{2}} {\frac{[N - 1] [κ - 1]}{N} e^{u_{1}} (1 - \frac{[N - 1 + κ]}{N} e^{u_{1}}) - \frac{[N - 1] [κ - 1] (2 + [N - 2] κ)}{N^{2}} e^{u_{1} + u_{2}} \\ - \frac{(1 + [N - 1] κ)}{N} e^{u_{2}} (1 - \frac{(1 + [N - 1] κ)}{N} e^{u_{2}})} \\ + 4 π \sum_{i = 1}^{d_{2}} m_{2, i} δ_{p_{2, i}} . \end{matrix} \end{matrix}$ where $N ⩾ 1$ and $p_{i, j_{k}} \neq p_{i, j_{l}}$ if $j_{k} \neq j_{l}$ for each $i = 1, 2$ . In [10], the authors showed the existence of solutions of (1.6) for two cases:

in $R^{2}$ with the topological boundary condition (i.e. ${lim}_{| x | \to + \infty} u_{i} (x) = 0$ for $i = 1, 2$ ),

in a two dimensional flat torus Ω.

Among the results in [10], we introduce the following theorem:

Theorem A ([10, Theorem 2.2]).

Let $κ > 1$ . For any configuration of vortex points $p_{i, j}$ , there exists $ε_{0} = ε_{0} (p_{i, j}) > 0$ such that for any $ε \in (0, ε_{0})$ , ( 1.6 ) has at least two distinct solutions on Ω. One of these two solutions, $(u_{1, ε}^{0}, u_{2, ε}^{0})$ , always satisfies $e^{u_{i, ε}^{0}} \to 1$ pointwise a.e. in Ω and strongly in $L^{p} (Ω)$ for any $p ⩾ 1$ and $i = 1, 2$ as $ε \to 0$ .

In order to prove Theorem A, the authors applied a constrained minimization approach and the mountain pass theorem. However, the asymptotic behavior of the other solution, which obtained from mountain pass theorem, still remains unknown.

When $N = 2$ and $κ = 3$ , (1.6) gives the same equations as those arising in the $SU (3)$ model (see [31]). The equation (1.6) for $SU (3)$ model is neither cooperative nor competitive system, that is, the nonlinear part is not a monotone function with respect each component. In view of this fact, $L^{1} (R^{2})$ boundedness of nonlinear terms is not easy even for radially symmetric solution [28], and it is still unknown for non-radial case.

Various kinds of solutions in $R^{2}$ of $SU (3)$ model which have different boundary condition at infinity were concerned in [11–14,54]. On a flat two torus Ω, in [43], Nolasco and Tarantello obtained doubly periodic solutions for $SU (3)$ model as minimizers of several functionals, and showed that if $\sum_{1 ⩽ j ⩽ 2, 1 ⩽ i ⩽ d_{j}} m_{j, i} = 1$ , then one of those minimizers $(u_{1, ε}, u_{2, ε})$ satisfies $u_{1, ε} \to \frac{1}{2}$ pointwise a.e. and $u_{2, ε} - 2 ln ε$ converges to a solution of a mean field type equation. These kinds of solutions are called mixed type (I) solutions, and the improved asymptotic behavior has been studied in [21]. Motivated by the works [21,43], we extend the notion of mixed type (I) solutions to the equation (1.6) in Ω. More precisely, we denote $\begin{matrix} (1.7) & C_{N, τ}^{(1)} = \frac{N}{[N - 1 + κ]} and C_{N, τ}^{(2)} = \frac{N}{(1 + [N - 1] κ)}, \end{matrix}$ and use the following definition:

Definition 1.1.
Let $(u_{1, ε}, u_{2, ε})$ be a sequence of solutions of (1.6) on Ω. Then $(u_{1, ε}, u_{2, ε})$ is called mixed-type (I) solution if one of followings hold: $\begin{matrix} (1.8) & \begin{matrix} (i) & u_{1, ε} \to ln \frac{N}{[N - 1 + κ]} pointwise a.e., sup_{Ω} (u_{2, ε} - 2 ln ε) ⩽ C_{}, or \\ (ii) & u_{2, ε} \to ln \frac{N}{(1 + [N - 1] κ)} pointwise a.e., sup_{Ω} (u_{1, ε} - 2 ln ε) ⩽ C_{}, \end{matrix} \end{matrix}$ some constant $C_{} \in R$ .

It turns out that (1.6) has a limiting problem, which consists of two decoupled equations. One of them is the following equation: $\begin{matrix} (1.9) & \{\begin{matrix} Δ U_{j} + e^{U_{j} (x)} (1 - e^{U_{j} (x)}) = 4 π m_{1, j} δ_{0} in R^{2}, \\ {lim}_{| x | \to + \infty} U_{j} (x) = 0, \end{matrix} \end{matrix}$ and the other one is the following equation: $\begin{matrix} (1.10) & Δ w_{2} + \frac{{(κ N)}^{2}}{{[N - 1 + κ]}^{2}} e^{w_{2}} = \frac{4 π [N - 1] [κ - 1]}{[N - 1 + κ]} \sum_{i = 1}^{d_{1}} m_{1, i} δ_{p_{1, i}} + 4 π \sum_{j = 1}^{d_{2}} m_{2, i} δ_{p_{2, i}} in Ω . \end{matrix}$ Here we say that $w_{2}$ is a non-degenerate solution of (1.10) if the following linearized equation $\begin{matrix} Δ ϕ + \frac{{(κ N)}^{2}}{{[N - 1 + κ]}^{2}} e^{w_{2}} ϕ = 0 in Ω \end{matrix}$ has only trivial solution $ϕ = 0$ . We refer the readers to [6–8,34,35] for recent works about the non-degeneracy of solution of (1.10).

