arXiv subjects

Yujin Guo

Publications and source records attributed to Yujin Guo.

At least 19 recordsLinked to original sources

arXiv2026

Qualitative Properties of Ground States for the Stationary Magnetopolaron with a Weak Magnetic Field

We investigate ground states of the stationary magnetopolaron in $\mathbb R^3$ with a constant magnetic field. When the strength $|b|$ of the magnetic field is sufficiently small, we prove the uniqueness and nondegeneracy of ground states, up to magnetic translations and phase shifts. Applying the uniqueness result, we further derive rigorously the symmetry and monotonicity of ground states for sufficiently small $|b|>0$. The second-order asymptotic expansion of the ground state energy is also derived as $b\to 0$.

math.AP↗

arXiv2026

Complete Classification and Nondegeneracy of $N$-Component Cubic Nonlinear Schr\"{o}dinger System in ${\mathbb R}$

We study the one-dimensional cubic nonlinear Schr\"{o}dinger system \[ u_i''+2\left(\sum_{k=1}^N u_k^2\right)u_i=-\mu_i u_i \quad \mbox{in } \ \mathbb R,\ \ i=1,2,\cdots,N, \] where $u=(u_1,\cdots,u_N)\in \big(H^1(\mathbb{R})\big)^N$, $\mu_1\leq\mu_2\leq\cdots\leq\mu_N<0$, and $N\geq 2$ is arbitrary. In this paper, we prove the following results for any $N\ge 2$: (i). All nontrivial solutions of the system can be completely classified; (ii). The linearized operator at any nontrivial solution of the system is non-degenerate; (iii). For all $i=1, 2,\cdots, N$, the exact $L^2$-mass identity of $u_i$ is derived in terms of $2\sqrt {|\mu_i|}$, which yields a complete characterization of normalized solutions satisfying $\int_{\mathbb{R}}u_i^2dx=1$. These settle some conjectures of [R. Frank, D. Gontier and M. Lewin, CMP, 2021] and [Y. Guo, Y. Luo and J. Wei, APDE, 2026], where the system was addressed specially for $N=2$ and $N=3$, respectively.

math.AP↗

arXiv2026

Ground States of One-Dimensional Fermionic Schr\"{o}dinger Systems Near a Critical Exponent

We study ground states of the fermionic nonlinear Schr\"{o}dinger system $J_2(p)$ in $\R$, where $p>1$ denotes a polynomial exponent of the nonlinear term. It is known that the system $J_2(p)$ admits ground states for any $1<p<2$, while there is no ground state for $J_2(2)$. We prove that there is no ground state of $J_2(p)$ as $p\searrow 2$, which addresses the special case of Conjecture 5 in [D. Gontier, M. Lewin and F. Q. Nazar, ARMA, 2021]. The refined limiting profile of ground states for $J_2(p)$ is also analyzed as $p\nearrow 2$, which shows that the corresponding density admits exactly two bumps whose distance goes up to infinity as $p\nearrow 2$.

math.AP↗

arXiv2026

On the Limiting Behavior of $L^2$-Critical Pseudo-Relativistic Fermi Systems

We consider ground states of a pseudo-relativistic Fermi system in the $L^2$-critical case. We prove that the system admits ground states, if and only if the attractive strength $a$ satisfies $0<a<D_{4/3,2}$, where $D_{4/3,2}\in(0, \infty)$ is the optimal constant of a dual fractional Lieb--Thirring inequality. The limiting behavior of ground states for the system is further analyzed as $a\nearrow D_{4/3,2}$. As a byproduct, the qualitative properties of optimizers for the dual fractional Lieb-Thirring inequality are also investigated.

math.AP↗

arXiv2026

Ground States of Attractive Fermi Schr\"{o}dinger Systems with Ring-Shaped Potentials

As an application of the finite-rank Lieb-Thirring inequality established in [R. L. Frank, D. Gontier and M. Lewin, Comm. Math. Phys., 2021], we study ground states of mass-critical N-coupled Fermi nonlinear Schr\"{o}dinger systems with attractive interactions in $\mathbb{R}^3$, which are trapped in ring-shaped potentials. For any given $N\in\mathbb{N}^+$, we prove that ground states exist if $0<a<a_N^*$, where $a$ denotes the strength of attractive interactions in the system, and $a_N^*$ is the best constant of a finite-rank Lieb-Thirring inequality. Moreover, for some $N\in\mathbb{N}^+$, we also prove the nonexistence of minimizers for the system as soon as $a\geq a_N^*$. Applying the energy estimates and the blow-up analysis, we further analyze the mass concentration behavior of ground states for the system as $a\nearrow a_N^*$.

