arXiv subjects

Guangze Gu

Publications and source records attributed to Guangze Gu.

7 recordsLinked to original sources

arXiv2026

Qualitative bifurcation diagram for Grad-Shafranov type equations

We study the qualitative behavior of solutions of Grad-Shafranov type equations arising in plasma physics with general differential operators and general nonlinearities. In particular, we extend recent estimates about threshold values for uniqueness, monotonicity and non-existence of the free boundary. The argument is based on a refined spectral analysis for weighted non-local problems together with comparison techniques and level set analysis.

math.AP↗

arXiv2026

Isolated Singularities for Fractional Hartree Equations

We study isolated singularities of positive solutions to a fractional Hartree equation with Riesz interaction, \[ (-\Delta)^s u = \left( \int_{\mathbb{R}^N\setminus\{0\}} \frac{u^p(y)}{|x-y|^\mu}\,dy \right)u^q \quad \text{in } \mathbb{R}^N\setminus\{0\}. \] The puncture changes the passage from the differential equation to its integral form: a fractional fundamental-solution term may occur at the singular point. For non-removable blow-up singularities satisfying a fundamental-order upper bound and a weighted source condition, we derive the corresponding Riesz decomposition and retain its nonnegative singular term in an off-center Kelvin moving-spheres argument. In the range determined by two nonnegative Kelvin weights, this yields radial symmetry and strict radial monotonicity. We also construct and classify positive radial homogeneous singular solutions in the corresponding convergence regime. If a nonzero positive radial homogeneous scaling limit is independently known to exist and to solve the limiting equation, then its coefficient is uniquely determined.

math.AP↗

arXiv2025

Nonlinear Dirac equations on noncompact quantum graphs with potentials: Multiplicity and Concentration

In this paper, we study the existence and multiplicity of solutions to the following class of nonlinear Dirac equations (NLDE) on noncompact quantum graphs: \[ -i\,\varepsilon c\,\sigma_1\,\partial_x u + m c^2 \sigma_3 u + V(x)\,u = f(|u|)\,u, \quad x\in \mathcal{G}, \tag{P} \] where $V:\mathcal{G}\to\mathbb{R}$ and $f:\mathbb{R}\to\mathbb{R}$ are continuous, $\varepsilon>0$ is a semiclassical parameter, $m>0$ denotes the mass, and $c>0$ the speed of light. Here $\sigma_1,\sigma_3$ are Pauli matrices, and $\mathcal{G}$ is a noncompact quantum graph. We prove that when $\varepsilon$ is sufficiently small, the number of solutions to $(P)$ is at least the number of global minima of $V$. Moreover, these solutions exhibit semiclassical concentration: as $\varepsilon\to0$, their concentration points approach the set of global minima of $V$.

math.AP↗

arXiv2025

Nonrelativistic limit of bound-state solutions for nonlinear Dirac equation on noncompact quantum graphs

In this paper, we investigate the nonrelativistic limit and qualitative properties of bound-state solutions for the nonlinear Dirac equation (NLDE) defined on noncompact quantum graphs: \[ -i c \frac{d}{d x} \sigma_1 \psi+m c^2 \sigma_3 \psi-\omega \psi=g(|\psi|) \psi, \quad \text { in } \mathcal{G} \] where $ g : \mathbb{R}\rightarrow\mathbb{R} $ is a continuous nonlinear function, $ c>0 $ represents the speed of light, $ m>0 $ is the particle's mass, $ \omega\in\mathbb{R} $ is related to the frequency, $ \sigma_1 $ and $ \sigma_3 $ denote the Pauli matrices, and $\mathcal{G}$ is a noncompact quantum graph. We establish the existence of bound-state solutions to the NLDE on $\mathcal{G}$, and prove that these solutions converge toward the corresponding bound-state solutions of a nonlinear Schr\"odinger equation (NLS) in the nonrelativistic limit (i.e., as the speed of light $ c \to \infty $) for particles of small mass. Furthermore, we prove uniform boundedness and exponential decay properties of the NLDE solutions, uniformly in $ c $, thereby offering insight into their asymptotic behavior.

math.AP↗

arXiv2025

Nonlocal problems with Hardy-Littlewood-Sobolev critical exponent and Hardy potential

We are concerned with a Brezis-Nirenberg type problem for a critical Choquard equation, in the sense of Hardy-Littlewood-Sobolev inequality, and with the Hardy potential in a smooth bounded domain. By exploiting variational methods we obtain existence results, which extend to different perturbation terms. Some estimates of independent interest about a nonlocal minimization problem are also derived.

math.AP↗

arXiv2025

On nonlinear elliptic problems with Hardy-Littlewood-Sobolev critical exponent and Sobolev-Hardy critical exponent

In this paper we use variational methods to establish the existence of solutions for a class of nonlinear elliptic problems involving a combined convolution-type and Hardy nonlinearity with subcritical and critical growth.

math.AP↗

arXiv2025

Symmetry and monotonicity of singular solutions for the Hartree equation

In this paper we are concerned with positive singular solutions of the following nonlocal Hartree equation $$-\Delta u\!=\Big( \int_{\mathbb{R}^N\setminus \Gamma}\frac{F(u(y))}{|x-y|^\mu}dy \Big)f (u(x)), \quad x\in \mathbb{R}^N\setminus\Gamma,$$ where $F$ is the primitive of $f$ and $\Gamma$ is the singular set. Under suitable assumptions, we prove that $u$ is symmetric and monotone with respect to the singular set by using moving plane methods. Furthermore, we complement this study by showing the existence, for a model problem, of a singular solution with the desired properties.

math.AP↗