In this paper we find a different solution from [10], which satisfies the following asymptotic behavior.
Theorem 1.1.
Let* $κ > 1$ . Suppose that $w_{2}$ is a non-degenerate solution of ( 1.10 ). Then for small $ε > 0$ , there exists a mixed-type (I) solutions $(u_{1, ε}, u_{2, ε})$ of ( 1.6 ) satisfying $\begin{array}{l} {‖ u_{1, ε} - ln C_{N, τ}^{(1)} - \sum_{j = 1}^{d_{1}} U_{j} (\frac{x - p_{1, j}}{ε}) χ_{d} (| x - p_{1, j} |) ‖}_{L^{\infty} (Ω)} = O (ε), \\ {‖ \frac{u_{1, ε} - ln C_{N, τ}^{(1)}}{2} + u_{2, ε} - 2 ln ε - w_{2} ‖}_{W^{2, 2} (Ω)} = O (ε) . \end{array}$

In order to prove Theorem 1.1, we use a suitable transformation defined in (2.1) below, and observe the related linearized operator is invertible. We would expect that the analysis for Theorem 1.1 would help to classify solutions to (1.6) according to its asymptotic behavior. We also remark that another mixed type (I) solutions satisfying Definition 1.1(ii) also can be constructed whenever the non-degeneracy condition for the following equation holds: $\begin{matrix} Δ w_{1} + \frac{{(κ N)}^{2}}{{(1 + [N - 1] κ)}^{2}} e^{w_{1}} = 4 π \sum_{i = 1}^{d_{1}} m_{1, i} δ_{p_{1, i}} + \frac{4 π [κ - 1]}{(1 + [N - 1] κ)} \sum_{j = 1}^{d_{2}} m_{2, i} δ_{p_{2, i}} in Ω . \end{matrix}$

We note that the solutions in this paper and [10,21,43] have no blow up phenomena. Here we say that $u_{i, ε}$ blow up if the nonlinear terms $\frac{e^{u_{i, ε}}}{ε^{2}}$ tends to a sum of Dirac measure as $ε \to 0$ . Recently, doubly periodic solutions for $SU (3)$ model with blow up phenomena has been studied in [32,36]. It would be also interesting to consider blow up solution for (1.6) in general case.

The rest of our paper is organized as follows. In Section 2, we introduce some basic estimates. In Section 3, we establish the existence result for mixed type (I) solution of (1.6).
2. Preliminaries

In this section, we prepare some basic estimates to construct mixed-type (I) solution $(u_{1, ε}, u_{2, ε})$ of (1.6) satisfying $u_{1, ε} \to ln C_{N, τ}^{(1)}$ pointwise a.e. on Ω and $u_{2, ε} - 2 ln ε ⩽ C_{*} in Ω$ for some constant $C_{*} \in R$ . We see that $(u_{1, ε}, u_{2, ε})$ is a solution of (1.6) if and only if $\begin{matrix} (2.1) & \{\begin{matrix} {\hat{u}}_{1, ε} (x) : = u_{1, ε} (x) - ln C_{N, τ}^{(1)}, \\ {\hat{w}}_{2, ε} (x) : = \frac{[N - 1] [κ - 1] (u_{1, ε} (x) - ln C_{N, τ}^{(1)})}{[N - 1 + κ]} + u_{2, ε} (x) - 2 ln ε \end{matrix} \end{matrix}$ satisfy $\begin{matrix} (2.2) & \{\begin{matrix} Δ {\hat{u}}_{1, ε} + \frac{1}{ε^{2}} e^{{\hat{u}}_{1, ε}} (1 - e^{{\hat{u}}_{1, ε}}) \\ = \frac{[κ - 1]}{N} e^{u_{2, ε} - 2 ln ε} (1 - \frac{(1 + [N - 1] κ)}{N} ε^{2} e^{u_{2, ε} - 2 ln ε} - \frac{(N - [N - 2] [κ - 1])}{[N - 1 + κ]} e^{{\hat{u}}_{1, ε}}) \\ + 4 π \sum_{i = 1}^{d_{1}} m_{1, i} δ_{p_{1, i}}, \\ Δ {\hat{w}}_{2, ε} + \frac{κ N}{[N - 1 + κ]} e^{u_{2, ε} - 2 ln ε} (1 - \frac{(1 + [N - 1] κ)}{N} ε^{2} e^{u_{2, ε} - 2 ln ε} + \frac{(κ - 1) (N - 1)}{[N - 1 + κ]} e^{{\hat{u}}_{1, ε}}) \\ = \frac{4 π [N - 1] [κ - 1]}{[N - 1 + κ]} \sum_{i = 1}^{d_{1}} m_{1, i} δ_{p_{1, i}} + 4 π \sum_{j = 1}^{d_{2}} m_{2, i} δ_{p_{2, i}} . \end{matrix} \end{matrix}$ By using a suitable scaling, the term $\frac{1}{ε^{2}} e^{{\hat{u}}_{1, ε}} (1 - e^{{\hat{u}}_{1, ε}})$ can be the main term in the first equation of (2.2). With this observation, we define the approximate solution $U_{1, ε}$ for the first component ${\hat{u}}_{1, ε}$ in (2.2) such that $\begin{matrix} (2.3) & U_{1, ε} (x) : = \sum_{j = 1}^{d_{1}} U_{j} (\frac{x - p_{1, j}}{ε}) χ_{d} (| x - p_{1, j} |), \end{matrix}$ where $U_{j}$ is the entire (topological) solution of (1.9), and $0 ⩽ χ_{d} ⩽ 1$ is a smooth function, $χ_{d} \equiv 1$ on $B_{\frac{d}{2}} (0)$ , $χ_{d} \equiv 0$ on $R^{2} ∖ B_{d} (0)$ for small fixed constant $d > 0$ , independent of $ε > 0$ . In [49, Lemma 4.13], it was shown that for any $δ \in (0, 1)$ , there exists a constant $C_{δ} > 0$ satisfying $\begin{matrix} (2.4) & (1 - e^{U_{j} (x)}) + | \nabla U_{j} (x) | + | U_{j} (x) | ⩽ C_{δ} e^{- (1 - δ) | x |} for any | x | ⩾ 1 . \end{matrix}$ We also note that if ${\hat{u}}_{1, ε} \to 0$ a.e. in Ω, the equation (1.10) would be a limiting problem for the second equation of (2.2). We assume that there is a non-degenerate solution $w_{2}$ of (1.10).

