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Yujin Guo

Publications and source records attributed to Yujin Guo.

At least 37 records · Page 2Linked to original sources
arXiv2024

Mass Concentration of Two-Spinless Fermi Systems with Attractive Interactions

We study the two-spinless mass-critical Fermi systems with attractive interactions and trapping potentials. We prove that ground states of the system exist, if and only if the strength $a$ of attractive interactions satisfies $0<a<a_2^*$, where $0<a_2^*<+\infty$ is the best constant of a dual finite-rank Lieb-Thirring inequality. By the blow-up analysis of many-fermion systems, we show that ground states of the system concentrate at the flattest minimum points of the trapping potential $V(x)$ as $a\nearrow a_2^*$.

math-ph↗
arXiv2023

Ground States of Fermionic Nonlinear Schrödinger Systems with Coulomb Potential II: The $L^2$-Critical Case

As a continuation of \cite{me}, we consider ground states of the $N$ coupled fermionic nonlinear Schrödinger system with a parameter $a $ and the Coulomb potential $V(x)$ in the $L^2$-critical case, where $a>0$ represents the attractive strength of the quantum particles. For any given $N\in\mathbb{N}^+$, we prove that the system admits ground states, if and only if the attractive strength $a$ satisfies $0<a<a^*_N$, where the critical constant $0<a^*_N<\infty$ is the same as the best constant of a dual finite-rank Lieb-Thirring inequality. By developing the so-called blow-up analysis of many-body fermionic problems, we also prove the mass concentration behavior of ground states for the system as $a\nearrow a_N^*$.

math-ph↗
arXiv2022

Existence and Uniqueness of Constraint Minimizers for the Planar Schrodinger-Poisson System with Logarithmic Potentials

In this paper, we study constraint minimizers $u$ of the planar Schrödinger-Poisson system with a logarithmic convolution potential $\ln |x|\ast u^2$ and a logarithmic external potential $V(x)=\ln (1+|x|^2)$, which can be described by the $L^2$-critical constraint minimization problem with a subcritical perturbation. We prove that there is a threshold $ρ^* \in (0,\infty)$ such that constraint minimizers exist if and only if $0<ρ<ρ^*$. In particular, the local uniqueness of positive constraint minimizers as $ρ\nearrowρ^*$ is analyzed by overcoming the sign-changing property of the logarithmic convolution potential and the non-invariance under translations of the logarithmic external potential.

math-ph↗
arXiv2022

Nonexistence of Vortices for Rotating Two-Component Focusing Bose Gases

This paper is concerned with ground states of two-component Bose gases confined in a harmonic trap $V(x)=x_1^2+Λ^2 x_2^2$ rotating at the velocity $Ω>0$, where $Λ\ge 1$ and $(x_1, x_2)\in R^2$. We focus on the case where the intraspecies interaction $(-a_1,-a_2)$ and the interspecies interaction $-β$ are both attractive, i.e, $a_1, a_2$ and $β$ are all positive. It is shown that for any $0<Ω<Ω^*:=2$, ground states exist if and only if $0 0$ is the unique positive solution of $-Δw+ w-w^3=0$ in $R^2$. By developing the argument of refined expansions, we further prove the nonexistence of vortices for ground states as $β\nearrowβ^*$, where $0<Ω<Ω^*$ and $0<a_1,\, a_2<a^*$ are fixed.

math-ph↗
arXiv2022

Ground States of Attractive Bose Gases in Rotating Anharmonic Traps

This paper is concerned with ground states of attractive Bose gases confined in an anharmonic trap $V(x)=ω(|x|^2+k|x|^4)$ rotating at the velocity $Ω>0$, where $ω>0$ denotes the trapping frequency, and $k>0$ represents the strength of the quartic term. It is known that for any $Ω>0$, ground states exist in such traps if and only if $0 0$ is the unique positive solution of $ΔQ-Q+Q^{3}=0$ in $\mathbb{R}^2$. By analyzing the refined energies and expansions of ground states, we prove that there exists a constant $C>0$, independent of $0 0$.

