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Huyuan Chen

Publications and source records attributed to Huyuan Chen.

At least 19 recordsLinked to original sources
arXiv2026

On finite-energy solutions of Kazan-Warner equations on the lattice graph

We investigate finite-energy solutions to Kazdan-Warner type equations in 2-dimensional integer lattice graph $$ - Δu= \varepsilon e^{κu} +βδ_0\quad {\rm in}\ \mathbb{Z}^2,$$ where $\varepsilon=\pm1$, $κ>0$ and $β\in\mathbb{R}$. When $\varepsilon=1$, we prove the existence of a continuous family of finite-energy solutions for some parameter $κ$. This provides a resolution of the open problem on the existence of finite-energy solutions to the Liouville equation. When $\varepsilon=-1$ and $β>\frac{4π}κ$, we prove that the set of finite-energy solutions exhibits a layer structure. Moreover, we derive the extremal solution in this case.

math.AP↗
arXiv2026

Sharp Lifespan Dichotomies and Threshold Phenomena for Semilinear Heat Equations Driven by the Logarithmic Laplacian

We investigate nonnegative mild solutions of $\partial_t u+(-\Delta)^{\ln}u=f(u)$ in $(0,T)\times\mathbb R^N$, with initial datum $u(0,\cdot)=\mu u_0$, $\mu>0$. Unlike the classical and fractional heat semigroups, the positive logarithmic heat kernel exists only for $0<t<N/2$, and the corresponding linear evolution may become singular at its terminal time, with lifespan and growth depending on the spatial decay of $u_0$. The behavior of $f$ near zero determines local solvability: if $\int_{0^+} d\sigma/f(\sigma)<\infty$, then no finite nonnegative solution exists on any positive time interval. Under suitable assumptions on $u_0$ and $f$, we establish well-posedness for the nonintegrable logarithmic heat kernel. If $f$ has at most global linear growth, the nonlinear solution attains the full linear lifespan, while the Osgood condition at infinity implies that the maximal existence time tends to zero as $\mu\to\infty$. We further distinguish slow-decay, fast-decay, and critical-tail initial data. In the noncritical regimes, a weighted Osgood tail condition yields blow-up strictly before the linear terminal time; if it fails, an amplitude threshold occurs under additional assumptions on $f$. In the critical regime, the dividing power is $3/2$: a square-root weighted Osgood condition yields premature blow-up, while its failure again leads to an amplitude threshold. Finally, we obtain terminal-time blow-up estimates and sharp rates for power nonlinearities.

math.AP↗
arXiv2026

On positive solutions of Lane-Emden equations on the integer lattice graphs

In this paper, we investigate the existence and nonexistence of positive solutions to the Lane-Emden equations $$ -Δu = Q |u|^{p-2}u $$ on the $d$-dimensional integer lattice graph $\mathbb{Z}^d$, as well as in the half-space and quadrant domains, under the zero Dirichlet boundary condition in the latter two cases. Here, $d \geq 2$, $p > 0$, and $Q$ denotes a Hardy-type positive potential satisfying $Q(x) \sim (1+|x|)^{-α}$ with $α\in [0, +\infty]$. \smallskip We identify the Sobolev super-critical regions of the parameter pair $(α, p)$ for which the existence of positive solutions is established via variational methods. In contrast, within the Serrin sub-critical regions of $(α, p)$, we demonstrate nonexistence by iteratively analyzing the decay behavior at infinity, ultimately leading to a contradiction. Notably, in the full-space and half-space domains, there exists an intermediate regions between the Sobolev critical line and the Serrin critical line where the existence of positive solutions remains an open question. Such an intermediate region does not exist in the quadrant domain.

math.AP↗
arXiv2026

Maximal hypersurfaces with prescribed light-like cones in Lorentz-Minkowski space

The purpose in this paper is to study the maximal hypersurfaces with multiple light-cones in Lorentz-Minkowski space by considering the weak solutions to the mean curvature equation with multiple Dirac masses. Such solutions are constructed via an approximation procedure, using regular solutions with smooth sources that converge weakly to the Dirac measures.

