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Huyuan Chen

Publications and source records attributed to Huyuan Chen.

At least 55 records · Page 3Linked to original sources
arXiv2021

Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian

We provide bounds for the sequence of eigenvalues $\{λ_i(Ω)\}_i$ of the Dirichlet problem $$ L_Δu=λu\ \ {\rm in}\ \, Ω,\quad\quad u=0\ \ {\rm in}\ \ \mathbb{R}^N\setminus Ω,$$ where $L_Δ$ is the logarithmic Laplacian operator with Fourier transform symbol $2\ln |ζ|$. The logarithmic Laplacian operator is not positively definitive if the volume of the domain is large enough. In this article, we obtain the upper and lower bounds for the sum of the first $k$ eigenvalues by extending the Li-Yau method and Kröger's method respectively. Moreover, we show the limit of the sum of the first $k$ eigenvalues, which is independent of the volume of the domain. Finally, we discuss the lower and upper bounds of the $k$-th principle eigenvalue, the asymptotic behavior of the limit of eigenvalues.

math.AP↗
arXiv2020

Positive supersolutions for the Lane-Emden system with inverse-square potentials

In this paper, we study the nonexistence of positive supersolutions for the following Lane-Emden system with inverse-square potentials \begin{equation}\label{0} \left\{ \begin{array}{lll} -Δu+\frac{μ_1}{|x|^2} u= v^p \quad {\rm in}\ \, Ω\setminus\{0\},\\[2mm] -Δv+\frac{μ_2}{|x|^2} v= u^q \quad {\rm in}\ \, Ω\setminus\{0\} \end{array} \right. \end{equation} for suitable $p,q>0$, $μ_1,μ_2\geq -(N-2)^2/4$, where $Ω$ is a smooth bounded domain containing the origin in $\mathbb{R}^N$ with $N\geq 3$. Precisely, we provide sharp supercritical regions of $(p,q)$ for the nonexistence of positive supersolutions to system (\ref{0}) in the cases $-(N-2)^2/4\leq μ_1,μ_2<0$ and $-(N-2)^2/4\leq μ_1<0\leq μ_2$. Due to the negative coefficients $μ_1,μ_2$ of the inverse-square potentials, an initial blowing-up at the origin could be derived and an iteration procedure could be applied in the supercritical case to improve the blowing-up rate until the nonlinearities are not admissible in some weighted $L^1$ spaces. In the subcritical case, we prove the existence of positive supersolutions for system (\ref{s 1.1}) by specific radially symmetric functions.

math.AP↗
arXiv2020

The Poisson problem for the fractional Hardy operator: Distributional identities and singular solutions

The purpose of this paper is to study and classify singular solutions of the Poisson problem $$ %\begin{equation}\label{eq 0.1} \left \{ \begin{aligned} {\mathcal L}^s_μu = f \quad\ {\rm in}\ \, Ω\setminus \{0\},\\ u =0 \quad\ {\rm in}\ \, {\mathbb R}^N \setminus Ω %\\ %\liminf_{x \to 0}\:|u(x)| /Φ_μ(x) = k. \end{aligned} \right. $$ for the fractional Hardy operator ${\mathcal L}_μ^s u= (-Δ)^s u +\fracμ{|x|^{2s}}u$ in a bounded domain $Ω\subset {\mathbb R}^N$ ($N \ge 2$) containing the origin. Here $(-Δ)^s$, $s\in(0,1)$, is the fractional Laplacian of order $2s$, and $μ\ge μ_0$, where $μ_0 = -2^{2s}\frac{Γ^2(\frac{N+2s}4)}{Γ^2(\frac{N-2s}{4})}<0$ is the best constant in the fractional Hardy inequality. The analysis requires a thorough study of fundamental solutions and associated distributional identities. Special attention will be given to the critical case $μ= μ_0$ which requires more subtle estimates than the case $μ>μ_0$.

math.AP↗
arXiv2020

Asymptotic behaviors of governing equation of Gauged Sigma model for Heisenberg ferromagnet

