We study an inhomogeneous semilinear heat inequality on the unit sphere \(\mathbb S^N\), \(N\ge3\), in an exterior geodesic domain associated with a fixed pole. The equation involves the singular Hardy-type potential \(λ/\sin^2 r\), where \(r=d(o,x)\), and the weighted nonlinearity \((\sin r)^α|u|^p\). For \(α>-2\) and \(0<λ\le λ^*=((N-2)/2)^2\), we prove the existence of a critical exponent \(p_{\mathrm{crit}}=p_{\mathrm{crit}}(α,N,λ)\) governing the existence and nonexistence of solutions. More precisely, we prove that no weak solution exists for any nontrivial nonnegative source in the range \(p>p_{\mathrm{crit}}\), whereas classical solutions exist for some positive continuous sources in the range \(1<p<p_{\mathrm{crit}}\). Under suitable additional assumptions, we also prove nonexistence at the critical exponent \(p=p_{\mathrm{crit}}\). If \(α\le -2\), we show that nonexistence holds for all \(p>1\). The analysis is based on the construction of radial Hardy barriers adapted to the antipodal singularity and on sharp integral estimates involving power and logarithmic cutoffs near \(r=π\).
In this paper we study existence and nonexistence of nonnegative distributional solutions for a class of semilinear fractional elliptic equations involving the Hardy potential.
In this paper we study nonexistence of non-negative distributional supersolutions for a class of semilinear elliptic equations involving inverse-square potentials.
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.
In this paper, we show global existence, in spatial dimensions greater than or equal to four, for semilinear wave equations with quadratic nonlinearities exterior to a nontrapping obstacle. This extends the previous work of Shibata-Tsutsumi and Hayashi. Shibata-Tsutsumi showed the result for spatial dimensions greater than or equal to 6, and Hayashi proved the result exterior to a ball.
The aim of this article is to prove an "almost" global existence result for some semilinear wave equations in the plane outside a bounded convex obstacle with the Neumann boundary condition.
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.
We study two-dimensional semilinear strongly damped wave equation with mixed nonlinearity $|u|^p+|u_t|^q$ in an exterior domain, where $p,q>1$. Assuming the smallness of initial data in exponentially weighted spaces and some conditions on powers of nonlinearity, we prove global (in time) existence of small data energy solution with suitable higher regularity by using a weighted energy method.
We consider the Cauchy-Dirichlet problem for semilinear wave equations in a three space dimensional domain exterior to a bounded and non-trapping obstacle. We obtain a detailed estimate for the lower bound of the lifespan of classical solutions when the size of initial data tends to zero, in a similar spirit to that of the works of John and Hörmander where the Cauchy problem was treated. We show that our estimate is sharp at least for some special case.
We study the semilinear elliptic inequality $-Δu\geqφ(δ_K(x))f(u)$ in $R^N\setminus K,$ where $φ, f$ are non-negative and continuous functions, $K\subset R^N$ $(N\geq 2)$ is a compact set and $δ_K(x)={\rm dist}(x,\partial K)$. We obtain optimal conditions in terms of $φ$ and $f$ for the existence of $C^2$ positive solutions. Our analysis emphasizes the role played by the geometry of the compact set $K$.
In this paper, we consider the initial boundary value problem in an exterior domain for semilinear strongly damped wave equations with power nonlinear term of the derivative-type $|u_t|^q$ or the mixed-type $|u|^p+|u_t|^q$, where $p,q>1$. On one hand, employing the Banach fixed-point theorem we prove local (in time) existence of mild solutions. On the other hand, under some conditions for initial data and the exponents of power nonlinear terms, the blow-up results are derived by applying the test function method.