math.AP↗
arXiv2026

The Fractional-Logarithmic Laplacian:Fundamental Properties and Eigenvalues

In this paper, we introduce, for the first time, the fractional--logarithmic Laplacian \( (-Δ)^{s+\log} \), defined as the derivative of the fractional Laplacian \( (-Δ)^t \) at \( t=s \). It is a singular integral operator with Fourier symbol \( |ξ|^{2s}(2\ln|ξ|) \), and we prove the pointwise integral representation \[ (-Δ)^{s+\log}u(x) = c_{n,s}\,\mathrm{PV}\!\int_{\mathbb{R}^n} \frac{u(x)-u(y)}{|x-y|^{n+2s}}\bigl(-2\ln|x-y|\bigr)\,dy + b_{n,s}(-Δ)^s u(x), \] where \( c_{n,s} \) is the normalization constant of the fractional Laplacian and \( b_{n,s}:=\frac{d}{ds}c_{n,s}.\) We also establish several equivalent formulations of \( (-Δ)^{s+\log} \), including the singular-integral representation, the Fourier-multiplier representation, the spectral-calculus definition, and an extension characterization. We develop the associated functional framework on both \( \mathbb{R}^n \) and bounded Lipschitz domains, introducing the natural energy spaces and proving embedding results. In particular, we obtain a compact embedding at the critical exponent \( 2_s^*=\frac{2n}{n-2s},\) a phenomenon that differs from the classical Sobolev and fractional Sobolev settings. We further study the Poisson problem, proving existence and \( L^\infty \)-regularity results. We then investigate the Dirichlet eigenvalue problem and establish qualitative spectral properties. Finally, we derive a Weyl-type asymptotic law for the eigenvalue counting function and for the \( k \)-th Dirichlet eigenvalue, showing that the high-frequency behavior combines the fractional Weyl scaling with a logarithmic growth factor, thereby interpolating between the fractional Laplacian and the logarithmic Laplacian.

math.AP↗
arXiv2026

The Conformal Fractional--Logarithmic Laplacian on the Sphere: Yamabe Problems and Sharp Inequalities

In this paper, we introduce the conformal fractional--logarithmic Laplacian on the unit sphere, defined as the derivative of the conformal fractional Laplacian with respect to the order parameter \(s\in(0,1)\). We investigate its fundamental analytic and spectral properties, including its relation to the conformal logarithmic Laplacian, its spectral representation, and the explicit form of its eigenvalues and eigenfunctions. We further establish its conformal covariance law and derive the associated Yamabe-type equation, proving its equivalence to the corresponding conformal equation in \(\mathbb R^N\) through stereographic projection. Finally, we apply this framework to sharp Sobolev-type inequalities, recovering the sharp logarithmic Sobolev inequality, revealing the failure of a naive fractional--logarithmic analogue, and establishing new sharp fractional--logarithmic inequalities.

math.AP↗
arXiv2026

Existence and nonexistence of solutions for fractional elliptic equations arising from closed MEMS model

The objective of our paper is to investigate fractional elliptic equations of the form $(-Δ)^s u=\frac{λ}{(a-u)^2}$ within a bounded domain $Ω$, subject to zero Dirichlet boundary conditions. Here, $s\in(0,1)$, $λ>0$, and the function $a$ vanishes at the boundary while satisfying additional conditions. This problem originates from Micro-Electromechanical Systems (MEMS) devices, particularly when the elastic membrane makes contact with the ground plate at the boundary. We establish both existence and nonexistence results, illustrating how the boundary decay of the membrane influences the solutions and pull-in voltage.

math.AP↗
arXiv2026

Existence and Convergence of Least-Energy Solutions Involving the Logarithmic Schrödinger Operator

