Huyuan Chen, Axander Quaas, Feng Zhou
In this paper, we present a new distributional identity for the solutions of elliptic equations involving Hardy potentials with singularities located on the boundary of the domain. Then we use it to obtain the boundary isolated singular solutions of nonhomogeneous problems.
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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.
The paper deals with existence and multiplicity of positive solutions to nonlocal equations with critical Hrardy-Sobolev nonlinearities and external terms. We establish the profile decomposition of the Palais-Smale sequences associated with the functional and existence of at least two positive solutions to the equation.
Nonhomogeneous div-curl type estimates for system of complex vector fields on local Hardy spaces
本文在局部Hardy空间$h^1$上提出了常系数复向量场椭圆系统的非齐次div-curl型估计，并借此实现了局部$bmo$空间的向量场分解。
We prove a general perturbation theorem that can be used to obtain pairs of nontrivial solutions of a wide range of local and nonlocal nonhomogeneous elliptic problems. Applications to critical $p$-Laplacian problems, $p$-Laplacian problems with critical Hardy-Sobolev exponents, critical fractional $p$-Laplacian problems, and critical $(p,q)$-Laplacian problems are given. Our results are new even in the semilinear case $p = 2$.
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.
We will present versions of the Rellich-Kondrachov theorem for pseudo-differential operators acting on localizable Hardy spaces. One of the techniques includes boundedness properties for pseudodifferential operators with symbols in the Hörmander class, that might be of interest in his own right, extending results previously obtained by Goldberg, Alvarez and Hounie, Taylor - among others. As application, we obtain compact embedding results for distributions in nonhomogeneous localizable Hardy-Sobolev spaces.
Let $Ø$ be a smooth bounded domain in $\R^N$ with $N\ge 1$. In this paper we study the Hardy-Poincaré inequality with weight function singular at the boundary of $Ø$. In particular we provide sufficient and necessary conditions on the existence of minimizers.
Let $Ø$ be a smooth bounded domain in $\R^N$ with $N\ge 1$. In this paper we study the Hardy-Poincaré inequalities with weight function singular at the boundary of $Ø$. In particular we give sufficient conditions so that the best constant is achieved.
Weighted fractional Hardy-Sobolev and Hardy-Sobolev-Maz'ya inequalities with singularities on flat submanifold
本文研究了带权重的分数Hardy-Sobolev和Hardy-Sobolev-Maz'ya不等式，探讨了奇点在平坦子流形上的应用，并证明了相关不等式在特定条件下的不成立性。
Let $Ø$ be a bounded domain in $\R^N$ with $0\in\deØ$ and $N\ge 2$. In this paper we study the Hardy-Poincaré inequality for maps in $H^1_0(Ω)$. In particular we give sufficient and some necessary conditions so that the best constant is achieved.
We study the Hardy inequality when the singularity is placed on the boundary of a bounded domain in $\mathbb{R}^n$ that satisfies both an interior and exterior ball condition at the singularity. We obtain the sharp Hardy constant $n^2/4$ in case the exterior ball is large enough and show the necessity of the large exterior ball condition. We improve Hardy inequality with the best constant by adding a sharp Sobolev term. We next produce criteria that lead to characterizing maximal potentials that improve Hardy inequality. Breaking the criteria one produces successive improvements with sharp con…
We consider the Cauchy problem for the Hardy parabolic equation $\partial_t u-Δu=|x|^{-γ}u^p$ with initial data $u_0$ singular at some point $z$. Our main results show that, if $z\neq 0$, then the optimal strength of the singularity of $u_0$ at $z$ for the solvability of the equation is the same as that of the Fujita equation $\partial_t u-Δu=u^p$. Moreover, if $z=0$, then the optimal singularity for the Hardy parabolic equation is weaker than that of the Fujita equation. We also obtain analogous results for a fractional case $\partial_t u+(-Δ)^{θ/2} u=|x|^{-γ}u^p$ with $0<θ<2$.