In conclusion, we want to construct solutions $({\hat{u}}_{1, ε} (x), {\hat{w}}_{2, ε} (x))$ for the system (2.2) satisfying $\begin{matrix} (2.5) & ({\hat{u}}_{1, ε} (x), {\hat{w}}_{2, ε} (x)) = (U_{1, ε} (x) + η_{1, ε} (x), w_{2} (x) + η_{2, ε} (x)), \end{matrix}$ where $(η_{1, ε}, η_{2, ε})$ are error terms. For this purpose, we define the following linear operator $\begin{matrix} L_{ε} (η_{1}, η_{2}) : = (L_{1, ε} (η_{1}), L_{2} (η_{2})), \end{matrix}$ where $\begin{matrix} \{\begin{matrix} L_{1, ε} (η_{1}) : = Δ η_{1} + [\sum_{j = 1}^{d_{1}} f_{j, ε} (x) χ_{d} (| x - p_{1, j} |) + f_{0, ε} (x) (1 - \sum_{j = 1}^{d_{1}} χ_{d} (| x - p_{1, j} |))] η_{1}, \\ L_{2} (η_{2}) : = (Δ + \frac{{(κ N)}^{2}}{{[N - 1 + κ]}^{2}} e^{w_{2}}) η_{2}, \end{matrix} \end{matrix}$ here $f_{j, ε} (x) = \frac{e^{U_{j} (\frac{x - p_{1, j}}{ε})} (1 - 2 e^{U_{j} (\frac{x - p_{1, j}}{ε})})}{ε^{2}}$ for $j = 0, 1, \dots, d_{1}$ , $p_{1, 0} \in Ω ∖ [⋃_{j = 1}^{d_{1}} {p_{1, j}}]$ is a fixed point, and $U_{0} \equiv 0$ . In [21], the invertibility of $L_{ε}$ was proved.

Theorem 2.1 ([21, Theorem 2.4, Theorem 2.5]).

The operator $L_{1, ε} : W^{2, 2} (Ω) \to L^{2} (Ω)$ is an isomorphism. Moreover, for any $(ϕ_{1}, h_{1}) \in [W^{2, 2} (Ω) \cap L^{\infty} (Ω)] \times [L^{2} (Ω) \cap L^{\infty} (Ω)]$ satisfying $L_{1, ε} (ϕ_{1}) = h_{1}$ , there exists a constant $C > 0$ such that $‖ ϕ_{1} ‖_{L^{\infty} (Ω)} ⩽ C ε^{2} ‖ h_{1} ‖_{L^{\infty} (Ω)}$ .

The operator $L_{2} : W^{2, 2} (Ω) \to L^{2} (Ω)$ is an isomorphism. Moreover, for any $ϕ_{2} \in W^{2, 2} (Ω)$ and $h_{2} \in L^{2} (Ω)$ satisfying $L_{2} (ϕ_{2}) = h_{2}$ , there exists a constant $C > 0$ such that $‖ ϕ_{2} ‖_{W^{2, 2} (Ω)} ⩽ C ‖ h_{2} ‖_{L^{2} (Ω)}$ .

We see that $({\hat{u}}_{1, ε}, {\hat{w}}_{2, ε})$ satisfies (2.2) is equivalent to $(η_{1, ε}, η_{2, ε})$ satisfies $\begin{matrix} \{\begin{matrix} L_{1, ε} (η_{1, ε}) = f_{ε} (η_{1, ε}, η_{2, ε}), \\ L_{2} (η_{2, ε}) = g_{ε} (η_{1, ε}, η_{2, ε}), \end{matrix} \end{matrix}$ where $\begin{matrix} \{\begin{matrix} f_{ε} (η_{1}, η_{2}) : = \frac{1}{ε^{2}} {\sum_{j = 1}^{d_{1}} e^{U_{j} (\frac{x - p_{1, j}}{ε})} (1 - e^{U_{j} (\frac{x - p_{1, j}}{ε})}) χ_{d} (| x - p_{1, j} |)} \\ - \frac{1}{ε^{2}} e^{U_{1, ε} + η_{1}} (1 - e^{U_{1, ε} + η_{1}}) \\ + [\sum_{j = 1}^{d_{1}} \frac{e^{U_{j} (\frac{x - p_{1, j}}{ε})} (1 - 2 e^{U_{j} (\frac{x - p_{1, j}}{ε})})}{ε^{2}} χ_{d} (| x - p_{1, j} |) - \frac{1}{ε^{2}} (1 - \sum_{j = 1}^{d_{1}} χ_{d} (| x - p_{1, j} |))] η_{1} \\ - 2 \sum_{j = 1}^{d_{1}} \nabla_{x} U_{j} (\frac{x - p_{1, j}}{ε}) \nabla_{x} χ_{d} (| x - p_{1, j} |) - \sum_{j = 1}^{d_{1}} U_{j} (\frac{x - p_{1, j}}{ε}) Δ_{x} χ_{d} (| x - p_{1, j} |) \\ + \frac{[κ - 1]}{N} e^{w_{2} + η_{2} - \frac{[N - 1] [κ - 1] (U_{1, ε} + η_{1})}{[N - 1 + κ]}} {1 - \frac{(1 + [N - 1] κ)}{N} ε^{2} e^{w_{2} + η_{2} - \frac{[N - 1] [κ - 1] (U_{1, ε} + η_{1})}{[N - 1 + κ]}} \\ - \frac{(N - [N - 2] [κ - 1])}{[N - 1 + κ]} e^{U_{1, ε} + η_{1}}}, \\ g_{ε} (η_{1}, η_{2}) : = \frac{{(κ N)}^{2}}{{[N - 1 + κ]}^{2}} e^{w_{2}} (1 + η_{2}) \\ - \frac{(κ N)}{[N - 1 + κ]} e^{w_{2} + η_{2} - \frac{[N - 1] [κ - 1] (U_{1, ε} + η_{1})}{[N - 1 + κ]}} {1 \\ - \frac{(1 + [N - 1] κ)}{N} ε^{2} e^{w_{2} + η_{2} - \frac{[N - 1] [κ - 1] (U_{1, ε} + η_{1})}{[N - 1 + κ]}} + \frac{(κ - 1) (N - 1)}{[N - 1 + κ]} e^{U_{1, ε} + η_{1}}} . \end{matrix} \end{matrix}$ If ${p_{1, j}}$ is not empty, i.e. $4 π \sum_{i = 1}^{d_{1}} m_{1, i} \neq 0$ , then we should estimate $w_{2} - \frac{[N - 1] [κ - 1]}{[N - 1 + κ]} U_{j} (\frac{x - p_{1, j}}{ε})$ near the vortex points $p_{1, j}$ . For this, we use the exponential decay property of $U_{j}$ and compare the singular parts of $w_{2}$ and $U_{j}$ as in [21].