In this paper, we study critical semilinear nonlocal elliptic equations involving the logarithmic Schrödinger operator and its fractional pseudo-relativistic counterpart, both arising in quantum models with nonlocal and relativistic effects. We first establish the existence, uniqueness, and regularity of weak solutions to equations involving the logarithmic operator \((I - Δ)^{\ln}\) with subcritical logarithmic nonlinearities. We then investigate a Brezis--Nirenberg-type problem involving the fractional pseudo-relativistic Schrödinger operator \((I - Δ)^s\), and prove the existence of least-energy solutions under both subcritical and critical nonlinearities. In particular, we show that these least-energy solutions converge, up to a subsequence, to a nontrivial least-energy solution of the limiting problem as \(s \to 0^+\). Our approach relies on variational methods, including the geometry of the Nehari manifold, uniform positive lower bounds, mountain-pass structure, the Palais--Smale condition, and delicate asymptotic analysis.

math.AP↗
arXiv2026

On Poincar\'e-Sobolev level involving fractional GJMS operators on hyperbolic space

This paper is devoted to a qualitative analysis of the Poincar\'e--Sobolev level associated with the fractional GJMS operators \(\mathcal{P}_s\) \(\bigl(s\in(0,\tfrac n2)\setminus\mathbb N\bigr)\) on the hyperbolic space \(\mathbb H^n\). In contrast to the integer-order case, when \(s\notin\mathbb N\) the operator \(\mathcal{P}_s\) does not enjoy the conformal covariance that allows one, in the upper half-space or ball model, to relate it to the Euclidean fractional Laplacian \((-\Delta)^s\); this link is crucial for importing Euclidean theory. We therefore introduce \(\widetilde{\mathcal{P}}_s\) (\(s>0\)), which is conformally related to \((-\Delta)^s\). Our purpose in the paper is to analyze the monotonicity, attainability, and strict-gap regions of the Poincar\'e--Sobolev levels associated with \(\mathcal{P}_s\) and \(\widetilde{\mathcal{P}}_s\). First, we reinterpret the Brezis--Nirenberg problem through the lens of Poincar\'e--Sobolev levels, connecting earlier results for the Euclidean Laplacian and for operators \(\mathcal{P}_k\) on \(\mathbb H^n\) with integer \(k\in(0,\tfrac n2)\). We then establish new, explicit lower bounds for the Hardy term in fractional Hardy--Sobolev--Maz'ya inequalities involving both \(\mathcal{P}_s\) and \(\widetilde{\mathcal{P}}_s\). By applying the concentration--compactness principle together with a detailed analysis of the strict-gap regions for the Poincar\'e--Sobolev levels, we prove the existence of solutions to the Brezis--Nirenberg problem on \(\mathbb H^n\) for both operators. Finally, combining the Hardy lower bounds with criteria for attainability, we obtain a complete characterization of the Poincar\'e--Sobolev levels \(H_{n,s}\) and \(\widetilde H_{n,s}\).

math.AP↗
arXiv2025

Classification and symmetry of global solutions for nonlinear elliptic equations with mixed reaction terms

In this paper, we describe the set of all positive distributional $C^1(\mathbb R^N\setminus \{0\})$-solutions of elliptic equations with mixed reaction terms of the form $$ \mathbb L_{ρ,λ,τ}[u]:= Δu-(N-2+2ρ) \frac{x\cdot \nabla u}{|x|^2} +λ\frac{u^τ|\nabla u|^{1-τ}}{|x|^{1+τ}}=|x|^θu^q\quad \mbox{in } \mathbb R^N\setminus \{0\}, $$ where $ρ,λ, θ\in \mathbb R$ are arbitrary, $N\geq 2$, $q>1$ and $τ\in [0,1)$. Defining $β=(θ+2)/(q-1)$ and $f_{ρ,λ,τ}(t)=t\left(t+2ρ\right) +λ|t|^{1-τ}$ for $t\in \mathbb R$, we show that the equation has positive solutions if and only if $f_{ρ,λ,τ}(β)>0$. Under this condition, we provide existence and the exact asymptotic behaviour near zero and at infinity for all positive solutions. We obtain that all such solutions are radially symmetric. When $θ<-2$ and $ρ,λ\in \mathbb R$, we also find the precise local behaviour near zero for all positive solutions of our equation in $Ω\setminus \{0\}$, where $Ω$ is an open set containing $0$. By introducing the second term in $\mathbb L_{ρ,λ,τ}[\cdot]$ with $ρ\in \mathbb R$, we reduce the study to $θ<-2$ via a modified Kelvin transform. We reveal new and surprising phenomena compared with the work of C\^ırstea and Fărcăşeanu (2021), where $ρ=(2-N)/2$ and $τ=1$.