Weighted fractional Hardy inequalities with singularity on any flat submanifold
本文研究了任意平坦子流形上的加权分数Hardy不等式，推导了带奇异性的不等式，并处理了临界情况下的对数权重函数问题。
Fractional Sobolev-Choquard critical systems with Hardy term and weighted singularities
研究全空间上带Hardy项和加权奇异性的分数阶Sobolev-Choquard临界p-Laplacian系统，利用Morrey空间中的Sobolev不等式改进和Caffarelli-Kohn-Nirenberg不等式，通过变分方法给出弱非平凡解存在的充分条件。
The paper is concerned with the existence and multiplicity of positive solutions of the nonhomogeneous Choquard equation over an annular type bounded domain. Precisely, we consider the following equation \[ -\De u = \left(\int_{\Om}\frac{|u(y)|^{2^*_μ}}{|x-y|^μ}dy\right)|u|^{2^*_μ-2}u+f \; \text{in}\; \Om,\quad u = 0 \; \text{ on } \pa \Om , \] where $\Om$ is a smooth bounded annular domain in $\mathbb{R}^N( N\geq 3)$, $2^*_μ=\frac{2N-μ}{N-2}$, $f \in L^{\infty}(\Om)$ and $f \geq 0$. We prove the existence of four positive solutions of the above problem using the Lusternik-Schnirelmann theory …
In this article we first establish a complete characterization of Hardy's inequalities in $\mathbb{R}^n$ involving distances to different codimension subspaces. In particular the corresponding potentials have strong interior singularities. We then provide necessary and sufficient conditions for the validity of Hardy-Sobolev-Maz'ya inequalities with optimal Sobolev terms.
We study existence and uniqueness of solutions of (E 1) --$Δ$u + $μ$ |x| ^{-2} u + g(u) = $ν$ in $Ω$, u = $λ$ on $\partial$$Ω$, where $Ω$ $\subset$ R N + is a bounded smooth domain such that 0 $\in$ $\partial$$Ω$, $μ$ $\ge$ -- N 2 4 is a constant, g a continuous nondecreasing function satisfying some integral growth condition and $ν$ and $λ$ two Radon measures respectively in $Ω$ and on $\partial$$Ω$. We show that the situation differs considerably according the measure is concentrated at 0 or not. When g is a power we introduce a capacity framework which provides necessary and sufficient cond…
Attainability of the best constant of Hardy-Sobolev inequality with full boundary singularities
研究Hardy-Sobolev不等式在全边界奇异情形下最佳常数的可达性，发现二维中非凸域可达，高维中负平均曲率或特殊非脐点情形可达，而接近球时不可达。
Let $N \geq 3$ and $Ω\subset \mathbb{R}^N$ be $C^2$ bounded domain. We study the existence of positive solution $u \in H^1(Ω)$ of \begin{align*} \left\{ \begin{array}{l} -Δu + λu = \frac{|u|^{2^*(s)-2}u}{|x-x_1|^s} + \frac{|u|^{2^*(s)-2}u}{|x-x_2|^s}\text{ in }Ω\\ \frac{\partial u}{\partial ν} = 0 \text{ on }\partialΩ, \end{array}\right. \end{align*} where $0 < s <2$, $2^*(s) = \frac{2(N-s)}{N-2}$ and $x_1, x_2 \in \overlineΩ$ with $x_1 \neq x_2$. First, we show the existence of positive solutions to the equation provided the positive $λ$ is small enough. In case that one of the singularities …
In this paper, we are devoted to studying the positive solutions of the following higher order Hardy-Hénon equation $$ (-Δ)^{m}u=|x|^αu^{p} \quad\mbox{in}~ B_{1}\setminus\{0\}\subset\mathbb{R}^{n} $$ with an isolated singularity at the origin, where $α>-2m$, $m\geq1$ is an integer and $n>2m$. For $1<p<\frac{n+2m}{n-2m}$, singularity and decay estimates of solutions will be given. For $\frac{n+α}{n-2m}<p<\frac{n+2m}{n-2m}$ with $-2m<α<2m$, we show the super polyharmonic properties of solutions near the singularity, which are essential tools in the study of polyharmonic equation. Using these pro…