Lemma 2.2.
[ 21 , Lemma 4.1]Fix a large constant $R > 0$ , independent of $ε > 0$ . Then for each $1 ⩽ j ⩽ d_{1}$ , we have $\begin{matrix} \{\begin{matrix} (i) & w_{2} - \frac{[N - 1] [κ - 1]}{[N - 1 + κ]} U_{j} (\frac{x - p_{1, j}}{ε}) = \frac{2 [N - 1] [κ - 1]}{[N - 1 + κ]} m_{1, j} ln ε + O (1) in B_{ε R} (p_{1, j}), \\ (ii) & w_{2} + \frac{(N - [N - 2] [κ - 1])}{[N - 1 + κ]} U_{j} (\frac{x - p_{1, j}}{ε}) ⩽ \frac{2 [N - 1] [κ - 1]}{[N - 1 + κ]} m_{1, j} ln ε + O (1) in B_{ε R} (p_{1, j}), \\ (iii) & w_{2} - \frac{[N - 1] [κ - 1]}{[N - 1 + κ]} U_{j} (\frac{x - p_{1, j}}{ε}) \\ = \frac{2 [N - 1] [κ - 1]}{[N - 1 + κ]} m_{1, j} ln | x - p_{1, j} | + O (1) in B_{d} (p_{1, j}) ∖ B_{ε R} (p_{1, j}), \\ (iv) & w_{2} + \frac{(N - [N - 2] [κ - 1])}{[N - 1 + κ]} U_{j} (\frac{x - p_{1, j}}{ε}) \\ = \frac{2 [N - 1] [κ - 1]}{[N - 1 + κ]} m_{1, j} ln | x - p_{1, j} | + O (1) in B_{d} (p_{1, j}) ∖ B_{ε R} (p_{1, j}) . \end{matrix} \end{matrix}$
Proof.
We have $\begin{matrix} (2.6) & w_{2} (x) = \frac{2 [N - 1] [κ - 1]}{[N - 1 + κ]} m_{1, j} ln | x - p_{1, j} | + O (1) in B_{d} (p_{1, j}) . \end{matrix}$ We also see that $U_{j} (\frac{x - p_{1, j}}{ε}) = 2 m_{1, j} ln | \frac{x - p_{1, j}}{ε} | + O (1)$ in $B_{ε R} (p_{1, j})$ . Thus we deduce that $\begin{array}{l} w_{2} - \frac{[N - 1] [κ - 1]}{[N - 1 + κ]} U_{j} (\frac{x - p_{1, j}}{ε}) \\ = \frac{2 [N - 1] [κ - 1]}{[N - 1 + κ]} m_{1, j} ln | x - p_{1, j} | \\ - \frac{2 [N - 1] [κ - 1]}{[N - 1 + κ]} m_{1, j} ln | \frac{x - p_{1, j}}{ε} | + O (1) \\ (2.7) & = \frac{2 [N - 1] [κ - 1]}{[N - 1 + κ]} m_{1, j} ln ε + O (1) in B_{ε R} (p_{1, j}), \end{array}$ which implies Lemma 2.2(i).

We note that $U_{j} ⩽ 0$ in $R^{2}$ . Then (2.7) implies that $\begin{array}{l} w_{2} + \frac{(N - [N - 2] [κ - 1])}{[N - 1 + κ]} U_{j} (\frac{x - p_{1, j}}{ε}) \\ = w_{2} - \frac{[N - 1] [κ - 1]}{[N - 1 + κ]} U_{j} (\frac{x - p_{1, j}}{ε}) + U_{j} (\frac{x - p_{1, j}}{ε}) \\ ⩽ \frac{2 [N - 1] [κ - 1]}{[N - 1 + κ]} m_{1, j} ln ε + O (1) in B_{ε R} (p_{1, j}), \end{array}$ which implies Lemma 2.2(ii). Since $U_{j}$ has exponential decay at infinity (see (2.4)), we see that $\begin{matrix} U_{j} (\frac{x - p_{1, j}}{ε}) = O (1) in B_{d} (p_{1, j}) ∖ B_{ε R} (p_{1, j}) . \end{matrix}$ Together with (2.6), we obtain Lemma 2.2(iii) and (iv). Now we complete the proof of Lemma 2.2. □

3. Construction of mixed-type (I) solutions

We are going to construct mixed-type (I) solutions of (1.6).

Proof of Theorem 1.1.

We claim that there exist $ε_{0} > 0$ such that for each $ε \in (0, ε_{0})$ , there exists $(η_{1, ε}, η_{2, ε}) \in W^{2, 2} (Ω) \times W^{2, 2} (Ω)$ such that $\begin{array}{l} (3.1) & \{\begin{matrix} L_{ε} (η_{1, ε}, η_{2, ε}) = (f_{ε} (η_{1, ε}, η_{2, ε}), g_{ε} (η_{1, ε}, η_{2, ε})), \\ ‖ η_{1, ε} ‖_{L^{\infty} (Ω)} + ‖ η_{2, ε} ‖_{W^{2, 2} (Ω)} = O (ε) . \end{matrix} \end{array}$ Define a mapping $F_{ε} : W^{2, 2} (Ω) \times W^{2, 2} (Ω) \to W^{2, 2} (Ω) \times W^{2, 2} (Ω)$ as following: $\begin{matrix} (3.2) & F_{ε} (η_{1}, η_{2}) : = (L_{1, ε}^{- 1} (f_{ε} (η_{1}, η_{2})), L_{2}^{- 1} (g_{ε} (L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}), η_{2}))) . \end{matrix}$ As in [21], we use $g_{ε} (L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}), η_{2})$ in (3.2) instead of $g_{ε} (η_{1}, η_{2})$ since the norm of $L_{1, ε}^{- 1}$ is small enough in Theorem 2.1.

