arXiv subjects

Xiaoming An

Publications and source records attributed to Xiaoming An.

7 recordsLinked to original sources

arXiv2026

Quantitative analysis of ground states for the fractional logarithmic Schr\"odinger equation

Let $N\geq1$ and $0<s<1$. We study positive ground states of the fractional logarithmic Schr\"odinger equation \begin{equation*} (-\Delta)^sQ=Q\log Q \quad\text{in }\mathbb{R}^N. \end{equation*} We prove that for every $N\geq1$ and $0<s<1$, the positive ground state is unique up to translations and nondegenerate. More precisely, for the linearized operator $L_Q=(-\Delta)^s-1-\log Q$, it holds that \begin{equation*} \ker L_Q=\operatorname{span}\{\partial_{x_1}Q,\cdots,\partial_{x_N}Q\}. \end{equation*} A main difficulty is that the potential $-1-\log Q$ is unbounded in $\mathbb{R}^N$, which prevents a direct application of the available radial oscillation theory for fractional Schr\"odinger operators with bounded potentials. We overcome this difficulty by a bounded-potential approximation. Using also the fact that the associated quadratic form has Morse index one and an angular decomposition, we obtain the nondegeneracy. Based on the isolation of logarithmic ground states and the uniqueness theory for the fractional power equation, we prove uniqueness by a variational approximation with subcritical power nonlinearities. As an application, we establish sharp fractional logarithmic Sobolev inequalities and characterize all cases of equality.

math.AP↗

arXiv2026

Qualitative analysis of energy ground states for magnetic focusing Gross-Pitaevskii equations

We prove the uniqueness, asymptotics, symmetry, and orbital stability of energy ground states for 3D magnetic Gross-Pitaevskii equations under mild conditions on the electric potential and magnetic field. In particular, both the electric potential and the magnetic field are allowed to have singularities, and we cover in a unified approach the physically relevant Aharonov-Bohm magnetic field as well as the constant magnetic field. Since we are in the 3D case, the unique energy ground state (up to a phase factor) is obtained as a local minimizer, rather than a global one, by restricting the kinetic energy of candidate critical points within a suitable range. The qualitative analysis of the energy ground state we carry on is mainly based on a related Pohozaev identity, the implicit function theorem, and variational methods.

math.AP↗

arXiv2022

Nonstandard solutions for a perturbed nonlinear Schrödinger system with small coupling coefficients\protect\thanks{A perturbed nonlinear Schrödinger system

In this paper, we consider the following weakly coupled nonlinear Schrödinger system \begin{equation*} \left\{ \begin{array}{ll} -ε^{2}Δu_1 + V_1(x)u_1 = |u_1|^{2p - 2}u_1 + β|u_1|^{p - 2}|u_2|^pu_1, & x\in \mathbb{R}^N,\\ -ε^{2}Δu_2 + V_2(x)u_2 = |u_2|^{2p - 2}u_2 + β|u_2|^{p - 2}|u_1|^pu_2, & x\in \mathbb{R}^N, \end{array} \right. \end{equation*} where $ε>0$, $β\in\mathbb{R}$ is a coupling constant, $2p\in (2,2^*)$ with $2^* = \frac{2N}{N - 2}$ if $N\geq 3$ and $+\infty$ if $N = 1,2$, $V_1$ and $V_2$ belong to $C(\mathbb{R}^N,[0,\infty))$. When $p\ge 2$ and $β>0$ is suitably small, we show that the problem has a family of nonstandard solutions $\{w_ε = (u^1_ε,u^2_ε):0<ε<ε_{0}\}$ concentrating synchronously at the common local minimum of $V_1$ and $V_2$. All decay rates of $V_i(i=1,2)$ are admissible and we can allow that $β>0$ is close to $0$ in this paper. Moreover, the location of concentration points is given by local Pohozaev identities. Our proofs are based on variational methods and the penalized technique.

math.AP↗

arXiv2022

Multi-peak semiclassical bound states for Fractional Schrödinger Equations with fast decaying potentials

We study the following fractional Schrödinger equation \begin{equation*}\label{eq0.1} \varepsilon^{2s}(-Δ)^s u + V(x)u = f(u), \,\,x\in\mathbb{R}^N, \end{equation*} where $s\in(0,1)$. Under some conditions on $f(u)$, we show that the problem has a family of solutions concentrating at any finite given local minima of $V$ provided that $V\in C(\R^N,[0,+\infty))$. All decay rates of $V$ are admissible. Especially, $V$ can be compactly supported. Different from the local case $s=1$ or the case of single-peak solutions, the nonlocal effect of the operator $(-Δ)^s$ makes the peaks of the candidate solutions affect mutually, which causes more difficulties in finding solutions with multiple bumps. The methods in this paper are penalized technique and variational method.

math.AP↗

arXiv2022

Semiclassical states for fractional logarithmic Schrödinger equations

In this paper, we consider the following fractional logarithmic Schrödinger equation \begin{equation*} \varepsilon^{2s}(-Δ)^s u + V(x)u=u\log |u|^2\ \ \text{in}\ \R^N, \end{equation*} where $\varepsilon>0$, $N\ge 1$, $V(x)\in C(\R^N,[-1,+\infty))$. By introducing an interesting penalized function, we show that the problem has a positive solution $u_{\varepsilon}$ concentrating at a local minimum of $V$ as $\varepsilon\to 0$. There is no restriction on decay rates of $V$, especially it can be compactly supported.

math.AP↗

arXiv2021

Semi-classical analysis for Fractional Schrödinger Equations with fast decaying potenials

We study the following fractional Schrödinger equation \begin{equation*}\label{eq0.1} ε^{2s}(-Δ)^s u + V(x)u = |u|^{p - 2}u, \,\,x\in\,\,\mathbb{R}^N, \end{equation*} where $s\in (0,\,1)$, $N>2s$, $p>1$ is subcritical and $V(x)$ is a nonnegative continuous potential. We use penalized technique to show that the problem has a family of solutions concentrating at a positive local minimum of $V(x)$ provided that $\frac{2s}{N-2s}+2<p<\frac{2N}{N-2s}$. The novelty is that $V$ can decay arbitrarily or even be compactly supported.

math.AP↗

arXiv2017

Semi-classical Solutions For Fractional Schrodinger Equations With Potential Vanishing At Infinity

We study the following fractional Schrödinger equation \begin{equation}\label{eq0.1} \varepsilon^{2s}(-Δ)^s u + Vu = |u|^{p - 2}u,\ \ x\in\,\,\mathbb{R}^N. \end{equation} We show that if the external potential $V\in C(\mathbb{R}^N;[0,\infty))$ has a local minimum and $p\in (2 + 2s/(N - 2s), 2^*_s)$, where $2^*_s=2N/(N-2s),\,N\ge 2s$, the problem has a family of solutions concentrating at the local minimum of $V$ provided that $\liminf_{|x|\to \infty}V(x)|x|^{2s} > 0$. The proof is based on variational methods and penalized technique. {\textbf {Key words}: } fractional Schrödinger; vanishing potential; penalized technique; variational methods.

math.AP↗