The Hardy Inequality (HI) for potentials with countably many singularities of the form $V=\sum_{k\in \mathbf{Z}}\frac{1}{|x-a_k|^2}$ is not a trivial issue. In principle, the more singular poles are, the less the Hardy constant is: it is well-known that in all the existing results about the HI with finite number of singularities the best constants converge to 0 with the number $n$ of singularities going to infinity. In this note we provide an example of nontrivial HI in right cylinders of fixed radius $R>0$ in $\mathbf{R}^d$, for a potential $V$ defined above having the singularities $\{a_k\}_…
We study a nonlinear equation in the half-space $\{x_1>0\}$ with a Hardy potential, specifically \[-Δu -\fracμ{x_1^2}u+u^p=0\quad\text{in}\quad \mathbb R^n_+,\] where $p>1$ and $-\infty<μ<1/4$. The admissible boundary behavior of the positive solutions is either $O(x_1^{-2/(p-1)})$ as $x_1\to 0$, or is determined by the solutions of the linear problem $-Δh -\fracμ{x_1^2}h=0$. In the first part we study in full detail the separable solutions of the linear equations for the whole range of $μ$. In the second part, by means of sub and supersolutions we construct separable solutions of the nonlinea…
In this paper, we study the asymptotic behavior of positive solutions of the fractional Hardy-Hénon equation $$ (-Δ)^σu = |x|^αu^p ~~~~~~~~~~~ in ~~ B_1 \backslash \{0\} $$ with an isolated singularity at the origin, where $σ\in (0, 1)$ and the punctured unit ball $B_1 \backslash \{0\} \subset \mathbb{R}^n$ with $n \geq 2$. When $-2σ< α< 2σ$ and $\frac{n+α}{n-2σ} < p < \frac{n+2σ}{n-2σ}$, we give a classification of isolated singularities of positive solutions, and in particular, this implies sharp blow up estimates of singular solutions. Further, we describe the precise asymptotic behavior of…
We study the boundary behaviour of the of (E) $-\Gd u-\myfrac{\xk }{d^2(x)}u+g(u)=0$, where $0<\xk <\frac{1}{4}$ and $g$ is a continuous nonndecreasing function in a bounded convex domain of $\BBR^N$. We first construct the Martin kernel associated to the the linear operator $\CL_{\xk }=-\Gd-\frac{\xk }{d^2(x)}$ and give a general condition for solving equation (E) with any Radon measure $\gm$ for boundary data. When $g(u)=|u|^{q-1}u$ we show the existence of a critical exponent $q_c=q_c(N,\xk )>1$: when $0<q<q_c$ any measure is eligible for solving (E) with $\gm$ for boundary data; if $q\geq …
In this paper we study the optimization problem $$μ^\star(Ω):=\inf_{u\in \semi}\frac{\into |\n u|^2 \dx}{\into V u^2 \dx}$$ in a suitable functional space $\semi$. Here, $V$ is the multi-singular potential given by $$V:=\sum_{1\leq i<j\leq n} \frac{|a_i-a_j|^2}{|x-a_i|^2|x-a_j|^2}$$ and all the singular poles $a_1, \ldots, a_n$, $n\geq 2$, arise either in the interior or at the boundary of a smooth open domain $Ω\subset \rr^N$, with $N\geq 3$ or $N \geq 2$, respectively. For a bounded domain $Ω$ containing all the singularities in the interior, we prove that $μ^\star(Ω)>μ^\star(\rr^N)$ when $n…
Fractional Sobolev-Chocard critical equation with Hardy term and weighted singularities
本文研究全空间中含 Hardy 项、局部 Sobolev 临界项与非局部 Choquard 临界项的分数阶 $p$-Laplacian 方程，借助加权 Morrey 嵌入和 CKN 不等式，以变分法证明非平凡弱解存在。
Singular Klein-Gordon equation on a bounded domain
本文研究有界域上带奇异势和奇异数据的Klein-Gordon方程，利用迹与延拓域建立弱解，并证明非常弱解的存在性、唯一性与相容性。
