arXiv subjects

Zhipeng Yang

Publications and source records attributed to Zhipeng Yang.

At least 19 recordsLinked to original sources

arXiv2026

Symmetry and Monotonicity for Subcritical Fractional Hartree Equations with Singular Sets of Zero Capacity

We study positive weak singular solutions of the subcritical fractional Hartree equation \[ (-Δ)^s u = \left( \int_{\mathbb{R}^N\setminusΓ} \frac{u^p(y)}{|x-y|^μ}d y \right)u^q \quad\text{in }\mathbb{R}^N\setminusΓ, \] where $0 2s$, $0<μ<N$, and $Γ\subset\{x_1=0\}$ is compact with zero $(s,2)$-capacity. For solutions with a nonremovable singular point on $Γ$, we prove hyperplane symmetry and strict transverse monotonicity under global Lebesgue integrability conditions. For $N\ge2$, we also prove these conclusions under upper bounds on $p$ and $q$, using only the integrability required for weak solutions. We also establish local regularity and construct solutions with logarithmic singularities along a line segment.

math.AP↗

arXiv2026

Uniqueness and nondegeneracy for a critical Choquard equation with Hardy potential

We study the critical Choquard equation \[ -Δu-μ|x|^{-2}u =\bigl(|\cdot|^{2-N}*|u|^p\bigr)|u|^{p-2}u \qquad\text{in }\mathbb R^N, \] where $|\cdot|^{2-N}$ is the Newton kernel, $p=\frac{N+2}{N-2}$, and $0<μ<\frac{(N-2)^2}{4}$. For every $N\ge3$, we prove that positive solutions in $D^{1,2}(\mathbb R^N)$ are radially symmetric about the origin, unique up to dilation, and nondegenerate. Our proof uses normalized logarithmic derivatives to obtain a cooperative system. The resulting monotonicity principle underlies both the uniqueness and nondegeneracy arguments, and remains effective in the subquadratic range $1<p<2$, corresponding to $N\ge7$. Consequently, the linearized quadratic form has Morse index one and the sharp inequality admits a global quadratic remainder estimate. The best constant is strictly concave in the Hardy parameter on $\bigl[0,\frac{(N-2)^2}{4}\bigr)$, is of class $C^2$ with strictly negative second derivative in the interior, and satisfies two-sided bounds that agree to first order at $μ=0$.

math.AP↗

arXiv2026

On symmetry and monotonicity of singular solutions to critical fractional Hartree equations

This paper continues the study initiated in \cite{GuJevnikarYang2026b} of fractional Hartree equations with zero capacity singular sets. We are concerned here with the critical case, i.e. \[ (-Δ)^s u=\left(\int_{\mathbb{R}^N\setminusΓ} \frac{u^{2_{μ,s}^*}(y)}{|x-y|^μ}\,d y\right)u^{2_{μ,s}^*-1} \quad\text{in }\mathbb{R}^N\setminusΓ, \] where $N\ge2$, $s\in(0,1)$, $μ\in(0,N)$ and $2_{μ,s}^*$ is the critical exponent associated to this class of problems. We prove symmetry and monotonicity with respect to the singular set $Γ$. The Kelvin transforms centered at regular points provide sufficient decay at infinity for both the transformed solutions and their associated Hartree coefficients, without imposing global integrability assumptions. Combined with a moving plane argument and a comparison estimate in critical Lebesgue spaces, this allows the moving plane method to cover the full interaction range.

math.AP↗

arXiv2026

Ground state uniqueness and quantitative stability for the mass-critical Hartree equation

We prove that the ground state of the mass-critical Hartree equation is unique up to phase rotations and translations and nondegenerate in every dimension $d\ge3$. This makes the recent conditional $L^2$ classification of minimal-mass blow-up and non-scattering solutions unconditional; we also give a direct $H^1$ proof of the finite-time blow-up classification. We further establish a global quadratic stability estimate for the sharp Hartree interpolation inequality. For the optimal stability constant, we rule out vanishing of minimizing sequences and describe their possible loss of compactness.

math.AP↗

arXiv2026

Nonrelativistic limit of nonlinear Dirac ground states on noncompact quantum graphs