Let $\begin{matrix} B_{ε} : = {(η_{1}, η_{2}) \in W^{2, 2} (Ω) \times W^{2, 2} (Ω) : ‖ η_{1} ‖_{L^{\infty} (Ω)} + ‖ η_{2} ‖_{W^{2, 2} (Ω)} < A ε}, \end{matrix}$ where $A > 0$ is a fixed large constant, independent of $ε > 0$ , which will be determined later. We use the following norm $\begin{matrix} ‖ (η_{1}, η_{2}) ‖ : = ‖ η_{1} ‖_{L^{\infty} (Ω)} + ‖ η_{2} ‖_{W^{2, 2} (Ω)} . \end{matrix}$ To find a solution of (2.2), we are going to find a fixed point of the map $F_{ε}$ by using contraction mapping theorem.

Step 1. We claim that for small $ε > 0$ , $F_{ε} : B_{ε} \to B_{ε}$ . To prove it, we need to estimate $‖ F_{ε} (η_{1}, η_{2}) ‖$ . By using Theorem 2.1, we have for some constant $C > 0$ , independent of $ε > 0$ , $\begin{array}{l} ‖ F_{ε} (η_{1}, η_{2}) ‖ \\ = {‖ L_{1, ε}^{- 1} (f_{ε} (η_{1}, η_{2})) ‖}_{L^{\infty} (Ω)} + {‖ L_{2}^{- 1} (g_{ε} (L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}), η_{2})) ‖}_{W^{2, 2} (Ω)} \\ (3.3) & ⩽ C (ε^{2} {‖ f_{ε} (η_{1}, η_{2}) ‖}_{L^{\infty} (Ω)} + {‖ g_{ε} (L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}), η_{2}) ‖}_{L^{2} (Ω)}) . \end{array}$ Now we are going to estimate $ε^{2} f_{ε} (η_{1}, η_{2})$ and $g_{ε} (L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}), η_{2})$ . Let $\begin{matrix} G (t) : = e^{t} (1 - e^{t}) . \end{matrix}$ We note that $\begin{matrix} (3.4) & G (0) = 0 and G^{'} (0) = - 1 . \end{matrix}$ Then we see that $\begin{array}{l} ε^{2} f_{ε} (η_{1}, η_{2}) \\ = \sum_{j = 1}^{d_{1}} {G (U_{j} (\frac{x - p_{1, j}}{ε})) χ_{d} (| x - p_{1, j} |)} - G (U_{1, ε} + η_{1}) \\ + [\sum_{j = 1}^{d_{1}} G^{'} (U_{j} (\frac{x - p_{1, j}}{ε})) χ_{d} (| x - p_{1, j} |) - (1 - \sum_{j = 1}^{d_{1}} χ_{d} (| x - p_{1, j} |))] η_{1} \\ - 2 ε^{2} \sum_{j = 1}^{d_{1}} \nabla_{x} U_{j} (\frac{x - p_{1, j}}{ε}) \nabla_{x} χ_{d} (| x - p_{1, j} |) \\ - ε^{2} \sum_{j = 1}^{d_{1}} U_{j} (\frac{x - p_{1, j}}{ε}) Δ_{x} χ_{d} (| x - p_{1, j} |) \\ + ε^{2} \frac{[κ - 1]}{N} e^{w_{2} + η_{2} - \frac{[N - 1] [κ - 1] (U_{1, ε} + η_{1})}{[N - 1 + κ]}} {1 - \frac{(1 + [N - 1] κ)}{N} ε^{2} e^{w_{2} + η_{2} - \frac{[N - 1] [κ - 1] (U_{1, ε} + η_{1})}{[N - 1 + κ]}} \\ - \frac{(N - [N - 2] [κ - 1])}{[N - 1 + κ]} e^{U_{1, ε} + η_{1}}} . \end{array}$ In view of Lemma 2.2 and $U_{1, ε} \equiv 0$ in $Ω ∖ ⋃_{j}^{d_{1}} B_{d} (p_{1, j})$ , we see that $\begin{array}{l} ε^{2} \frac{[κ - 1]}{N} e^{w_{2} + η_{2} - \frac{[N - 1] [κ - 1] (U_{1, ε} + η_{1})}{[N - 1 + κ]}} {1 - \frac{(1 + [N - 1] κ)}{N} ε^{2} e^{w_{2} + η_{2} - \frac{[N - 1] [κ - 1] (U_{1, ε} + η_{1})}{[N - 1 + κ]}} \\ - \frac{(N - [N - 2] [κ - 1])}{[N - 1 + κ]} e^{U_{1, ε} + η_{1}}} \\ (3.5) & = O (ε^{2}) (1 + | η_{1} | + | η_{2} |) in Ω . \end{array}$ From (3.5) and $U_{j} ⩽ 0$ , we see that $\begin{array}{l} ε^{2} f_{ε} (η_{1}, η_{2}) \\ = G (U_{j} (\frac{x - p_{1, j}}{ε})) - G (U_{j} (\frac{x - p_{1, j}}{ε}) + η_{1}) + G^{'} (U_{j} (\frac{x - p_{1, j}}{ε})) η_{1} \\ + O (ε^{2}) (1 + | η_{1} | + | η_{2} |) \\ (3.6) & = O (| η_{1} |^{2}) + O (ε^{2}) (1 + | η_{1} | + | η_{2} |) in B_{\frac{d}{2}} (p_{1, j}) . \end{array}$ By using (3.5), (2.4), and (3.4), we see that there is a constant $c > 0$ satisfying $\begin{array}{l} ε^{2} f_{ε} (η_{1}, η_{2}) \\ = {G (U_{j} (\frac{x - p_{1, j}}{ε})) χ_{d} (| x - p_{1, j} |)} - G (U_{j} (\frac{x - p_{1, j}}{ε}) χ_{d} (| x - p_{1, j} |) + η_{1}) \\ + [G^{'} (U_{j} (\frac{x - p_{1, j}}{ε})) χ_{d} (| x - p_{1, j} |) - (1 - χ_{d} (| x - p_{1, j} |))] η_{1} \\ + O (ε^{2}) (1 + | η_{1} | + | η_{2} |) + O (e^{- \frac{c}{ε}}) \\ (3.7) & = O (| η_{1} |^{2}) + O (ε^{2}) (1 + | η_{1} | + | η_{2} |) in B_{d} (p_{1, j}) ∖ B_{\frac{d}{2}} (p_{1, j}) . \end{array}$ From (3.5), (3.4), and $χ_{d} (| x - p_{1, j} |) \equiv 0$ in $Ω ∖ ⋃_{j}^{d_{1}} B_{d} (p_{1, j})$ , we see that $\begin{matrix} (3.8) & ε^{2} f_{ε} (η_{1}, η_{2}) = O (| η_{1} |^{2}) + O (ε^{2}) (1 + | η_{1} | + | η_{2} |) in Ω ∖ ⋃_{j = 1}^{d_{1}} B_{d} (p_{1, j}) . \end{matrix}$ By Theorem 2.1, and (3.6)–(3.8), we deduce that for some constants $c, C > 0$ , independent of $ε > 0$ , $\begin{array}{l} {‖ L_{1, ε}^{- 1} (f_{ε} (η_{1}, η_{2})) ‖}_{L^{\infty} (Ω)} & ⩽ c ε^{2} {‖ (f_{ε} (η_{1}, η_{2})) ‖}_{L^{\infty} (Ω)} \\ (3.9) & ⩽ C (‖ η_{1} ‖_{L^{\infty} (Ω)}^{2} + ε^{2} (‖ η_{1} ‖_{L^{\infty} (Ω)} + ‖ η_{2} ‖_{W^{2, 2} (Ω)}) + ε^{2}) . \end{array}$ We see that $\begin{matrix} g_{ε} (L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}), η_{2}) = I (η_{1}, η_{2}) + II (η_{1}, η_{2}) + III (η_{1}, η_{2}), \end{matrix}$ where $\begin{array}{l} \begin{matrix} I (η_{1}, η_{2}) & = - \frac{(κ N)}{[N - 1 + κ]} [e^{w_{2} + η_{2} - \frac{[N - 1] [κ - 