math.AP↗
arXiv2021

Local Uniqueness of Ground States for Rotating Bose-Einstein Condensates with Attractive Interactions

We study ground states of two-dimensional Bose-Einstein condensates with attractive interactions in a trap $V(x)$ rotating at the velocity $Ω$. It is known that there exist a critical rotational velocity $0<Ω^*:=Ω^*(V)\leq \infty$ and a critical number $0<a^*<\infty$ such that for any rotational velocity $0\le Ω<Ω^*$, ground states exist if and only if the coupling constant $a$ satisfies $a<a^*$. For a general class of traps $V(x)$, which may not be symmetric, we prove in this paper that up to a constant phase, there exists a unique ground state as $a\nearrow a^*$, where $Ω\in(0,Ω^*)$ is fixed. This result extends essentially our recent uniqueness result, where only the radially symmetric traps $V(x)$ could be handled with.

math.AP↗
arXiv2021

The Nonexistence of Vortices for Rotating Bose-Einstein Condensates in Non-Radially Symmetric Traps

We consider ground states of rotating Bose-Einstein condensates with attractive interactions in non-radially harmonic traps $V(x)=x_1^2+Λ^2x_2^2 $, where $0<Λ\not =1$ and $x=(x_1, x_2)\in R^2$. For any fixed rotational velocity $0\le Ω<Ω^*:=2\min \{1, Λ\}$, it is known that ground states exist if and only if $ a 0$ denotes the product for the number of particles times the absolute value of the scattering length. We analyze the asymptotic expansions of ground states as $a\nearrow a^*$, which display the visible effect of $Ω$ on ground states. As a byproduct, we further prove that ground states do not have any vortex in the region $R(a):=\{x\in R^2:\,|x|\le C (a^*-a)^{-\frac{1}{12}}\}$ as $a\nearrow a^*$ for some constant $C>0$, which is independent of $0<a<a^*$.

math.AP↗
arXiv2021

Existence and Asymptotic Behavior of Ground States for Rotating Bose-Einstein Condensates

We study ground states of two-dimensional Bose-Einstein condensates with repulsive ($a>0$) or attractive ($a<0$) interactions in a trap $V (x)$ rotating at the velocity $Ω$. It is known that there exist critical parameters $a^*>0$ and $Ω^*:=Ω^*(V(x))>0$ such that if $Ω>Ω^*$, then there is no ground state for any $a\in\R$; if $0\le Ω<Ω^*$, then ground states exist if and only if $a\in(-a^*,+\infty)$. As a completion of the existing results, in this paper, we focus on the critical case where $0<Ω=Ω^*<+\infty$ and classify the existence and nonexistence of ground states for $a\in\R$. Moreover, for a suitable class of radially symmetric traps $V(x)$, employing the inductive symmetry method, we prove that up to a constant phase, the ground states must be real-valued, unique and free of vortices as $Ω\searrow 0$, no matter whether the interactions of the condensates are repulsive or not.

math.AP↗
arXiv2020

The Nonexistence of Vortices for Rotating Bose-Einstein Condensates with Attractive Interactions

This article is devoted to studying the model of two-dimensional attractive Bose-Einstein condensates in a trap $V(x)$ rotating at the velocity $Ω$. This model can be described by the complex-valued Gross-Pitaevskii energy functional. It is shown that there exists a critical rotational velocity $0<Ω^*:=Ω^*(V)\leq \infty$, depending on the general trap $V(x)$, such that for any rotational velocity $0\leq Ω<Ω^*$, minimizers (i.e., ground states) exist if and only if $a 0$ denotes the absolute product for the number of particles times the scattering length, and $w>0$ is the unique positive solution of $Δw-w+w^3=0$ in $\mathbb{R}^2$. If $V(x)=|x|^2$ and $ 0<Ω<Ω^*(=2)$ is fixed, we prove that, up to a constant phase, all minimizers must be real-valued, unique and free of vortices as $a \nearrow a^*$, by analyzing the refined limit behavior of minimizers and employing the non-degenerancy of $w$.

