We study the existence and nonexistence of singular solutions to the equation $u_t-Δu - \fracκ{|x|^2}u+|x|^αu|u|^{p-1}=0$, $p>1$, in $\R^N\times[0,\infty)$, $N\ge 3$, with a singularity at the point $(0,0)$, that is, nonnegative solutions satisfying $u(x,0)=0$ for $x\ne0$, assuming that $\a>-2$ and $κ<\left(\frac{N-2}2\right)^2$. The problem is transferred to the one for a weighted Laplace-Beltrami operator with a non-linear absorbtion, absorbing the Hardy potential in the weight. A classification of a singular solution to the weighted problem either as a {\it source solution} with a multiple of the Dirac mass as initial datum, or as a unique {\it very singular solution}, leads to a complete classification of singular solutions to the original problem, which exist if and only if $p<1+\frac{2(2+α)}{N+2+\sqrt{(N-2)^2-4κ}}$.
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.
The aim of this paper is to obtain new Hardy inequalities with double singular weights - at an interior point and on the boundary of the domain. These inequalities give us the possibility to derive estimates from below of the first eigenvalue of the p-Laplacian with Dirichlet boundary conditions.
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.
Our aim in this article is to study semilinear elliptic equations involving a fractional Hardy operator, an absorption and a Radon source in a weighted distributional sense. We show various scenarios, produced by the combined effect of the fractional Hardy potential, the growth of the absorption term and the concentration of the measure, in which existence and uniqueness results holds.
Let $Ω$ be a smooth bounded domain in $\mahbb R^N$ with $N\ge 3$ and let $Σ_k$ be a closed smooth submanifold of $δΩ$ of dimension $1\le k\le N-2$. In this paper we study the weighted Hardy inequality with weight function singular on $Σ_k$. In particular we provide sufficient and necessary conditions for existence of minimizers.
We consider the Hardy-Hénon parabolic equation $u_t-Δu =|x|^a |u|^{p-1}u$ with $p>1$ and $a\in {\mathbb R}$. We establish the space-time singularity and decay estimates, and Liouville-type theorems for radial and nonradial solutions. As applications, we study universal and a priori bound of global solutions as well as the blow-up estimates for the corresponding initial boundary value problem.
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 $0<θ<2$.
In this note we address the continuity of strongly singular Calderón-Zygmund operators on Hardy-Morrey spaces $\mathcal{HM}_{q}^λ(\mathbb{R}^n)$, assuming weaker integral conditions on the associated kernel. Important examples that falls into this scope are pseudodifferential operators on the Hörmander classes $OpS^{m}_{σ,μ}(\mathbb{R}^n)$ with $0<σ\leq 1$, $0 \leq μ<1$, $μ\leq σ$ and $m\leq -n(1-σ)/2$.
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 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 …
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 = -…
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…
For $N\geq 4$, we let $Ω$ to be a smooth bounded domain of $\mathbb{R}^N$, $Γ$ a smooth closed submanifold of $Ω$ of dimension $k$ with $1\leq k \leq N-2$ and $h$ a continuous function defined on $Ω$. We denote by $ρ_Γ\left(\cdot\right):=\dist_g\left(\cdot, Γ\right)$ the distance function to $Γ$. For $σ\in (0,2)$, we study existence of positive solutions $u \in H^1_0\left(Ω\right)$ to the nonlinear equation $$ -Δu+h u=ρ_Γ^{-σ} u^{2^*(σ)-1} \qquad \textrm{in } Ω, $$ where $2^*(σ):=\frac{2(N-σ)}{N-2}$ is the critical Hardy-Sobolev exponent. In particular, we provide existence of solution under t…
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 consider a bounded domain $Ω$ of $\mathbb{R}^N$, $N\ge3$, $h$ and $b$ continuous functions on $Ω$. Let $Γ$ be a closed curve contained in $Ω$. We study existence of positive solutions $u \in H^1_0\left(Ω\right)$ to the perturbed Hardy-Sobolev equation: $$ -Δu+h u+bu^{1+δ}=ρ^{-σ}_Γu^{2^*_σ-1} \qquad \textrm{ in } Ω, $$ where $2^*_σ:=\frac{2(N-σ)}{N-2}$ is the critical Hardy-Sobolev exponent, $σ\in [0,2)$, $0< δ<\frac{4}{N-2}$ and $ρ_Γ$ is the distance function to $Γ$. We show that the existence of minimizers does not depend on the local geometry of $Γ$ nor on the potential $h$. For $N=3$, th…
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…
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…
The present work is concerned with existence of positive solutions for a class of fractional equation involving a Kirchhoff term and singular potential.
