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Youngae Lee

Publications and source records attributed to Youngae Lee.

17 recordsLinked to original sources
arXiv2021

Infinitely many segregated vector solutions of Schrodinger system

We consider the following system of Schrödinger equations \begin{equation*}\left.\begin{cases} -ΔU + λU = α_0 U^3+ βUV^2 -ΔV + μ(y) V = α_1 V^3+βU^2V \end{cases}\right. \text{in} \quad \mathbb{R}^N, \ N=2, 3,\end{equation*} where $λ$, $α_0$, $α_1>0$ are positive constants, $β\in \mathbb{R}$ is the coupling constant, and $μ: \mathbb{R}^N \rightarrow \mathbb{R}$ is a potential function. Continuing the work of Lin and Peng \cite{lin_peng_2014}, we present a solution of the type where one species has a peak at the origin and the other species has many peaks over a circle, but as seen in the above, coupling terms are nonlinear.

math.AP↗
arXiv2020

Blow up at infinity in the SU(3) Chern-Simons model, part I

We consider non-topological solutions of a nonlinear elliptic system problem derived from the $SU(3)$ Chern-Simons models in $\mathbb{R}^2$. The existence of non-topological solutions even for radial symmetric case has been a long standing open problem. Recently, [Choe, Kim, Lin (2015, 2016)] showed the existence of radial symmetric non-topological solution when the vortex points collapse. However, the arguments in [Choe, Kim, Lin (2015, 2016)] cannot work for an arbitrary configuration of vortex points. In this paper, we develop a new approach by using different scalings for different components of the system to construct a family of non-topological solutions, which blows up at infinity.

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arXiv2020

The asymptotic behavior of Chern-Simons Vortices for Gudnason Model

We consider an elliptic system arising from a supersymmetric gauge field theory. In this paper, we complete to classify all possible solutions according to their asymptotic behavior under a weak coupling effect. Interestingly, it turns out that one of components does not follow the feature of condensate solutions for the classical Chern-Simons-Higgs model. Moreover, in order to prove the concentration property of blow up component, we need to improve the convergence rate and the gradient estimation for the other component, which converges to a constant. We expect that this result would provide an insight for the study of general elliptic system problems, which are even neither cooperative nor competitive.

math.AP↗
arXiv2019

Periodic Maxwell-Chern-Simons vortices with concentrating property

In order to study electrically and magnetically charged vortices in fractional quantum Hall effect and anyonic superconductivity, the Maxwell-Chern-Simons (MCS) model was introduced by [Lee, Lee, Min (1990)] as a unified system of the classical Abelian-Higgs model (AH) and the Chern-Simons (CS) model. In this article, the first goal is to obtain the uniform (CS) limit result of (MCS) model with respect to the Chern-Simons parameter without any restriction on either a particular class of solutions or the number of vortex points. The most important step for this purpose is to derive the relation between the Higgs field and the neutral scalar field. Our (CS) limit result also provides the critical clue to answer the open problems raised by [Ricciardi,Tarantello (2000)] and [Tarantello (2004)], and we succeed to establish the existence of periodic Maxwell-Chern-Simons vortices satisfying the concentrating property of the density of superconductive electron pairs. Furthermore, we expect that the (CS) limit analysis in this paper would help to study the stability, multiplicity, and bubbling phenomena for solutions of the (MCS) model.

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arXiv2018

Existence of bubbling solutions without mass concentration

The seminal work \cite{bm} by Brezis and Merle has been pioneering in studying the bubbling phenomena of the mean field equation with singular sources. When the vortex points are not collapsing, the mean field equation possesses the property of the so-called "bubbling implies mass concentration". Recently, Lin and Tarantello in \cite{lt} pointed out that the "bubbling implies mass concentration" phenomena might not hold in general if the collapse of singularities occurs. In this paper, we shall construct the first concrete example of non-concentrated bubbling solution of the mean field equation with collapsing singularities.