math.AP↗
arXiv2020

Singular Behavior of an Electrostatic--Elastic Membrane System with an External Pressure

We analyze nonnegative solutions of the nonlinear elliptic problem $Δu=\frac{λf(x)}{u^2}+P$, where $λ>0$ and $P\geq0$, on a bounded domain $Ω$ of $\mathbb{R}^N$ ($N\geq 1$) with a Dirichlet boundary condition. This equation models an electrostatic--elastic membrane system with an external pressure $P\geq 0$, where $λ>0$ denotes the applied voltage. First, we completely address the existence and nonexistence of positive solutions. The classification of all possible singularities at $|x|=0$ for nonnegative solutions $u(x)$ satisfying $u(0)=0$ is then analyzed for the special case where $Ω=B_1(0)\subset \mathbb{R}^2$ and $f(x)=|x|^α$ with $α\geq0$. In particular, we show that for some $α,$ $u(x)$ admits only the "isotropic" singularity at $|x|=0$, and otherwise $u(x)$ may admit the "anisotropic" singularity at $|x|=0$. When $u(x)$ admits the "isotropic" singularity at $|x|=0$, the refined singularity of $u(x)$ at $|x|=0$ is further investigated, depending on whether $P>0$, by applying Fourier analysis.

math.AP↗
arXiv2018

Limit Behavior of Mass Critical Hartree Minimization Problems with Steep Potential Wells

We consider minimizers of the following mass critical Hartree minimization problem: \[ e_λ(N):=\underset{\{u\in H^1(R^d),\,\|u\|^2_2=N\}}{\inf} E_λ(u),\,\ d\ge 3, \] where the Hartree energy functional $E_λ(u)$ is defined by \[ E_λ(u):=\int_{R ^d}|\nabla u(x)|^2dx+λ\int_{R ^d}g(x)u^2(x)dx-\frac{1}{2} \int_{R ^d}\int_{R ^d} \frac{u^2(x)u^2(y)}{|x-y|^2}dxdy,\,\ λ>0,\] and the steep potential $g(x)$ satisfies $0=g(0)=\inf _{R^d}g(x)\le g(x)\le 1$ and $1-g(x)\in L^{\frac{d}{2}}(R^d)$. We prove that there exists a constant $N^*>0$, independent of $λg(x)$, such that if $N\ge N^*$, then $e_λ(N)$ does not admit minimizers for any $λ>0$; if $0 0$ such that $e_λ(N)$ admits minimizers for any $λ>λ^*(N)$, and $e_λ(N)$ does not admit minimizers for $0<λ<λ^*(N)$. For any given $0<N<N^*$, the limit behavior of positive minimizers for $e_λ(N)$ is also studied as $λ\to\infty$, where the mass concentrates at the bottom of $g(x)$.

math.FA↗
arXiv2018

Ground States of Two-Component Attractive Bose-Einstein Condensates II: Semi-trivial Limit Behavior

As a continuation of [14], we study new pattern formations of ground states $(u_1,u_2)$ for two-component Bose-Einstein condensates (BEC) with homogeneous trapping potentials in $R^2$, where the intraspecies interaction $(-a,-b)$ and the interspecies interaction $-β$ are both attractive, $i.e,$ $a$, $b$ and $β$ are all positive. If $0<b<a^*:=\|w\|^2_2$ and $0<β<a^*$ are fixed, where $w$ is the unique positive solution of $Δw-w+w^3=0$ in $R^2$, the semi-trivial behavior of $(u_1,u_2)$ as $a\nearrow a^*$ is proved in the sense that $u_1$ concentrates at a unique point and while $u_2\equiv 0$ in $R^2$. However, if $0<b<a^*$ and $a^*\leβ<β^*=a^*+\sqrt{(a^*-a)(a^*-b)}$, the refined spike profile and the uniqueness of $(u_1,u_2)$ as $a\nearrow a^*$ are analyzed, where $(u_1,u_2)$ must be unique, $u_1$ concentrates at a unique point, and meanwhile $u_2$ can either blow up or vanish, depending on how $β$ approaches to $a^*$.