In this note, we study weak solutions of equation \begin{equation}\label{eq 00.1} Δu =\frac{4e^u}{1+e^u} -4π\sum^{N}_{i=1}δ_{p_i}+4π\sum^{M}_{j=1}δ_{q_j} \quad{\rm in}\;\; \mathbb{R}^2, \end{equation} where $\{δ_{p_i}\}_{i=1}^N$ (resp. $\{δ_{q_j}\}_{j=1}^M$ ) are Dirac masses concentrated at the points $p_i, i=1,\cdots, N$, (resp. $q_j, i=1,\cdots, M$) %$δ_{p_j}$ is Dirac mass concentrated at the point $p_j$ and $N-M>1$. This equation presents a governing equation of Gauged Sigma model for Heisenberg ferromagnet and we prove that it has a sequence of solutions $u_β$ having behaviors as $-2πβ\ln |x|+O(1)$ at infinity with a free parameter $β\in(2,2(N-M))$, and our concern in this paper is to study the asymptotic behavior's estimates in the extremal case that $β$ near $2$ and $2(N-M)$.

math.AP↗
arXiv2019

Schr{ö}dinger operators with Leray-Hardy potential singular on the boundary

We study the kernel function of the operator u $\rightarrow$ L $μ$ u = --$Δ$u + $μ$ |x| 2 u in a bounded smooth domain $Ω$ $\subset$ R N + such that 0 $\in$ $\partial$$Ω$, where $μ$ $\ge$ -- N 2 4 is a constant. We show the existence of a Poisson kernel vanishing at 0 and a singular kernel with a singularity at 0. We prove the existence and uniqueness of weak solutions of L $μ$ u = 0 in $Ω$ with boundary data $ν$ + k$δ$ 0 , where $ν$ is a Radon measure on $\partial$$Ω$ \ {0}, k $\in$ R and show that this boundary data corresponds in a unique way to the boundary trace of positive solution of L $μ$ u = 0 in $Ω$.

math.AP↗
arXiv2019

The Dirichlet Problem for the Logarithmic Laplacian

In this paper, we study the logarithmic Laplacian operator $L_Δ$, which is a singular integral operator with symbol $2\log |ζ|$. We show that this operator has the integral representation $$L_Δu(x) = c_{N} \int_{\mathbb{R}^N } \frac{ u(x)1_{B_1(x)}(y)-u(y)}{|x-y|^{N} } dy + ρ_N u(x) $$ with $c_N = π^{- \frac{N}{2}} Γ(\frac{N}{2})$ and $ρ_N=2 \log 2 + ψ(\frac{N}{2}) -γ$, where $Γ$ is the Gamma function, $ψ= \frac{Γ'}Γ$ is the Digamma function and $γ= -Γ'(1)$ is the Euler Mascheroni constant. This operator arises as formal derivative $\partial_s \Big|_{s=0} (-Δ)^s$ of fractional Laplacians at $s= 0$. We develop the functional analytic framework for Dirichlet problems involving the logarithmic Laplacian on bounded domains and use it to characterize the asymptotics of principal Dirichlet eigenvalues and eigenfunctions of $(-Δ)^s$ as $s \to 0$. As a byproduct, we then derive a Faber-Krahn type inequality for the principal Dirichlet eigenvalue of $L_Δ$. Using this inequality, we also establish conditions on domains giving rise to the maximum principle in weak and strong forms. This allows us to also derive regularity up to the boundary of solutions to corresponding Poisson problems.

math.AP↗
arXiv2019

On Gaussian curvature equations in $\mathbb{R}^2$ with prescribed non-positive curvature