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arXiv2018

Local uniqueness of $m$-bubbling sequences for the Gel'fand equation

We consider the Gel'fand problem, $$ \begin{cases} Δw_{\varepsilon}+\varepsilon^2 h e^{w_{\varepsilon}}=0\quad&\mbox{in}\quadΩ, w_{\varepsilon}=0\quad&\mbox{on}\quad\partialΩ, \end{cases} $$ where $h$ is a nonnegative function in ${Ω\subset\mathbb{R}^2}$. Under suitable assumptions on $h$ and $Ω$, we prove the local uniqueness of $m-$bubbling solutions for any $\varepsilon>0$ small enough.

math.AP↗
arXiv2018

Non degeneracy, Mean Field Equations and the Onsager theory of 2D turbulence

The understanding of some large energy, negative specific heat states in the Onsager description of 2D turbulence, seems to require the analysis of a subtle open problem about bubbling solutions of the mean field equation. Motivated by this application we prove that, under suitable non degeneracy assumptions on the associated $m$-vortex Hamiltonian, the $m$-point bubbling solutions of the mean field equation are non degenerate as well. Then we deduce that the Onsager mean field equilibrium entropy is smooth and strictly convex in the high energy regime on domains of second kind.

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arXiv2018

Uniqueness of bubbling solutions of mean field equations

We prove uniqueness of blow up solutions of the mean field equation as $ρ_n \rightarrow 8πm$, $m\in\mathbb{N}$. If $u_{n,1}$ and $u_{n,2}$ are two sequences of bubbling solutions with the same $ρ_n$ and the same (non degenerate) blow up set, then $u_{n,1}=u_{n,2}$ for sufficiently large $n$. The proof of the uniqueness requires a careful use of some sharp estimates for bubbling solutions of mean field equations [24] and a rather involved analysis of suitably defined Pohozaev-type identities as recently developed in [51] in the context of the Chern-Simons-Higgs equations. Moreover, motivated by the Onsager statistical description of two dimensional turbulence, we are bound to obtain a refined version of an estimate about $ρ_n-8πm$ in case the first order evaluated in [24] vanishes.

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arXiv2017

Uniqueness for bubbling solutions with collapsing singularities

The seminal work \cite{bm} by Brezis and Merle showed that the bubbling solutions of the mean field equation have the property of mass concentration. Recently, Lin and Tarantello in \cite{lt} found that the "bubbling implies mass concentration" phenomena might not hold if there is a collapse of singularities. Furthermore, a sharp estimate \cite{llty} for the bubbling solutions has been obtained. In this paper, we prove that there exists at most one sequence of bubbling solutions if the collapsing singularity occurs. The main difficulty comes from that after re-scaling, the difference of two solutions locally converges to an element in the kernel space of the linearized operator. It is well-known that the kernel space is three dimensional. So the main technical ingredient of the proof is to show that the limit after re-scaling is orthogonal to the kernel space.

math.AP↗
arXiv2017

Degree counting for Toda system with simple singularity : one point blow up

In this paper, we study the degree counting formula of the rank two Toda system with simple singular source when $ρ_1\in(0,4π)\cup(4π,8π)$ and $ρ_2\notin 4π\mathbb{N}.$ The key step is to derive the degree formula of the shadow system, which arises from the bubbling solutions as $ρ_1$ tends to $4π$. In order to compute the topological degree of the shadow system, we need to find some suitable deformation. During this deformation, we shall deal with \textit{new} difficulty arising from the new phenomena: blow up does not necessarily imply concentration of mass. This phenomena occurs due to the collapsing of singularities. This is a continuation of the previous work Lee, Lin, Wei and Yang.

math.AP↗
arXiv2017

Existence of mixed type solutions in the Chern-Simons gauge theory of rank two in $\mathbb{R}^2$