We study the nonrelativistic limit of nonlinear Dirac ground states on finite noncompact metric graphs with Kirchhoff-type conditions. For a nondecreasing nonlinearity of arbitrary finite superquadratic growth, a sufficiently long pendant path guarantees ground states whose nonrelativistic limits are Schrödinger ground states. Every choice of these states is relatively compact in $H^1$ and satisfies a uniform exponential decay estimate. On general graphs, we prove a compactness alternative: loss of compactness requires at least the action of a full-line ground state. Translated line profiles show that the loss constants are sharp. The proof combines an $H^1$ estimate in terms of the action with convergence of Dirac fiber maxima to Schrödinger ray maxima. For power nonlinearities, we give an integral criterion in terms of the dimensionless pendant length and obtain the same conclusions for arbitrary edge lengths on four graph topologies.

math.AP↗

arXiv2026

Existence and concentration of multiple solutions for the fractional logarithmic Schrödinger-Poisson system via the penalization method

We study positive solutions of the fractional logarithmic Schrödinger-Poisson system \[ \begin{cases} \varepsilon^{2α}(-Δ)^αu+V(x)u+ϕu=u\log u^2+|u|^{q-2}u, & \text{in } \mathbb{R}^3, \varepsilon^{2α}(-Δ)^αϕ=u^2, & \text{in } \mathbb{R}^3, \end{cases} \] where $\varepsilon>0$, $3/4<α<1$, and $4<q<6/(3-2α)$. We assume that the potential is continuous and nonnegative and attains its global minimum inside a bounded domain, with strictly larger values on the boundary. We work in the intersection of a weighted fractional Sobolev space and an Orlicz space, so that the logarithmic energy is finite and the associated functional is of class $C^1$. For sufficiently small $\varepsilon$, we obtain at least as many distinct positive solutions as the relative Lusternik-Schnirelmann category of the minimum set in a small neighborhood. The global maximum points of these solutions approach that minimum set as $\varepsilon\to0$. The proof combines penalization, an autonomous Nehari-Pohozaev argument, a local natural constraint, and category theory.

math.AP↗

arXiv2026

Multiplicity of bound states for a class of nonlinear Dirac equations on generalized star graphs with potentials

In this paper we investigate the existence and multiplicity of bound states for a class of nonlinear Dirac equations on a generalized star graph $\mathcal{G}$: \[ -ic\,σ_1\,\partial_xu+mc^2σ_3u+V(x)u=f(|u|)u,\qquad x\in\mathcal{G}, \] where $V:\mathcal{G}\to\mathbb{R}$ is continuous, $f\in C^1([0,\infty))$, $m>0$, $c>0$, and $σ_1,σ_3$ are the Pauli matrices. Under suitable monotonicity and growth assumptions on $f$, a uniform spectral gap and sufficiently long constant-potential intervals on $k$ terminal edges yield at least $2^k-1$ bound states, distinct up to a constant phase. The constant potentials on these intervals may differ. The construction uses half-line solutions and a uniform inverse estimate. For potentials tending to a constant at infinity, we prove compactness below the full-line ground level. In the flat-well setting, the solutions localized near individual terminal vertices lie below this level and provide a ground state through the generalized Nehari minimization. In particular, one localized solution is obtained for each minimum component containing a long terminal interval.

math.AP↗

arXiv2026

Isolated Singularities for Fractional Hartree Equations

We study isolated singularities of positive solutions to a fractional Hartree equation with Riesz interaction, \[ (-Δ)^s u = \left( \int_{\mathbb{R}^N\setminus\{0\}} \frac{u^p(y)}{|x-y|^μ}\,dy \right)u^q \quad \text{in } \mathbb{R}^N\setminus\{0\}. \] The puncture changes the passage from the differential equation to its integral form: a fractional fundamental solution term may occur at the singular point. Under an upper bound at the scale of the fundamental solution and a weighted integrability condition on the source, we derive a Riesz decomposition with a nonnegative singular term. Under the same assumptions, if in addition the solution blows up at the puncture, $p,q\ge1$, and both Kelvin weights are nonnegative, an off-center moving spheres argument yields radial symmetry and strict radial decrease. We also classify positive radial homogeneous singular solutions under the corresponding convergence conditions. If a nonzero positive radial homogeneous scaling limit exists and solves the limiting equation, this classification uniquely determines its coefficient.

math.AP↗

arXiv2026

Sharp stability for mixed bubbles on hyperbolic space

We prove sharp residual stability estimates for the critical Poincaré-Sobolev equation near sums of one positive hyperbolic solution and one projected Euclidean bubble in dimensions at least four. The estimates are uniform in the relative positions of the centers. The sharp rate is linear except in dimension six, where it contains a square root of a logarithm. For nonnegative functions, the same estimates hold under an energy constraint near the sum of the two profile energies for $λ$ outside a finite set, and an additional mass constraint removes this restriction. For arbitrary finite numbers of profiles, we identify and realize all nonnegative Palais-Smale levels. We characterize the parameters at which different pairs of profile numbers have the same energy and show that the limiting energy and mass determine both profile numbers.