The purpose of this paper is to study the solutions of $$ Δu +K(x) e^{2u}=0 \quad{\rm in}\;\; \mathbb{R}^2 $$ with $K\le 0$. We introduce the following quantity: $$α_p(K)=\sup\left\{α\in \mathbb{R}:\, \int_{\mathbb{R}^2} |K(x)|^p(1+|x|)^{2αp+2(p-1)} dx<+\infty\right\}, \quad \forall\; p \ge 1.$$ Under the assumption $({\mathbb H}_1)$: $α_p(K)> -\infty$ for some $p>1$ and $α_1(K) > 0$, we show that for any $0 < α< α_1(K)$, there is a unique solution $u_α$ with $u_α(x) = α\ln |x|+ c_α+o\big(|x|^{-\frac{2β}{1+2β}} \big)$ at infinity and $β\in (0,\,α_1(K)-α)$. Furthermore, we show an example $K_0 \leq 0$ such that $α_p(K_0) = -\infty$ for any $p>1$ and $α_1(K_0) > 0$, for which we study the asymptotic behavior of solutions. In particular, we prove the existence of a solution $u_*$ such that $u_* -α_*\ln|x| = O(1)$ at infinity for some $α_* > 0$, but who does not converge to a constant at infinity. This example exhibits a new phenomenon of solutions with logarithmic growth and non-uniform behavior at infinity.

math.AP↗
arXiv2019

Weak solutions of semilinear elliptic equations with Leray-Hardy potential and measure data

We study existence and stability of solutions of (E 1) --$Δ$u + $μ$ |x| 2 u + g(u) = $ν$ in $Ω$, u = 0 on $\partial$$Ω$, where $Ω$ is a bounded, smooth domain of R N , N $\ge$ 2, containing the origin, $μ$ $\ge$ -- (N --2) 2 4 is a constant, g is a nondecreasing function satisfying some integral growth assumption and $ν$ is a Radon measure on $Ω$. We show that the situation differs according $ν$ is diffuse or concentrated at the origin. When g is a power we introduce a capacity framework to find necessary and sufficient condition for solvability.

math.AP↗
arXiv2018

Nonexistence of Positive Supersolution to a Class of Semilinear Elliptic Equations and Systems in an Exterior Domain

In this paper, we primarily consider the following semilinear elliptic equation \begin{eqnarray*} \arraycolsep=1pt\left\{ \begin{array}{lll} \displaystyle -Δu= h(x,u)\quad \ &{\rm in}\ Ω,\\[1.5mm] \phantom{ -Δ} \displaystyle u\ge 0\qquad &{\rm on}\ \partialΩ, \end{array}\right. \end{eqnarray*} where $Ω$ is an exterior domain in $R^N$ with $N\ge 3$, and derive optimal nonexistence results of positive supersolution. Our argument is based on a nonexistence result of positive supersolution of a linear elliptic problem with Hardy potential. We also establish sharp nonexistence results of positive supersolution to an elliptic system.

math.AP↗
arXiv2018

Initial trace of positive solutions to fractional diffusion equation with absorption

In this paper, we prove the existence of an initial trace T u of any positive solution u of the semilinear fractional diffusion equation (H) $\partial$ t u + (--$Δ$) $α$ u + f (t, x, u) = 0 in R * + $\times$ R N , where N $\ge$ 1 where the operator (--$Δ$) $α$ with $α$ $\in$ (0, 1) is the fractional Laplacian and f : R + $\times$ R N $\times$ R + $\rightarrow$ R is a Caratheodory function satisfying f (t, x, u)u $\ge$ 0 for all (t, x, u) $\in$ R + $\times$ R N $\times$ R +. We define the regular set of the trace T u as an open subset of R u $\subset$ R N carrying a nonnegative Radon measive $ν$ u such that lim t$\rightarrow$0 Ru u(t, x)$ζ$(x)dx = Ru $ζ$d$ν$ $\forall$$ζ$ $\in$ C 2 0 (R u), and the singular set S u = R N \ R u as the set points a such that lim sup t$\rightarrow$0 B$ρ$(a) u(t, x)dx = $\infty$ $\forall$$ρ$ \> 0. We study the reverse problem of constructing a positive solution to (H) with a given initial trace (S, $ν$) where S $\subset$ R N is a closed set and $ν$ is a positive Radon measure on R = R N \ S and develop the case f (t, x, u) = t $β$ u p where $β$ \> --1 and p \> 1.