We consider the Chern-Simons gauge theory of rank $2$ such as $SU(3)$, $SO(5)$, and $G_2$ Chern-Simons model in $\mathbb{R}^2$. There may exist three types of solutions in these theories, that is, topological, nontopological, and mixed type solutions. Among others, mixed type solutions can only exist in non-Abelian Chern-Simons models. We show the existence of mixed type solutions with an arbitrary configuration of vortex points which has been a long-standing open problem. To show it, as the first step, we need to find when a priori bound would fail. For the purpose, we shall find partially blowing up mixed type solutions by using different scalings for different components. Due to the different scalings, we should control the mass contribution from infinity which is one of the important parts in this paper.

math.AP↗
arXiv2017

Sharp estimates for solutions of mean field equation with collapsing singularity

The pioneering work of Brezis-Merle [7], Li-Shafrir [27], Li [26] and Bartolucci-Tarantello [4] showed that any sequence of blow up solutions for (singular) mean field equations of Liouville type must exhibit a "mass concentration" property. A typical situation of blow-up occurs when we let the singular (vortex) points involved in the equation (see (1.1) below) collapse together. However in this case Lin-Tarantello in [30] pointed out that the phenomenon: "bubbling implies mass concentration" might not occur and new scenarios open for investigation. In this paper, we present two explicit examples which illustrate (with mathematical rigor) how a "non-concentration" situation does happen and its new features. Among other facts, we show that in certain situations, the collapsing rate of the singularities can be used as blow up parameter to describe the bubbling properties of the solution-sequence. In this way we are able to establish accurate estimates around the blow-up points which we hope to use towards a degree counting formula for the shadow system (1.34) below.

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arXiv2015

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.

math.AP↗
arXiv2014

Uniqueness of topological multi-vortex solutions for a skew-symmetric Chern-Simons system

Consider the following skew-symmetric Chern-Simons system \begin{equation*}\left \{ \begin{split} &Δu_{1}+\frac{1}{\varepsilon^2} e^{u_{2}}(1-e^{u_{1}})=4π\sum^{N_1}_{j=1}δ_{p_{j,1}}\\ &Δu_{2}+\frac{1}{\varepsilon^2} e^{u_{1}}(1-e^{u_{2}})=4π\sum^{N_2}_{j=1}δ_{p_{j,2}} \end{split}\right.\quad\text{ in }\quadΩ, \end{equation*} where $Ω$ is a flat 2-dimensional torus $\mathbb{T}^2$ or $\mathbb{R}^2$, $\varepsilon> 0$ is a coupling parameter, and $δ_p$ denotes the Dirac measure concentrated at $p$. In this paper, we prove that, when the coupling parameter $\varepsilon$ is small, the topological type solutions to the above system are uniquely determined by the location of their vortex points. This result follows by the bubbling analysis and the non-degency of linearized equations.

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arXiv2013

Stable solutions and finite Morse index solutions of nonlinear elliptic equations with Hardy potential

We are concerned with Liouville-type results of stable solutions and finite Morse index solutions for the following nonlinear elliptic equation with Hardy potential: \begin{displaymath} Δu+\dfracμ{|x|^2}u+|x|^l |u|^{p-1}u=0 \qquad \textrm{in}\ \ Ω, \end{displaymath} where $Ω=\RN$, $\RN\setminus\{0\}$ for $N\geq3$, $p>1$, $l>-2$ and $μ<(N-2)^2/4$. Our results depend crucially on a new critical exponent $p=p_c(l,μ)$ and the parameter $μ$ in Hardy term. We prove that there exist no nontrivial stable solution and finite Morse index solution for $1<p<p_c(l,μ)$. We also observe a range of the exponent $p$ larger than $p_c(l,μ)$ satisfying that our equation admits a positive radial stable solution.

math.AP↗
arXiv2013

Asymptotic analysis of solutions to a gauged O(3) sigma model

We analyze an elliptic equation arising in the study of the gauged O(3) sigma model with the Chern-Simons term. In this paper, we study the asymptotic behavior of solutions and apply it to prove the uniqueness of stable solutions. However, one of the features of this nonlinear equation is the existence of stable nontopological solutions in $\RN$, which implies the possibility that a stable solution which blows up at a vortex point exists. To exclude this kind of blow up behavior is one of the main difficulties which we have to overcome.

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