math.AP↗
arXiv2017

Ground States of Two-Component Attractive Bose-Einstein Condensates I: Existence and Uniqueness

We study ground states of two-component Bose-Einstein condensates (BEC) with trapping potentials in $R^2$, where the intraspecies interaction $(-a_1,-a_2)$ and the interspecies interaction $-β$ are both attractive, $i.e,$ $a_1$, $a_2$ and $β$ are all positive. The existence and non-existence of ground states are classified completely by investigating equivalently the associated $L^2$-critical constraint variational problem. The uniqueness and symmetry-breaking of ground states are also analyzed under different types of trapping potentials as $β\nearrow β^*=a^*+\sqrt{(a^*-a_1)(a^*-a_2)}$, where $0<a_i<a^*:=\|w\|^2_2$ ($i=1,2$) is fixed and $w$ is the unique positive solution of $Δw-w+w^3=0$ in $R^2$. The semi-trivial limit behavior of ground states is tackled in the companion paper [12].

math.AP↗
arXiv2017

Blow-up behavior of ground states for a nonlinear Schrödinger system with attractive and repulsive interactions

We consider a nonlinear Schrödinger system arising in a two-component Bose-Einstein condensate (BEC) with attractive intraspecies interactions and repulsive interspecies interactions in $\mathbb{R}^2$. We get ground states of this system by solving a constrained minimization problem. For some kinds of trapping potentials, we prove that the minimization problem has a minimizer if and only if the attractive interaction strength $a_i (i=1,2)$ of each component of the BEC system is strictly less than a threshold $a^*$. %attractive intraspecies interactions satisfies $a_i< %a^*= \|Q\|_2^2,\ i=1,\,2$, where $Q$ is the unique positive radial solution of $Δu-u+u^3=0$ in $\mathbb{R}^2$; in contrast, there is no minimizer if either $a_i > a^*$ for $i=1$ or $2$, or $a_1=a_2=a^*$. Furthermore, as $(a_1, a_2)\nearrow (a^*, a^*)$, the asymptotical behavior for the minimizers of the minimization problem is discussed. Our results show that each component of the BEC system concentrates at a global minimum of the associated trapping potential.

math-ph↗
arXiv2017

Local Uniqueness and Refined Spike Profiles of Ground States for Two-Dimensional Attractive Bose-Einstein Condensates

We consider ground states of two-dimensional Bose-Einstein condensates in a trap with attractive interactions, which can be described equivalently by positive minimizers of the $L^2-$critical constraint Gross-Pitaevskii energy functional. It is known that ground states exist if and only if $a< a^*:= \|w\|_2^2$, where $a$ denotes the interaction strength and $w$ is the unique positive solution of $Δw-w+w^3=0$ in $R^2$. In this paper, we prove the local uniqueness and refined spike profiles of ground states as $a\nearrow a^*$, provided that the trapping potential $h(x)$ is homogeneous and $H(y)=\int_{R^2} h(x+y)w^2(x)dx$ admits a unique and non-degenerate critical point.

math.AP↗
arXiv2017

Blow-up solutions for two coupled Gross-Pitaevskii equations with attractive interactions

The paper is concerned with a system of two coupled time-independent Gross-Pitaevskii equations in $\mathbb{R}^2$, which is used to model two-component Bose-Einstein condensates with both attractive intraspecies and attractive interspecies interactions. This system is essentially an eigenvalue problem of a stationary nonlinear Schrödinger system in $\mathbb{R}^2$, solutions of the problem are obtained by seeking minimizers of the associated variational functional with constrained mass (i.e. $L^2-$norm constaints). Under certain type of trapping potentials $V_i(x)$ ($i=1,2$), the existence, non-existence and uniqueness of this kind of solutions are studied. Moreover, by establishing some delicate energy estimates, we show that each component of the solutions blows up at the same point (i.e., one of the global minima of $V_i(x)$) when the total interaction strength of intraspecies and interspecies goes to a critical value. An optimal blowing up rate for the solutions of the system is also given.

math.AP↗