math.AP↗
arXiv2018

On nonhomogeneous elliptic equations with the Hardy-Leray potentials

In this paper, we present some suitable distributional identities of the solutions for nonhomogeneous elliptic equations involving the Hardy-Leray potentials and study qualitative properties of the solutions to the corresponding nonhomogeneous problems by the distributional identities. We address some applications on the nonexistence of some nonhomogeneous problems with the Hardy-Leray potentials and the nonexistence principle eigenvalue with some indefinite potentials.

math.AP↗
arXiv2017

On isolated singularities of Kirchhoff--type Laplacian problems

In this paper, we study isolated singular positive solutions for the following Kirchhoff--type Laplacian problem: \begin{equation*} -\left(θ+\int_Ω |\nabla u| dx\right)Δu =u^p \quad{\rm in}\quad Ω\setminus \{0\},\qquad u=0\quad {\rm on}\quad \partial Ω, \end{equation*} where $p>1$, $θ\in \R$, $Ω$ is a bounded smooth domain containing the origin in $\R^N$ with $N\ge 2$. In the subcritical case: $1 0$. To estimate $M_θ(u)$, we make use of the rearrangement argument. Furthermore, we obtain a sequence of isolated singular solutions such that $M_θ(u)<0$, by analyzing relationship between the parameter $λ$ and the unique solution $u_λ$ of $$-Δu+λu^p=kδ_0\quad{\rm in}\quad B_1(0),\qquad u=0\quad {\rm on}\quad \partial B_1(0).$$ In the supercritical case: $N/(N-2)\le p<(N+2)/(N-2)$ with $N\ge3$, we obtain two isolated singular solutions $u_i$ with $i=1,2$ such that $M_θ(u_i)>0$ under some appropriate assumptions.

math.AP↗
arXiv2017

Isolated singularities for elliptic equations with Hardy operator and source nonlinearity

In this paper, we concern the isolated singular solutions for semi-linear elliptic equations involving the Hardy-Leray potentials \begin{equation}\label{0} -Δu+\fracμ{|x|^2} u=u^p\quad {\rm in}\quad Ω\setminus\{0\},\qquad u=0\quad{\rm on}\quad \partialΩ. \end{equation} We classify the isolated singularities and obtain the existence, the stability of positive solutions of (\ref{0}). Our results are based on the study of nonhomogeneous Hardy problem in a new distributional sense.

math.AP↗
arXiv2017

Anisotropic singularities to semilienar elliptic equations in a measure framework

The purpose of this article is to study very weak solutions of elliptic equation $$ -Δu+g(u)=2k\frac{\partial δ_0}{\partial x_N }+jδ_0\quad {\rm in}\quad\ \ B_1(0),\qquad u=0\quad {\rm on}\quad\ \ \partial B_1(0), $$ where $k>0$, $j\ge0$, $B_1(0)$ denotes the unit ball centered at the origin in $\mathbb{R}^N$ with $N\geq2$, $g:\mathbb{R}\to\mathbb{R}$ is an odd, nondecreasing and $C^1$ function, $δ_0$ is the Dirac mass concentrated at the origin and $\frac{\partialδ_0}{\partial x_N}$ is defined in the distribution sense that $$ \langle\frac{\partial δ_0}{\partial x_N},ζ\rangle=\frac{\partialζ(0)}{\partial x_N} , \qquad \forall ζ\in C^1_0(B_1(0)). $$ We obtain that the above problem admits a unique very weak solution $u_{k,j}$ under the integral subcritical assumption $$\int_1^{\infty}g(s)s^{-1-\frac{N+1}{N-1}}ds<+\infty.$$ Furthermore, we prove that $u_{k,j}$ has anisotropic singularity at the origin and we consider the odd property $u_{k,0}$ and limit of $\{u_{k,0}\}_k$ as $k\to\infty$. We pose the constraint on nonlienarity $g(u)$ that we only require integrability in the principle value sense, due to the singularities only at the origin. This makes us able to search the very weak solutions in a larger scope of the nonlinearity.

math.AP↗