math.AP↗

arXiv2026

Pseudo-relativistic fermionic systems with attractive Yukawa potential

We study the Hartree-Fock and Hartree-Fock-Bogoliubov theories for a large fermionic system with the pseudo-relativistic kinetic energy and an attractive Yukawa interaction potential. We prove that the system is stable if and only if the total mass does not excess a critical value, and investigate the existence and properties of ground states in both sub-critical and critical mass regimes.

math-ph↗

arXiv2026

Asymptotic Behavior of the Principal Eigenvalue Problems with Large Divergence-Free Drifts

In this paper, we consider the following principal eigenvalue problem with a large divergence-free drift: \begin{equation}\label{0.1} -\varepsilon\Delta \phi-2\alpha\nabla m(x)\cdot\nabla \phi+V(x)\phi=\lambda_\alpha \phi\ \,\ \text{in}\, \ H_0^1(\Omega),\tag{0.1} \end{equation} where the domain $\Omega\subset \mathbb{R}^N (N\ge 1)$ is bounded with smooth boundary $\partial\Omega$, the constants $\varepsilon>0$ and $\alpha>0$ are the diffusion and drift coefficients, respectively, and $m(x)\in C^{2}(\bar{\Omega})$, $V (x)\in C^{\gamma}(\bar{\Omega})~(0<\gamma<1)$ are given functions. For a class of divergence-free drifts where $m$ is a harmonic function in $\Omega$ and has no first integral in $H_{0}^{1}(\Omega)$, we prove the convergence of the principal eigenpair $(\lambda_\alpha, \phi)$ for (0.1) as $\alpha\rightarrow+\infty$, which addresses a special case of the open question proposed in [H. Berestycki, F. Hamel and N. Nadirashvili, CMP, 2005]. Moreover, we further investigate the refined limiting profiles of the principal eigenpair $(\lambda_\alpha, \phi)$ for (0.1) as $\alpha\rightarrow+\infty$, which display the visible effects of the large divergence-free drifts on the principal eigenpair $(\lambda_\alpha, \phi)$.

math.AP↗

arXiv2025

Refined Limiting Profiles of the Principal Eigenvalue Problems with Large Advection

In this paper, we are concerned with the following eigenvalue problem with an advection term: \begin{equation}\label{0.1} \left\{ \begin{split} -\epsilon\Delta \phi-2\alpha\nabla m(x)\cdot\nabla \phi+V(x)\phi&=\lambda \phi\ \ \text{in}\ \ \Omega,\\ \phi&=0\ \ \hbox{on}\ \ \partial\Omega, ~~~\text{(0.1)} \end{split} \right. \end{equation} where $\Omega\subset\mathbb{R}^N~(N\geq1)$ satisfying $\partial\Omega\in C^{2}$ is a bounded domain and contains the origin as an interior point, the constants $\epsilon>0$ and $\alpha>0$ are the diffusive and advection coefficients, respectively, and $m(x)\in C^{2}(\bar{\Omega})$, $V (x)\in C^{\gamma}(\bar{\Omega})~(0<\gamma<1)$ are given functions. We analyze the refined limiting profiles of the principal eigenpair $(\lambda, \phi)$ for (0.1) as $\alpha\rightarrow\infty$, which display the visible effect of the large advection on $(\lambda, \phi)$. It expects that our argument is applicable to investigating the refined expansions of the general principal eigenvalue problems.

math.AP↗

arXiv2025

Optimizers of the Finite-Rank Hardy-Lieb-Thirring Inequality for Hardy-Schr\"odinger Operator