math.AP↗
arXiv2025

Existence of solutions to elliptic equations involving regional fractional Laplacian with order $(0,\frac12]$

Our purpose of this paper is to investigate positive solutions of the elliptic equation with regional fractional Laplacian $$ ( - Δ)_{B_1}^s u +u= h(x,u) \quad {\rm in} \ \, B_1,\qquad u\in C_0(B_1), $$ where $( - Δ)_{B_1}^s$ with $s\in(0,\frac12]$ is the regional fractional Laplacian and $h$ is the nonlinearity. Ordinarily, positive solutions vanishing at the boundary are not anticipated to be derived for the equations with regional fractional Laplacian of order $s\in(0,\frac12]$. Positive solutions are obtained when the nonlinearity assumes the following two models: $h(x,t)=f(x)$ or $h(x,t)=h_1(x)\, t^p+ εh_2(x)$, where $p>1$, $ε>0$ small and $f, h_1, h_2$ are Hölder continuous, radially symmetric and decreasing functions under suitable conditions.

math.AP↗
arXiv2025

On positive solutions of critical semilinear equations involving the Logarithmic Laplacian

In this paper, we classify the solutions of the critical semilinear problem involving the logarithmic Laplacian $$(E)\qquad \qquad\qquad\qquad\qquad \mathcal{L}_Δu= k u\log u,\qquad u\geq0 \quad \ {\rm in}\ \ \mathbb{R}^n, \qquad\qquad\qquad\qquad\qquad\qquad$$ where $k\in(0,+\infty)$, $\mathcal{L}_Δ$ is the logarithmic Laplacian in $\mathbb{R}^n$ with $n\in\mathbb{N}$, and $s\log s=0$ if $s=0$. When $k=\frac4n$, problem $(E)$ only has the solutions with the form $$u_{\tilde x,t}(x)=β_n \Big(\frac{t}{t^2+|x-\tilde x|^2)}\Big)^{\frac{n}{2}}\quad \text{ for any $t>0$, $\tilde x\in\mathbb{R}^n$},$$ where $n\in\mathbb{N}$, $β_n=2^{\frac n2} e^{\frac n2ψ(\frac n2) }>0$. When $k\in(0,+\infty)\setminus\{\frac 4n\}$, problem $(E)$ has no any positive solution.

math.AP↗
arXiv2024

On m-order logarithmic Laplacians and related propeties

In this article, we study $m$-order logarithmic Laplacian $\mathcal{L}_m$, which is a singular integro-differential operator with symbol $\big(2\ln |\cdot|\big)^m$ by the Fourier transform. With help of these logarithmic Laplacians, we build the $n$-th order Taylor expansion for fractional Laplacian with respect to the order and the Riesz operators: for $u \in C^\infty_c(\mathbb{R}^N)$ and $x \in \mathbb{R}^N$, $$ (-Δ)^s u (x) = u(x) + \sum^{n}_{m=1} \frac{s^m}{m!}\mathcal{L}_mu(x) + o(s^n) \quad {\rm as}\ \, s\to 0^+ $$ and $$ \big(Φ_s\ast u\big)(x) = u(x) + \sum^{n}_{m=1}(-1)^m\frac{s^m}{m!}\mathcal{L}_mu(x) + o(s^n) \quad {\rm as}\ \, s\to 0^+, $$ where $ (-Δ)^s$ is the $s$-fractional Laplacian, $Φ_s\ast u$ is $s$-order of Riesz operator with the form $Φ_s(x)=κ_{N,s}|x|^{2s-N}$ in $\mathbb{R}^N\setminus\{0\}$. Moreover, we analyze qualitative properties of these operators based on the order $m$, such as basic regularity and the Dirichlet eigenvalues.