1] (U_{1, ε} + L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}))}{[N - 1 + κ]}} - e^{w_{2}} \\ - e^{w_{2}} (η_{2} - \frac{[N - 1] [κ - 1] (U_{1, ε} + L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}))}{[N - 1 + κ]}) \\ - e^{w_{2}} (\frac{[N - 1] [κ - 1] (U_{1, ε} + L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}))}{[N - 1 + κ]})], \end{matrix} \\ \begin{matrix} II (η_{1}, η_{2}) & = - \frac{(κ - 1) κ (N - 1) N}{{[N - 1 + κ]}^{2}} [e^{w_{2} + η_{2} + \frac{(N - [N - 2] [κ - 1]) (U_{1, ε} + L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}))}{[N - 1 + κ]}} - e^{w_{2}} \\ - e^{w_{2}} (η_{2} + \frac{(N - [N - 2] [κ - 1]) (U_{1, ε} + L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}))}{[N - 1 + κ]}) \\ + e^{w_{2}} (\frac{(N - [N - 2] [κ - 1]) (U_{1, ε} + L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}))}{[N - 1 + κ]})], \end{matrix} \\ III (η_{1}, η_{2}) = \frac{κ (1 + [N - 1] κ)}{[N - 1 + κ]} ε^{2} e^{2 w_{2} + 2 η_{2} - \frac{2 [N - 1] [κ - 1] (U_{1, ε} + L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}))}{[N - 1 + κ]}} . \end{array}$ Fix a large constant $R > 0$ , independent of $ε > 0$ . Then from Lemma 2.2 and (3.9), we see that for some constant $c, C > 0$ , independent of $ε > 0$ , $\begin{array}{l} {‖ g_{ε} (L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}), η_{2}) ‖}_{L^{2} (B_{ε R} (p_{1, j}))} \\ ⩽ c ({‖ ε^{\frac{2 [N - 1] [κ - 1]}{[N - 1 + κ]} m_{1, j}} ‖}_{L^{2} (B_{ε R} (p_{1, j}))} \\ + ε^{2} (1 + {‖ L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}) ‖}_{L^{2} (B_{ε R} (p_{1, j}))} + ‖ η_{2} ‖_{L^{2} (B_{ε R} (p_{1, j}))})) \\ (3.10) & ⩽ C (ε^{\frac{2 [N - 1] [κ - 1]}{[N - 1 + κ]} m_{1, j} + 1} + ε^{2} (1 + {‖ L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}) ‖}_{L^{2} (Ω)} + ‖ η_{2} ‖_{L^{2} (Ω)})) . \end{array}$ Since $U_{j}$ has exponential decays to zero at infinity in (2.4), we see that $\begin{array}{l} {‖ e^{w_{2}} U_{1, ε} ‖}_{L^{2} (B_{d} (p_{1, j}) ∖ B_{ε R} (p_{1, j}))} \\ = O {(\int_{ε R}^{d} r^{\frac{4 [N - 1] [κ - 1]}{[N - 1 + κ]} m_{1, j} + 1} exp (- c \frac{r}{ε}) d r)}^{1 / 2} \\ = O (ε^{\frac{2 [N - 1] [κ - 1]}{[N - 1 + κ]} m_{1, j} + 1}) {(\int_{R}^{\frac{d}{ε}} r^{\frac{4 [N - 1] [κ - 1]}{[N - 1 + κ]} m_{1, j} + 1} exp (- c t) d t)}^{1 / 2} \\ = O (ε^{\frac{2 [N - 1] [κ - 1]}{[N - 1 + κ]} m_{1, j} + 1}), \end{array}$ which implies $\begin{array}{l} {‖ g_{ε} (L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}), η_{2}) ‖}_{L^{2} (B_{d} (p_{1, j}) ∖ B_{ε R} (p_{1, j}))} \\ = O (‖ η_{2} ‖_{W^{2, 2} (Ω)}^{2}) + O (ε^{2} (1 + ‖ η_{2} ‖_{L^{2} (Ω)})) + O ({‖ L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}) ‖}_{L^{2} (Ω)}) \\ + O ({‖ e^{w_{2}} U_{1, ε} ‖}_{L^{2} (B_{d} (p_{1, j}) ∖ B_{ε R} (p_{1, j}))}) \\ = O (‖ η_{2} ‖_{W^{2, 2} (Ω)}^{2}) + O (ε^{2} (1 + ‖ η_{2} ‖_{L^{2} (Ω)})) + O ({‖ L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}) ‖}_{L^{2} (Ω)}) \\ (3.11) & + O (ε^{\frac{2 [N - 1] [κ - 1]}{[N - 1 + κ]} m_{1, j} + 1}) . \end{array}$ Since $U_{1, ε} \equiv 0$ on $Ω ∖ ⋃_{j = 1}^{d_{1}} B_{d} (p_{1, j})$ , we see that $\begin{array}{l} {‖ g_{ε} (L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}), η_{2}) ‖}_{L^{2} (Ω ∖ ⋃_{j = 1}^{d_{1}} B_{d} (p_{1, j}))} \\ (3.12) & = O (‖ η_{2} ‖_{W^{2, 2} (Ω)}^{2}) + O (ε^{2} (1 + ‖ η_{2} ‖_{L^{2} (Ω)})) + O ({‖ L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}) ‖}_{L^{2} (Ω)}) . \end{array}$ In view of Theorem 2.1, we remind that for some constant $c > 0$ , independent of $ε > 0$ , $\begin{matrix} (3.13) & {‖ L_{1, ε}^{- 1} (f_{ε} (η_{1}, η_{2})) ‖}_{L^{\infty} (Ω)} ⩽ c ε^{2} {‖ (f_{ε} (η_{1}, η_{2})) ‖}_{L^{\infty} (Ω)} . \end{matrix}$ By using (3.10)–(3.13), we get that for some constant $C > 0$ , independent of $ε > 0$ , $\begin{array}{l} {‖ g_{ε} (L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}), η_{2}) ‖}_{L^{2} (Ω)} \\ ⩽ C {ε^{2} {‖ (f_{ε} (η_{1}, η_{2})) ‖}_{L^{\infty} (Ω)} + ‖ η_{2} ‖_{W^{2, 2} (Ω)}^{2} \\ (3.14) & + ε^{\frac{2 [N - 1] [κ - 1]}{[N - 1 + κ]} m_{1, j} + 1} + ε^{2} (1 + ‖ η_{2} ‖_{W^{2, 2} (Ω)})} . \end{array}$ We note that the estimation also holds even when $4 π \sum_{i = 1}^{d_{1}} m_{1, i} \equiv 0$ . From (3.3), (3.9) and (3.14), we have for some constant $C > 0$ , independent of $ε > 0$ , $\begin{array}{rcl} ‖ F_{ε} (η_{1}, η_{2}) ‖ & ⩽ & C (‖ η_{1} ‖_{L^{\infty} (Ω)}^{2} + ‖ η_{2} ‖_{W^{2, 2} (Ω)}^{2} + ε^{\frac{2 [N - 1] [κ - 1]}{[N - 1 + κ]} m_{1, j} + 1} \\ (3.15) & + ε^{2} (1 + ‖ η_{1} ‖_{L^{\infty} (Ω)} + ‖ η_{2} ‖_{W^{2, 2} (Ω)})) . \end{array}$ Since $N ⩾ 1$ and $κ > 1$ , by letting $A = C + 1$ , we get that $‖ F_{ε} (η_{1}, η_{2}) ‖ < A ε$ and $F_{ε} (B_{ε}) \subset B_{ε}$ for small $ε > 0$ .