We investigate the Hardy-Schrödinger operator $L_γ=-Δ-\fracγ{|x|^2}$ on domains $Ω\subset\rn$, whose boundary contain the singularity $0$. The situation is quite different from the well-studied case when $0$ is in the interior of $Ω$. For one, if $0\inΩ$, then $L_γ$ is positive if and only if $γ<\frac{(n-2)^2}{4}$, while if $0\in\partialΩ$ the operator $L_γ$ could be positive for larger value of $γ$, potentially reaching the maximal constant $\frac{n^2}{4}$ on convex domains. We prove optimal regularity and a Hopf-type Lemma for variational solutions of corresponding linear Dirichlet boundary …
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 …
The main results in the paper are the weighted multipolar Hardy inequalities \begin{equation*} c\int_{\R^N}\sum_{i=1}^n\frac{u^2}{|x-a_i|^2}\,dμ\leq\int_{\R^N}|\nabla u |^2dμ+ K\int_{\R^N} u^2dμ, \end{equation*} in $\R^N$ for any $u$ in a suitable weighted Sobolev space, with $0<c\le c_{o,μ}$, $a_1,\dots,a_n\in \R^N$, $K$ constant. The weight functions $μ$ are of a quite general type. The paper fits in the framework of the study of Kolmogorov operators \begin{equation*} Lu=Δu+\frac{\nabla μ}μ\cdot\nabla u, \end{equation*} perturbed by multipolar inverse square potentials, and of the related ev…
In this paper we study the Hardy problem in R^N with N>2 and in a ball B of R^N. Using a suitable map we transform the Hardy problem into another one without the singular term. Then we obtain some bifurcation results from the radial positive solutions corresponding to some explicit values of the parameter lambda.
We consider the remaining unsettled cases in the problem of existence of energy minimizing solutions for the Dirichlet value problem $L_γu-λu=\frac{u^{2^*(s)-1}}{|x|^s}$ on a smooth bounded domain $Ω$ in $\mathbb{R}^n$ ($n\geq 3$) having the singularity $0$ in its interior. Here $γ<\frac{(n-2)^2}{4}$, $0\leq s <2$, $2^*(s):=\frac{2(n-s)}{n-2}$ and $0\leq λ<λ_1(L_γ)$, the latter being the first eigenvalue of the Hardy-Schrödinger operator $L_γ:=-Δ-\fracγ{|x|^2}$. There is a threshold $λ^*(γ, Ω) \geq 0$ beyond which the minimal energy is achieved, but below which, it is not. It is well known tha…
We establish magnetic improvements upon the classical Hardy inequality for two specific choices of singular magnetic fields. First, we consider the Aharonov-Bohm field in all dimensions and establish a sharp Hardy-type inequality that takes into account both the dimensional as well as the magnetic flux contributions. Second, in the three-dimensional Euclidean space, we derive a non-trivial magnetic Hardy inequality for a magnetic field that vanishes at infinity and diverges along a plane.
We establish sharp two-sided bounds on the heat kernel of the fractional Laplacian, perturbed by a drift having critical-order singularity, by transferring it to appropriate weighted space with singular weight.
In this paper we study the asymptotic behavior of solutions to an elliptic equation near the singularity of an inverse square potential with a coefficient related to the best constant for the Hardy inequality. Due to the presence of a borderline Hardy potential, a proper variational setting has to be introduced in order to provide a weak formulation of the equation. An Almgren-type monotonicity formula is used to determine the exact asymptotic behavior of solutions.
On a compact Riemannian manifold, we study a singular elliptic equation with critical Sobolev exponent and critical Hardy potential. In a first part, we prove an $H^2_1$ type decomposition result for Palais-Smale sequences of the associated energy functional. In a second part, we apply the decomposition result to obtain solutions of different energy levels.
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…
Upper bounds are obtained for the heat content of an open set D in a geodesically complete Riemannian manifold M with Dirichlet boundary condition on bd(D), and non-negative initial condition. We show that these upper bounds are close to being sharp if (i) the Dirichlet-Laplace-Beltrami operator acting in $L^2(D)$ satisfies a strong Hardy inequality with weight $r^2$, (ii) the initial temperature distribution, and the specific heat of D are given by $r^{-a}$ and $r^{-b}$ respectively, where $r$ is the distance to the boundary, and 1<a<2, 1<b<2.
In this paper, we show a weighted Hardy inequality in a limiting case for functions in weighted Sobolev spaces with respect to an invariant measure. We also prove that the constant in the left-hand side of the inequality is optimal. As applications, we establish the existence and nonexistence of positive exponentially bounded weak solutions to a parabolic problem involving the Ornstein-Uhlenbeck operator perturbed by a critical singular potential in two dimensional case, according to the size of the coefficient of the critical potential. These results can be considered as counterparts in the l…
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $N\geq 3$ and we let $Σ$ to be a closed submanifold of dimension $1 \leq k \leq N-2. $ In this paper we study existence and non-existence of minimizers of Hardy inequality with weight function singular on $Σ$ within the framework of Brezis-Marcus-Shafrir. In particular we provide necessary and sufficient conditions for existence of minimizers.