math.AP↗

arXiv2026

A sharp spectral multiplier theorem on the Engel group

We prove a spectral multiplier theorem for the standard sub-Laplacian on the Engel group under a scale-invariant local $H^s$ condition with $s>2$. The multiplier operators $F(\mathcal L)$ are of weak type $(1,1)$ and bounded on $L^p$ for every $1<p<\infty$. Together with the known lower bound, this identifies the sharp threshold as $2$, half the topological dimension. The proof combines a weighted Plancherel estimate for quartic oscillators with a decomposition into rank-two spectral subspaces, which controls parameter derivatives uniformly as one well escapes to infinity. For fixed smooth multipliers compactly supported in $(0,\infty)$, we also determine the leading term and first correction for the localized kernels. Whether the weak type $(1,1)$ estimate holds under the local $H^2$ condition remains open.

math.AP↗

arXiv2026

Extremals and stability for the Folland-Stein inequality on H-type groups

We classify all real extremals of the sharp $L^2$ Folland-Stein inequality on H-type groups and prove a global Bianchi-Egnell stability estimate. The linearized kernel at the standard bubble consists exactly of its translation and dilation derivatives. We also obtain explicit bounds for the spectral gap above the symmetry modes and characterize the optimal coefficient in the local stability asymptotics. The results apply to arbitrary orthogonal Clifford modules.

math.AP↗

arXiv2026

Strict ground-level gaps on long-pendant generalized star graphs for nonlinear Dirac equations

We study strict ground-level gaps for autonomous nonlinear Dirac equations on noncompact metric graphs. The relevant level at infinity is the autonomous full-line level $d_λ^\infty=d_{λ,\mathbb R}$, since a concentrating sequence escaping along a half-line sees the real line after recentering far from all vertices. We prove that, uniformly for $λ$ in a compact subinterval $I\Subset(-mc^2,mc^2)$, the presence of a sufficiently long pendant path in the essential reduction of the graph implies \[ d_λ(\mathcal G)<d_λ^\infty. \] The proof uses a half-line restriction of a symmetric full-line ground state, uniform exponential decay, resolvent localization for massive Dirac-Kirchhoff spectral projections, and a compactness argument for generalized Nehari fibres. We also identify elementary graph classes for which no such gap can hold: pure transmission graphs whose essential reduction is isometric to the full line satisfy $d_λ(\mathcal G)=d_λ^\infty$.

math.AP↗

arXiv2026

A Sharp Mass Threshold for Fractional Choquard Equations with Lower Hardy-Littlewood-Sobolev and $L^2$-Critical Terms

For $\frac12\le s<1$, we study a fractional Choquard equation with prescribed mass and nonlinearities at the lower Hardy-Littlewood-Sobolev and $L^2$-critical exponents. The sharp Hardy-Littlewood-Sobolev and Choquard Gagliardo-Nirenberg inequalities determine an explicit critical mass $a_*$. For $0 a_*$, the energy is unbounded from below on the mass sphere, while the Pohozaev set is nonempty.

math.AP↗

arXiv2026

Sub-Laplacians on Compact Lie Groups: Heat Kernels, Distance, and Zeta Determinants

We study heat kernels, sub-Riemannian distances, and spectral zeta functions of sub-Laplacians determined by closed connected subgroups of compact Lie groups. Combining Hall's inversion formula with the affine lattice expansion of the compact group heat kernel, we derive a Cartan integral representation involving the group's exponential lattice. For two-step compact Lie pairs, the full algebraic small-time heat trace expansion is determined, up to an exponentially small remainder, by two explicit constants $C_{G,L}$ and $β_{G,L}$. This expansion determines all heat coefficients, the poles and residues of the reduced spectral zeta function, and its values at nonpositive integers. For the transvective symmetric subclass, we prove uniform vertical asymptotics for the Carnot-Carathéodory distance; the leading coefficient $\mathfrak F_{G,K}(Z)$ is the attained minimum of a finite-dimensional singular value problem. For simply connected two-step pairs, we obtain an exact decomposition of the zeta-regularized determinant into local, lattice, and spectral terms, with exponential truncation estimates. We specialize these results to block subgroups of $\mathrm{SU}(N)$, recovering the classical $\mathrm{SU}(2)$ and CR sphere spectra.