Step 2. We claim that if $(η_{1}, η_{2}), (η_{1}^{'}, η_{2}^{'}) \in B_{ε}$ for small $ε > 0$ , then $\begin{matrix} ‖ F_{ε} (η_{1}, η_{2}) - F_{ε} (η_{1}^{'}, η_{2}^{'}) ‖ < \frac{1}{2} ‖ (η_{1}, η_{2}) - (η_{1}^{'}, η_{2}^{'}) ‖ . \end{matrix}$ By Theorem 2.1, we have for some constant $C > 0$ , independent of $ε > 0$ , $\begin{array}{l} ‖ F_{ε} (η_{1}, η_{2}) - F_{ε} (η_{1}^{'}, η_{2}^{'}) ‖ \\ = {‖ L_{1, ε}^{- 1} (f_{ε} (η_{1}, η_{2})) - L_{1, ε}^{- 1} (f_{ε} (η_{1}^{'}, η_{2}^{'})) ‖}_{L^{\infty} (Ω)} \\ + {‖ L_{2}^{- 1} (g_{ε} (L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}), η_{2})) - L_{2}^{- 1} (g_{ε} (L_{1, ε}^{- 1} \circ f_{ε} (η_{1}^{'}, η_{2}^{'}), η_{2}^{'})) ‖}_{W^{2, 2} (Ω)} \\ ⩽ C (ε^{2} {‖ f_{ε} (η_{1}, η_{2}) - f_{ε} (η_{1}^{'}, η_{2}^{'}) ‖}_{L^{\infty} (Ω)} \\ (3.16) & + {‖ g_{ε} (L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}), η_{2}) - g_{ε} (L_{1, ε}^{- 1} \circ f_{ε} (η_{1}^{'}, η_{2}^{'}), η_{2}^{'}) ‖}_{L^{2} (Ω)}) . \end{array}$ We see that $\begin{array}{l} ε^{2} (f_{ε} (η_{1}, η_{2}) - f_{ε} (η_{1}^{'}, η_{2}^{'})) \\ = G (U_{1, ε} + η_{1}^{'}) - G (U_{1, ε} + η_{1}) \\ + [\sum_{j = 1}^{d_{1}} G^{'} (U_{j} (\frac{x - p_{1, j}}{ε})) χ_{d} (| x - p_{1, j} |) - (1 - \sum_{j = 1}^{d_{1}} χ_{d} (| x - p_{1, j} |))] (η_{1} - η_{1}^{'}) \\ + ε^{2} \frac{[κ - 1]}{N} {e^{w_{2} + η_{2} - \frac{[N - 1] [κ - 1] (U_{1, ε} + η_{1})}{[N - 1 + κ]}} (1 - \frac{(1 + [N - 1] κ)}{N} ε^{2} e^{w_{2} + η_{2} - \frac{[N - 1] [κ - 1] (U_{1, ε} + η_{1})}{[N - 1 + κ]}} \\ - \frac{(N - [N - 2] [κ - 1])}{[N - 1 + κ]} e^{U_{1, ε} + η_{1}})} \\ - ε^{2} \frac{[κ - 1]}{N} {e^{w_{2} + η_{2}^{'} - \frac{[N - 1] [κ - 1] (U_{1, ε} + η_{1}^{'})}{[N - 1 + κ]}} (1 - \frac{(1 + [N - 1] κ)}{N} ε^{2} e^{w_{2} + η_{2}^{'} - \frac{[N - 1] [κ - 1] (U_{1, ε} + η_{1}^{'})}{[N - 1 + κ]}} \\ - \frac{(N - [N - 2] [κ - 1])}{[N - 1 + κ]} e^{U_{1, ε} + η_{1}^{'}})} . \end{array}$ By similar way in Step 1, we can deduce that if $(η_{1}, η_{2}), (η_{1}^{'}, η_{2}^{'}) \in B_{ε}$ , then as $ε \to 0$ , $\begin{matrix} (3.17) & ε^{2} {‖ f_{ε} (η_{1}, η_{2}) - f_{ε} (η_{1}^{'}, η_{2}^{'}) ‖}_{L^{\infty} (Ω)} ⩽ o (1) ({‖ η_{1} - η_{1}^{'} ‖}_{L^{\infty} (Ω)} + {‖ η_{2} - η_{2}^{'} ‖}_{W^{2, 2} (Ω)}) . \end{matrix}$ We see that $\begin{array}{l} g_{ε} (L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}), η_{2}) - g_{ε} (L_{1, ε}^{- 1} \circ f_{ε} (η_{1}^{'}, η_{2}^{'}), η_{2}^{'}) \\ = I (η_{1}, η_{2}) + II (η_{1}, η_{2}) + III (η_{1}, η_{2}) - (I (η_{1}^{'}, η_{2}^{'}) + II (η_{1}^{'}, η_{2}^{'}) + III (η_{1}^{'}, η_{2}^{'})) . \end{array}$ By similar way in Step 1, we deduce that for some constant $C > 0$ , independent of $ε > 0$ , $\begin{array}{l} {‖ g_{ε} (L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}), η_{2}) - g_{ε} (L_{1, ε}^{- 1} \circ f_{ε} (η_{1}^{'}, η_{2}^{'}), η_{2}^{'}) ‖}_{L^{2} (Ω)} \\ ⩽ C ({‖ L_{1, ε}^{- 1} \circ f_{ε} (η_{1}, η_{2}) - L_{1, ε}^{- 1} \circ f_{ε} (η_{1}^{'}, η_{2}^{'}) ‖}_{L^{\infty} (Ω)} \\ (3.18) & + (‖ η_{2} ‖_{W^{2, 2} (Ω)} + {‖ η_{2}^{'} ‖}_{W^{2, 2} (Ω)} + o (1)) {‖ η_{2} - η_{2}^{'} ‖}_{W^{2, 2} (Ω)}) . \end{array}$ In view of (3.16)–(3.18), and Theorem 2.1, we get $\begin{matrix} ‖ F_{ε} (η_{1}, η_{2}) - F_{ε} (η_{1}^{'}, η_{2}^{'}) ‖ = o (1) ({‖ η_{1} - η_{1}^{'} ‖}_{L^{\infty} (Ω)} + {‖ η_{2} - η_{2}^{'} ‖}_{W^{2, 2} (Ω)}) as ε \to 0, \end{matrix}$ and complete the proof of Step 2.