This paper examines the decay properties of positive solutions for a family of fully nonlinear systems of integral equations containing Wolf potentials and Hardy weights. This class of systems includes examples which are closely related to the Euler-Lagrange equations for several classical inequalities such as the Hardy-Sobolev and Hardy-Littlewood-Sobolev inequalities. In particular, a complete characterization of the fast decaying ground states in terms of their integrability is provided in that bounded and fast decaying solutions are shown to be equivalent to the integrable solutions. In ge…
We study the asymptotic behavior of a sequence of positive solutions $(u_ε)_{ε>0}$ as $ε\to 0$ to the family of equations \begin{equation*} \left\{\begin{array}{ll} Δu_ε+a(x)u_ε= \frac{u_ε^{2^*(s_ε)-1}}{|x|^{s_ε}}& \hbox{ in }Ω\\ u_ε=0 & \hbox{ on }\partialΩ. \end{array}\right. \end{equation*} where $(s_ε)_{ε>0}$ is a sequence of positive real numbers such that $\lim \limits_{ε\rightarrow 0} s_ε=0$, $2^{*}(s_ε):= \frac{2(n-s_ε)}{n-2}$ and $Ω\subset \mathbb{R}^{n}$ is a bounded smooth domain such that $0 \in \partial Ω$. When the sequence $(u_ε)_{ε>0}$ is uniformly bounded in $L^{\infty}$, then…
In this paper we study the positive solutions of sub linear elliptic equations with a Hardy potential which is singular at the boundary. By means of ODE techniques a fairly complete picture of the class of radial solutions is given. Local solutions with a prescribed growth at the boundary are constructed by means of contraction operators. Some of those radial solutions are then used to construct ordered upper and lower solutions in general domains. By standard iteration arguments the existence of positive solutions is proved. An important tool is the Hardy constant.
In this work we prove sharp $L^p$ versions of multipolar Hardy inequalities in the case of a bipolar potential and $p\geq 2$, which were first developed in the case $p=2$ by Cazacu (CCM 2016) and Cazacu&Zuazua (Studies in phase space analysis with applications to PDEs, 2013). Our results are sharp and minimizers do exist in the energy space. New features appear when $p>2$ compared to the linear case $p=2$ at the level of criticality of the p-Laplacian $-Δ_p$ perturbed by a singular Hardy bipolar potential.
Inhomogeneous parabolic equations with Hardy potential and memory on the Heisenberg group
Indian Institute of Technology (BHU)（印度理工学院（贝拿勒斯 Hindu 大学））；Birla Institute of Technology and Science Pilani, Dubai Campus（比拉理工科学研究所 pilani 迪拜校区）；Nazarbayev University（纳扎尔巴耶夫大学）
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 …
This article concerns about the existence of a positive SOLA (Solutions Obtained as Limits of Approximations) for the following singular critical Choquard problem involving fractional power of Laplacian and a critical Hardy potential.
In this paper, we study the following singular nonlinear elliptic problem \begin{equation}\label{eq:1} \left\{ \begin{array}{ll} \displaystyle (-Δ)^{\frac α2} u=λ|u|^{r-2}u+μ\frac{|u|^{q-2}u}{|x|^{s}}\quad &{\rm in }\quad Ω, \\ \\ u=0 &{\rm on }\quad \partialΩ, \end{array} \right. \end{equation} where $Ω$ is a smooth bounded domain in $\mathbb R^N$ with $0\in Ω$, $λ,μ>0,0<s\leqα$, $(-Δ)^{\frac α2}$ is the fractional Laplacian operator with $0<α<2$. We establish existence results of problem \eqref{eq:1} for subcritical, Sobolev critical and Hardy-Sobolev critical cases.
Let $Ω\subset \mathbb{R}^N$ be a bounded domain and $δ(x)$ be the distance of a point $x\in Ω$ to the boundary. We study the positive solutions of the problem $Δu +\fracμ{δ(x)^2}u=u^p$ in $Ω$, where $p>0, \,p\ne 1$ and $μ\in \mathbb{R},\,μ\ne 0$ is smaller then the Hardy constant. The interplay between the singular potential and the nonlinearity leads to interesting structures of the solution sets. In this paper we first give the complete picture of the radial solutions in balls. In particular we establish for $p>1$ the existence of a unique large solution behaving like $δ^{- \frac2{p-1}}$ at …
An analogy of nonexistence result by Baras and Goldstein (1984), for the heat equation with inverse singular potential, is proved for 2mth-order linear parabolic equations with Hardy-supercritical singular potentials. Extensions to other linear and nonlinear singular PDEs are discussed.
