arXiv subjects

Fulin Zhong

Publications and source records attributed to Fulin Zhong.

4 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

On the uniqueness of the critical point of $\psi_\Omega$

We prove that for any bounded convex domain $\Omega \subset \mathbb{R}^n$, the function \begin{equation*} \psi_\Omega(\xi) = \int_{\mathbb{R}^n\setminus\Omega} \frac{\mathrm{d}x}{|x-\xi|^{2n}}, \quad \xi\in\Omega, \end{equation*} has exactly one critical point. This confirms an conjecture proposed by Clapp, Pistoia and Salda\~na in [J. Math. Pures Appl. 205 (2026), 103783]. The proof uses a spherical coordinates representation to write $\psi_\Omega$ as an integral of the distance function $\rho(\xi,\omega)$. This approach is not limited to $\psi_\Omega$. Instead, it provides a general framework for analyzing a broad class of functionals involving the boundary distance. We also examine non-convex domains. In particular, a single annulus exhibits a full circle of critical points, while multiple concentric annuli produce finitely many critical spheres. These examples show that the convexity hypothesis is essential for the uniqueness conclusion. The method developed here for handling spherical integrals involving the distance function is likely to be useful in other geometric and analytic contexts.

math.AP↗

arXiv2025

Brouwer degree for Chern-Simons Higgs models on finite graphs

Let $G=(V, E)$ be a finite connected graph, where $V$ denotes the set of vertices and $E$ denotes the set of edges. We revisit the following Chern-Simons Higgs model, \begin{equation*} Δu=λ\mathrm{e}^u\left(\mathrm{e}^u-1\right)+f \ \text {in} \ V, \end{equation*} where $Δ$ is the graph Laplacian, $λ$ is a real number and $f$ is a function defined on $V$. Firstly, when $λ\int_V f \mathrm{d} μ\neq 0$, we find that the odevity of the number of vertices in the graph affects the number of solutions. Then by calculating the topological degree and using the relationship between the degree and the critical group of a related functional, we obtain the existence of multiple solutions. Also we study the existence of solutions when $λ\int_V f \mathrm{d} μ=0$. These findings extend the work of Huang et al. [Comm Math Phys 377:613-621 (2020)], Hou and Sun [Calc Var 61:139 (2022)] and Li et al. [Calc Var 63:81 (2024)]. Similarly, for the generalized Chern-Simons Higgs model, we obtain the same results. Moreover, this method is also applied to the Chern-Simons Higgs system, yielding partial results for the existence of multiple solutions. To our knowledge, this is the first instance where it has been concluded that an equation on graphs can have at least three distinct solutions. We think that our results will be valuable for studying the multiplicity of solutions to analogous equations on graphs.

math.AP↗

arXiv2025

Existence of periodic solutions for the Grushin critical problem

We study a Grushin critical problem in a strip domain which satisfies the periodic boundary conditions. By applying the finite-dimensional reduction method, we construct a periodic solution when the prescribed curvature function is periodic. Furthermore, we also consider the Grushin critical problem in $\mathbb{R}^{N} (N \geq 5)$. Compared with Billel et al. (Differential Integral Equations 32: 49-90, 2019), we use the method by Guo and Yan (Math. Ann. 388: 795-830, 2024) to construct periodic solutions under some weaker conditions, avoiding the complicated estimates and uniqueness proof. Notably, Guo and Yan (Math. Ann. 388: 795-830, 2024) obtained solutions periodic with respect to some of the first variables, while the solutions in this paper are periodic with respect to some intermediate variables.

math.AP↗