We study the following finite-rank Hardy-Lieb-Thirring inequality of Hardy-Schr\"odinger operator: \begin{equation*} \sum_{i=1}^N\left|\lambda_i\Big(-\Delta-\frac{c}{|x|^2}-V\Big)\right|^s\leq C_{s,d}^{(N)}\int_{\mathbb R^d}V_+^{s+\frac d2}dx, \end{equation*} where $N\in\mathbb N^+$, $d\geq3$, $0 0$ is the best constant of Hardy's inequality, and $V\in L^{s+\frac d2}(\mathbb R^d)$ holds for $s>0$. Here $\lambda_i\big(-\Delta-{c}{|x|^{-2}}-V\big)$ denotes the $i$-th min-max level of Hardy-Schr\"odinger operator $H_{c,V}:=-\Delta-{c}{|x|^{-2}}-V $ in $\mathbb R^d$, which equals to the $i$-th negative eigenvalue (counted with multiplicity) of $H_{c,V}$ in $\mathbb R^d$ if it exists, and vanishes otherwise. We analyze the existence and analytical properties of the optimizers for the above inequality.

math.AP↗

arXiv2025

Mass-Critical Neutron Stars in the Hartree-Fock and Hartree-Fock-Bogoliubov Theories

We investigate the ground states of neutron stars and white dwarfs in the Hartree-Fock (HF) and Hartree-Fock-Bogoliubov (HFB) theories. It is known that the system is stable below a critical mass, which depends on the gravitational constant, while it becomes unstable if the total mass exceeds the critical mass. We prove that if the total mass is at the critical mass, then the HFB minimizers do not exist for any gravitational constant, while the HF minimizers exist for every gravitational constant except for a countable set, which is fully characterized by the Gagliardo-Nirenberg inequality for orthonormal systems. Our results complement the existence results in the sub-critical mass case established in [E. Lenzmann and M. Lewin, Duke Math. J., 2010].

math.AP↗

arXiv2024

Classification and Nondegeneracy of Cubic Nonlinear Schr\"{o}dinger System in $\mathbb{R}$

We study the following one-dimensional cubic nonlinear Schr\"{o}dinger system: \[ u_i''+2\Big(\sum_{k=1}^Nu_k^2\Big)u_i=-\mu_iu_i \ \,\ \mbox{in}\, \ \mathbb{R} , \ \ i=1, 2, \cdots, N, \] where $\mu_1\leq\mu_2\leq\cdots\leq\mu_N<0$ and $N\ge 2$. In this paper, we mainly focus on the case $N=3$ and prove the following results: (i). The solutions of the system can be completely classified; (ii). Depending on the explicit values of $\mu_1\leq\mu_2\leq\mu_3<0$, there exist two different classes of normalized solutions $u=(u_1, u_2, u_3)$ satisfying $\int _{R}u_i^2dx=1$ for all $i=1, 2, 3$, which are completely different from the case $N=2$; (iii). The linearized operator at any nontrivial solution of the system is non-degenerate. The conjectures on the explicit classification and nondegeneracy of solutions for the system are also given for the case $N>3$. These address the questions of [R. Frank, D. Gontier and M. Lewin, CMP, 2021], where the complete classification and uniqueness results for the system were already proved for the case $N=2$.

math.AP↗

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 Schr\"{o}dinger 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 Schr\"{o}dinger 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↗

arXiv2023

Axial Symmetry of Normalized Solutions for Magnetic Gross-Pitaevskii Equations with Anharmonic Potentials

This paper is concerned with normalized solutions of the magnetic focusing Gross-Pitaevskii equations with anharmonic potentials in $\mathbb{R}^N$, where $N=2$ or $3$. We construct axially symmetric normalized concentrating solutions as the parameter $a>0$ approaches $a_*(N)$, where $a_*(N)\geq0$ is a critical constant depending only on $N$. We further prove that up to a constant phase (and a rotational transformation for $N=2$), normalized concentrating solutions are unique and axially symmetric as $a\to a_*(N)$. When $N=3$, we also prove that the corresponding unique normalized concentrating solution is free of vortices as $a\to a_*(3)$, even if the anharmonic potential is non-radially symmetric.

math.AP↗

arXiv2023

Ground States of Fermionic Nonlinear Schr\"{o}dinger Systems with Coulomb Potential I: The $L^2$-Subcritical Case

We consider ground states of the $N$ coupled fermionic nonlinear Schr\"{o}dinger systems with the Coulomb potential $V(x)$ in the $L^2$-subcritical case. By studying the associated constraint variational problem, we prove the existence of ground states for the system with any parameter $\alpha>0$, which represents the attractive strength of the non-relativistic quantum particles. The limiting behavior of ground states for the system is also analyzed as $\alpha\to\infty$, where the mass concentrates at one of the singular points for the Coulomb potential $V(x)$.

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↗