We consider the problem of attainability of the best constant in the following critical fractional Hardy-Sobolev inequality: \begin{equation*} μ_{γ,s}(\R^n):= \inf\limits_{u \in H^{\fracα{2}} (\R^n)\setminus \{0\}} \frac{ \int_{\R^n} |({-}{ Δ})^{\fracα{4}}u|^2 dx - γ\int_{\R^n} \frac{|u|^2}{|x|^α}dx }{(\int_{\R^n} \frac{|u|^{2_α^*(s)}}{|x|^{s}}dx)^\frac{2}{2_α^*(s)}}, \end{equation*} where $0\leq s<α<2$, $n>α$, ${2_α^*(s)}:=\frac{2(n-s)}{n-α},$ and $γ\in \mathbb{R}$. This allows us to establish the existence of nontrivial weak solutions for the following doubly critical problem on $\R^n$, \beg…
Semilinear elliptic equations involving power nonlinearities and hardy potentials with boundary singularities
本文研究带边界奇异 Hardy 势的半线性椭圆方程测度边值问题，针对吸收与源两类情形用容量刻画可解性条件，并处理 p 的超临界范围及 μ 的临界情形。
We establish -among other things- existence and multiplicity of solutions for the Dirichlet problem $\sum_i\partial_{ii}u+\frac{|u|^{\crit-2}u}{|x|^s}=0$ on smooth bounded domains $Ω$ of $ \rn$ ($n\geq 3$) involving the critical Hardy-Sobolev exponent $\crit =\frac{2(n-s)}{n-2}$ where $0<s<2$, and in the case where zero (the point of singularity) is on the boundary $\partial Ω$. Just as in the Yamabe-type non-singular framework (i.e., when s=0), there is no nontrivial solution under global convexity assumption (e.g., when $Ω$ is star-shaped around 0). However, in contrast to the non-satisfacto…
Let $Ω\subset\BBR^N$ be a bounded $C^2$ domain and $\CL_\gk=-\Gd-\frac{\gk}{d^2}$ the Hardy operator where $d=\dist (.,\prt\Gw)$ and $0<\gk\leq\frac{1}{4}$. Let $\ga_{\pm}=1\pm\sqrt{1-4\gk}$ be the two Hardy exponents, $\gl_\gk$ the first eigenvalue of $\CL_\gk$ with corresponding positive eigenfunction $ϕ_\gk$. If $g$ is a continuous nondecreasing function satisfying $\int_1^\infty(g(s)+|g(-s)|)s^{-2\frac{2N-2+\ga_+}{2N-4+\ga_+}}ds<\infty$, then for any Radon measures $\gn\in \GTM_{ϕ_\gk}(\Gw)$ and $\gm\in \GTM(\prt\Gw)$ there exists a unique weak solution to problem $P_{\gn,\gm}$: $\CL_\gk u…
The aim of this paper is to employ a strategy known from fluid dynamics in order to provide results for the linear heat equation $u_{t}-Δu-V(x)u=0$ in $\mathbb{R}^{n}$ with singular potentials. We show well-posedness of solutions, without using Hardy inequality, in a framework based in the Fourier transform, namely $PM^{k}$-spaces. For arbitrary data $u_{0}\in PM^{k}$, the approach allows to compute an explicit smallness condition on $V$ for global existence in the case of $V$ with finitely many inverse square singularities. As a consequence, well-posedness of solutions is obtained for the cas…
In this paper, we give a classification of the isolated singularities of positive solutions to the semilinear fractional elliptic equations $$(E) \quad\quad (-Δ)^s u = |x|^θ u^{p}\quad {\rm in}\ \ B_1\setminus\{0\},\quad u= h\quad{\rm in}\ \ \mathbb{R}^N\setminus B_1,\quad $$ where $s\in(0,1)$, $θ>-2s$, $p>\frac{N+θ}{N-2s}$, $B_1$ is the unit ball centered at the origin of $\mathbb{R}^N$ with $N>2s$. $h$ is a nonnegative Hölder continuous function in $\mathbb{R}^N\setminus B_1$. Our analysis of isolated singularities of $(E)$ is based on an integral upper bounds and the study of the Poisson pr…
We establish quantitative asymptotic behavior of positive solutions of a family of nonlinear elliptic equations on the half cylinder near the end. This unifies the study of isolated singularities of some semilinear elliptic equations, such as the Yamabe equation, Hardy-Hénon equation etc.