math.CA↗

arXiv2026

Topological Effects on Bubbling in the Critical Dirichlet Problem on Hyperbolic Domains: $3\leq N \leq5$

Let $Ω\Subset\mathbb H^N$, $N\in\{3,4,5\}$, be a bounded connected $C^2$ domain. We prove that the pure critical Dirichlet problem \[ -Δ_{\mathbb H}u=u^{\frac{N+2}{N-2}} \quad\text{in }Ω, \qquad u=0 \quad\text{on }\partialΩ \] admits a positive solution whenever $H_d(Ω;\mathbb F_2)\neq0$ for some $1\le d\le N-1$. This gives a hyperbolic Bahri-Coron theorem for $3\le N\le5$ under $C^2$ boundary regularity. Under conformal reduction, the hyperbolic geometry produces a positive potential and leads to dimension-dependent bubbling mechanisms. In dimension three, the required energy drop follows from the balance between diagonal corrections and pair interactions at fixed large multiplicity. In dimensions four and five, it is obtained at the matched scale through a normalized defect estimate and an all-pairs source-transfer bound. A unified Thom-barycenter construction converts these analytic estimates into the topological contradiction. Thus nontrivial domain topology forces existence despite critical loss of compactness and the additional geometric potential.

math.AP↗

arXiv2026

N2NMatcher: Towards Inlining-Resilient Binary Decomposition and Module Matching

Program-level Binary Code Similarity Analysis (BCSA) aims to identify semantically similar code regions across binary programs, serving as a fundamental technique for software plagiarism detection, vulnerability search, and malware analysis. Existing approaches often decompose binaries into modules following the structure of function call graphs (FCGs) and then match these modules by their contained functions. However, function inlining changes both FCG structures and binary function semantics, making existing decomposition and module matching methods less effective. In this work, we propose N2NMatcher, an inlining-resilient framework for binary decomposition and module matching. We first conduct an empirical study to examine whether binaries still contain functions that provide stable module boundaries across compilation settings. N2NMatcher learns to predict such functions as anchors using a hierarchical graph neural network that encodes binary ACFG-FCG representations built from opcode sequences, control-flow structures, and FCG calling context. It then performs anchor-bounded decomposition and matches the generated modules using learned module graph embeddings. Experimental results show that N2NMatcher improves both the decomposition quality and module matching accuracy compared to existing works, enabling more effective program-level BCSA.

cs.SE↗

arXiv2026

Word Recovery in Large Language Models Enables Character-Level Tokenization Robustness

Large language models (LLMs) trained with canonical tokenization exhibit surprising robustness to non-canonical inputs such as character-level tokenization, yet the mechanisms underlying this robustness remain unclear. We study this phenomenon through mechanistic interpretability and identify a core process we term word recovery. We first introduce a decoding-based method to detect word recovery, showing that hidden states reconstruct canonical word-level token identities from character-level inputs. We then provide causal evidence by removing the corresponding subspace from hidden states, which consistently degrades downstream task performance. Finally, we conduct a fine-grained attention analysis and show that in-group attention among characters belonging to the same canonical token is critical for word recovery: masking such attention in early layers substantially reduces both recovery scores and task performance. Together, our findings provide a mechanistic explanation for tokenization robustness and identify word recovery as a key mechanism enabling LLMs to process character-level inputs.

cs.CL↗

arXiv2026

Normalized Solutions and Semiclassical Concentration for Upper-Critical Fractional Choquard Equations

We study a fractional Choquard equation with an upper-critical Hartree term, an $L^2$-supercritical Hartree perturbation, and a semiclassical potential under a prescribed $L^2$-mass constraint. The potential is bounded and nonnegative, has a nonempty zero set, and has a positive lower limit at infinity. For every prescribed mass and all sufficiently small semiclassical parameters, we prove the existence of a pair of normalized solutions $\pm u_\varepsilon$ with a negative Lagrange multiplier. The proof combines a strict energy bound below the critical one-bubble level, compactness modulo translations for the autonomous ground-state set, a simultaneous cutoff of both Hartree terms, and a localized constrained mountain-pass argument. Moreover, suitable translates of $u_\varepsilon$ converge strongly in $H^s(\mathbb{R}^N)$ to a positive autonomous ground state, and the corresponding concentration points approach the zero set of the potential as $\varepsilon\to0$.

math.AP↗