Semilinear elliptic equations which give rise to solutions blowing up at the boundary are perturbed by a Hardy potential. The size of this potential effects the existence of a certain type of solutions (large solutions): if the potential is too small, then no large solution exists. The presence of the Hardy potential requires a new definition of large solutions, following the pattern of the associated linear problem. Nonexistence and existence results for different types of solutions will be given. Our considerations are based on a Phragmen-Lindelof type theorem which enables us to classify th…
We consider equations of the form $-L_μu +f(u)=0$ in a smooth domain $Ω$, where $L_μ=Δ+ μδ^{-2}$ and $δ(x)$ denotes the distance of the point $x$ to the boundary of the domain. The nonlinear term $f$ is positive, increasing and convex on $(0,\infty)$, satisfies the Keller-Osserman condition and some additional technical assumptions. The conditions are satisfied, in particular, by power and exponential nonlinearities. We discuss the question of existence and uniqueness of large solutions when $μ>0$.
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 s…
We consider the initial-boundary value problem of semilinear wave equation with nonlinearity $|u|^p$ in exterior domain in $\mathbf{R}^N$ $(N\geq 3)$. Especially, the lifespan of blowup solutions with small initial data are studied. The result gives upper bounds of lifespan which is essentially the same as the Cauchy problem in $\mathbf{R}^N$. At least in the case $N=4$, their estimates are sharp in view of the work by Zha--Zhou (2015). The idea of the proof is to use special solutions to linear wave equation with Dirichlet boundary condition which are constructed via an argument based on Waka…
On a bounded smooth domain we study solutions of a semilinear elliptic equation with an exponential nonlinearity and a Hardy potential depending on the distance to the boundary of the domain. We derive global a priori bounds of the Keller-Osserman type. Using a Phragmen-Lindelof alternative for generalized sub and super-harmonic functions we discuss existence, nonexistence and uniqueness of so-called large solutions, i.e., solutions which tend to infinity at the boundary. The approach develops the one used by the same authors for a problem with a power nonlinearity instead of the exponential n…
In this paper, we give a full classification of the nonexistence of positive weak solutions to the semilinear elliptic inequality involving the fractional Hardy potential in punctured and in exterior domains. Our methods are self-contained and new. The main ideas and key ingredients will be discussed in the next section after Theorem 1.1 for punctured domains, and after Theorem 1.4 for exterior domains. We will also explain why all the previous methods and techniques do not apply to our general setting. Let us give here a foretaste of our line of attack: Based on the imbalance between the Hard…
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$$Ω$, where $Ω$ is a bounded, smooth domain of R N , N $\ge$ 2, containing the origin, $μ$ $\ge$ -- (N --2) 2 4 is a constant, g is a nondecreasing function satisfying some integral growth assumption and $ν$ is a Radon measure on $Ω$. We show that the situation differs according $ν$ is diffuse or concentrated at the origin. When g is a power we introduce a capacity framework to find necessary and sufficient condition for solvability.
In this paper we study the initial boundary value problem for two-dimensional semilinear wave equations with small data, in asymptotically Euclidean exterior domains. We prove that if $1<p\le p_c(2)$, the problem admits almost the same upper bound of the lifespan as that of the corresponding Cauchy problem, only with a small loss for $1<p\le 2$. It is interesting to see that the logarithmic increase of the harmonic function in $2$-D has no influence to the estimate of the upper bound of the lifespan for $2<p\le p_c(2)$. One of the novelties is that we can deal with the problem with flat metric…
We study existence and uniqueness of solutions of (E 1) --$Δ$u + $μ$ |x| ^{-2} u + g(u) = $ν$ in $Ω$, u = $λ$ on $\partial$$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…
We consider a class of semilinear nonlocal problems with vanishing exterior condition and establish a Ambrosetti-Prodi type phenomenon when the nonlinear term satisfies certain conditions. Our technique makes use of the probabilistic tools and heat kernel estimates.