Step 3. From Steps 1–2, $F_{ε}$ is a contraction map from $B_{ε}$ to $B_{ε}$ . Then there exists a fixed point of the map $F_{ε}$ on $B_{ε}$ by contraction mapping theorem. Thus we find a unique solution $(η_{1, ε}, η_{2, ε})$ of (3.1) satisfying $\begin{matrix} L_{ε} (η_{1, ε}, η_{2, ε}) = (f_{ε} (η_{1, ε}, η_{2, ε}), g_{ε} (η_{1, ε}, η_{2, ε})), where ‖ η_{1, ε} ‖_{L^{\infty} (Ω)} + ‖ η_{2, ε} ‖_{W^{2, 2} (Ω)} < A ε . \end{matrix}$ In other words, $({\hat{u}}_{1, ε} (x), {\hat{w}}_{2, ε} (x)) = (U_{1, ε} (x) + η_{1, ε} (x), w_{2} (x) + η_{2, ε} (x))$ is a solution of (2.2). Then from our setting, $\begin{matrix} \{\begin{matrix} u_{1, ε} (x) = {\hat{u}}_{1, ε} (x) + ln C_{N, τ}^{(1)} \\ = U_{1, ε} (x) + η_{1, ε} (x) + ln C_{N, τ}^{(1)}, \\ u_{2, ε} (x) = {\hat{w}}_{2, ε} (x) - \frac{[N - 1] [κ - 1] (u_{1, ε} (x) - ln C_{N, τ}^{(1)})}{[N - 1 + κ]} + 2 ln ε \\ = w_{2} (x) + η_{2, ε} (x) - \frac{[N - 1] [κ - 1] (U_{1, ε} (x) + η_{1, ε} (x))}{[N - 1 + κ]} + 2 ln ε . \end{matrix} \end{matrix}$ satisfies the system (1.6). It is easy to see that $u_{1, ε} \to ln C_{N, τ}^{(1)}$ pointwise a.e. in Ω. In view of Lemma 2.2 and (2.4), we see that $u_{2, ε} (x) - 2 ln ε ⩽ C_{*}$ on Ω for some constant $C_{*} > 0$ , independent of $ε > 0$ . Now we complete the proof of Theorem 1.1. □

Footnotes

Acknowledgements

The authors wish to thank an anonymous referee very much for careful reading and valuable comments. The first author was supported by Young Researcher Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. NRF-2016R1C1B2014942). The first author was supported by the research grant of the Chungbuk National University in 2015.

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Doubly periodic mixed type solution of Non-Abelian Chern–Simons–Higgs vortices with flavor

Abstract

Keywords

1. Introduction

Theorem A ([10, Theorem 2.2]).

Theorem 2.1 ([21, Theorem 2.4, Theorem 2.5]).

Footnotes

Acknowledgements

References