math.AP↗
arXiv2024

The Cauchy problem associated to the logarithmic Laplacian with an application to the fundamental solution

Let $\lnlap$ be the logarithmic Laplacian operator with Fourier symbol $2\ln |ζ|$, we study the expression of the diffusion kernel which is associated to the equation $$\partial_tu+ \lnlap u=0 \ \ {\rm in}\ \, (0,\tfrac N2) \times \R^N,\quad\quad u(0,\cdot)=0\ \ {\rm in}\ \, \R^N\setminus \{0\}.$$ We apply our results to give a classification of the solutions of $$\left\{ \arraycolsep=1pt \begin{array}{lll} \displaystyle \partial_tu+ \lnlap u=0\quad \ &{\rm in}\ \ (0,T)\times \R^N\\[2.5mm] \phantom{ \lnlap \ \, } \displaystyle u(0,\cdot)=f\quad \ &{\rm{in}}\ \ \R^N \end{array} \right. $$ and obtain an expression of the fundamental solution of the associated stationary equation in $\R^N$, and of the fundamental solution in a bounded domain, i.e. $$\lnlap u=kδ_0\quad {\rm in}\ \ \cD'(Ω)\quad \text{such that }\,u=0\quad {\rm in}\ \ \R^N\setminusΩ. $$

math.AP↗
arXiv2023

An extension problem for the logarithmic Laplacian

The logarithmic Laplacian on the (whole) N-dimensional Euclidean space is defined as the first variation of the fractional Laplacian of order 2s at s=0 or, alternatively, as a singular Fourier integral operator with logarithmic symbol. While this operator has attracted fastly growing attention in recent years due to its relevance in the study of order-dependent problems, a characterization via a local extension problem on the (N+1)-dimensional upper half-space in the spirit of the Cafferelli-Sivestre extension for the fractional Laplacian has been missing so far. In this paper, we establish such a characterization. More precisely, we show that, up to a multiplicative constant, the logarithmic Laplacian coincides with the boundary-value operator associated with a weighted second-order operator on the upper half-space, which maps inhomogeneous Neumann data to a Robin boundary-value of the corresponding distributional solution with a singular excess term. This extension property of the logarithmic Laplacian leads to a new energy functional associated with this operator. By doubling the extension-variable, we show that distributional solutions of the extension problem are actually harmonic in the (N+2)-dimensional Euclidean space away from the boundary. As an application of these results, we establish a weak unique continuation principle for the (stationary) logarithmic Laplace equation.

math.AP↗
arXiv2023

Improved Lerey inequality and Trudinger-Moser type inequality involving the Leray potential

We obtain three types of results in this paper. Firstly we improve Leray's inequality by providing several types of reminder terms, secondly we introduce several Hilbert spaces based on these improved Leray inequalities and discuss their embedding properties, thirdly we obtain some Trudinger-Moser type inequalities in the unit ball of R2 associated with the norms of these Hilbert spaces where the Leray potential is used. Our approach is based on analysis of radially symmetric functions.

math.AP↗
arXiv2023

Singularities of fractional Emden's equations via Caffarelli-Silvestre extension

We study the isolated singularities of functions satisfying (E) (--$Δ$) s v$\pm$|v| p--1 v = 0 in $Ω$\{0}, v = 0 in R N \$Ω$, where 0 < s < 1, p > 1 and $Ω$ is a bounded domain containing the origin. We use the Caffarelli-Silvestre extension to R + x R N. We emphasize the obtention of a priori estimates, analyse the set of self-similar solutions via energy methods to characterize the singularities.

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