We study semilinear elliptic equations with Hardy potential $\mathrm{(E)} \; -L_μu+u^q=0$ in a bounded smooth domain $Ω\subset \mathbb R^N$. Here $q>1$, $L_μ=Δ+\fracμ{δ_Ω^2}$ and $δ_Ω(x)=\mathrm{dist}(x,\partialΩ)$. Assuming that $0\leq μ<C_H(Ω)$, boundary value problems with measure data and discrete boundary singularities for positive solutions of $\mathrm{(E)}$ have been studied earlier. In the present paper we study these problems, in arbitrary domains, assuming only $-\infty<μ<1/4$, even if $C_H(Ω)<1/4$. We recall that $C_H(Ω)\leq 1/4$ and, in general, strict inequality holds. The key to …
In this paper, we solve the fractional Lane-Emden equation in the Serrin's critical case for the fractional Laplacian by developing an innovative and self-contained approach that also applies to the classical setting ( Laplacian). We give a classification of the isolated singularities of positive solutions to the semilinear fractional elliptic equations $$(E) \qquad\qquad (-Δ)^s u = u^{\frac{N}{N-2s}}\quad {\rm in}\ \ Ω\setminus\{0\},\qquad u\geq 0\quad{\rm in}\ \ \mathbb{R}^N\setminusΩ,\qquad\qquad\quad $$ where $s\in(0,1)$, $Ω$ is a bounded domain containing the origin in $\mathbb{R}^N$ with…
A Doubly Critical Elliptic Problem with Submanifold Singularities
研究在有界域内具有子流形奇点的双重临界椭圆问题，通过变分方法和构造测试函数证明正解的存在性，分析子流形局部几何和势函数行为对解的影响。
In this paper, we are concerned with the fractional and higher order Hénon-Hardy type equations \begin{equation*} (-Δ)^{\fracα{2}}u(x)=f(x,u(x)) \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n}, \,\,\, \mathbb{R}^{n}_{+} \,\,\, \text{or} \,\,\, Ω\end{equation*} with $n>α$, $0<α<2$ or $α=2m$ with $1\leq m<\frac{n}{2}$. We first consider the typical case $f(x,u)=|x|^{a}u^{p}$ with $a\in(-α,\infty)$ and $0<p<p_{c}(a):=\frac{n+α+2a}{n-α}$. By using the method of scaling spheres, we prove Liouville theorems for the above Hénon-Hardy equations and equivalent integral equations in $\mathbb{R}…
This paper concerns with the heat equation in the half-space $\mathbb{R}_{+}^{n}$ with nonlinearity and singular potential on the boundary $\partial\mathbb{R}_{+}^{n}$. We develop a well-posedness theory (without using Kato and Hardy inequalities) that allows us to consider critical potentials with infinite many singularities and anisotropy. Motivated by potential profiles of interest, the analysis is performed in weak $L^{p}$-spaces in which we prove key linear estimates for some boundary operators arising from the Duhamel integral formulation in $\mathbb{R}_{+}^{n}$. Moreover, we investigate…
Local well-posedness of the higher order nonlinear Schrödinger equation on the half-line: single boundary condition case
本文利用 Fokas 方法建立半直线上高阶非线性薛定谔方程在单 Dirichlet 边界条件下的局部适定性，通过分数阶 Sobolev 空间框架和边界型 Strichartz 估计同时处理高、低正则性解，首次完整解决多项线性微分算子对应的非齐次初边值问题。
Maximal regularity and caloric trace estimates in mixed Lebesgue norms for the heat equation
研究半空间上非齐次Dirichlet边界数据的热方程，建立了边界数据g的混合Besov和Lizorkin-Triebel型范数与热延拓v的混合Lebesgue范数之间的最大正则性估计和逆热迹估计，覆盖端点情形。
Local regularity for anisotropic magnetic operators with general codimension singularities
研究静磁各向异性磁Schrödinger方程在余维一般奇异集上的解的局部正则性，通过吹胀分析建立Hölder和Schauder估计，揭示各向异性和奇异磁势的联合影响。
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