We study the existence and nonexistence of positive (super) solutions to the nonlinear $p$-Laplace equation $$-Δ_p u-\fracμ{|x|^p}u^{p-1}=\frac{C}{|x|^σ}u^q$$ in exterior domains of ${\R}^N$ ($N\ge 2$). Here $p\in(1,+\infty)$ and $μ\le C_H$, where $C_H$ is the critical Hardy constant. We provide a sharp characterization of the set of $(q,σ)\in\R^2$ such that the equation has no positive (super) solutions. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the $p$-Laplace operator wi…
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 …
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…
Let $Ω\subset {\mathbb R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $δ$ be the distance to $\partial Ω$. We study positive solutions of equation (E) $-L_μu+ g(|\nabla u|) = 0$ in $Ω$ where $L_μ=Δ+ \fracμ{δ^2} $, $μ\in (0,\frac{1}{4}]$ and $g$ is a continuous, nondecreasing function on ${\mathbb R}_+$. We prove that if $g$ satisfies a singular integral condition then there exists a unique solution of (E) with a prescribed boundary datum $ν$. When $g(t)=t^q$ with $q \in (1,2)$, we show that equation (E) admits a critical exponent $q_μ$ (depending only on $N$ and $μ$). In the subcritical case…
Consider the equation u_t=Δu-Vu +au^p \text{in} R^n\times (0,T); u(x,0)=ϕ(x)\gneq0, \text{in} R^n, where $p>1$, $n\ge2$, $T\in(0,\infty]$, $V(x)\sim\fracω{|x|^2}$ as $|x|\to\infty$, for some $ω\neq0$, and $a(x)$ is on the order $|x|^m$ as $|x|\to\infty$, for some $m\in (-\infty,\infty)$. A solution to the above equation is called global if $T=\infty$. Under some additional technical conditions, we calculate a critical exponent $p^*$ such that global solutions exist for $p>p^*$, while for $1<p\le p^*$, all solutions blow up in finite time. We also show that when $V\equiv0$, the blow-up/global s…
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…
In this paper, we discuss the global existence of weak solutions to the semilinear damped wave equation \begin{equation*} \begin{cases} \partial_t^2u-Δu + \partial_tu = f(u) & \text{in}\ Ω\times (0,T), \\ u=0 & \text{on}\ \partialΩ\times (0,T), \\ u(0)=u_0, \partial_tu(0)=u_1 & \text{in}\ Ω, \end{cases} \end{equation*} in an exterior domain $Ω$ in $\mathbb{R}^N$ $(N\geq 2)$, where $f:\mathbb{R}\to \mathbb{R}$ is a smooth function behaves like $f(u)\sim |u|^p$. From the view point of weighted energy estimates given by Sobajima--Wakasugi \cite{SoWa4}, the existence of global-in-time solutions wi…
Let $Ω$ be a bounded smooth domain in $\mathbb{R}^N$. We study positive solutions of equation (E) $-L_μu+ u^q = 0$ in $Ω$ where $L_μ=Δ+ \fracμ{δ^2}$, $0<μ$, $q>1$ and $δ(x)=\mathrm{dist}\,(x,\partialΩ)$. A positive solution of (E) is moderate if it is dominated by an $L_μ$-harmonic function. If $μ<C_H(Ω)$ (the Hardy constant for $Ω$) every positive $L_μ$- harmonic functions can be represented in terms of a finite measure on $\partialΩ$ via the Martin representation theorem. However the classical measure boundary trace of any such solution is zero. We introduce a notion of normalized boundary t…
Let $Ω$ be a smooth bounded domain in $\mathbb{R}^N$ and $δ(x)=\text{dist}\,(x,\partial Ω)$. Assume $μ>0$, $ν$ is a nonnegative finite measure on $\partial Ω$ and $g \in C(Ω\times \mathbb{R}_+)$. We study positive solutions of $$ (P)\qquad -Δu - \fracμ{δ^2} u = g(x,u) \text{ in } Ω, \qquad \text{tr}^*(u)=ν. $$ Here $\text{tr}^*(u)$ denotes the normalized boundary trace of $u$ which was recently introduced by M. Marcus and P. T. Nguyen. We focus on the case $0<μ< C_H(Ω)$ (the Hardy constant for $Ω$) and provide some qualitative properties of solutions of (P). When $g(x,u)=u^q$ with $q>1$, we pr…
Let $Ω\subset \mathbb{R}^N$ ($N \geq 3$) be a bounded smooth domain and $δ(x)=\text{dist}(x,\partial Ω)$. In this paper, we provide various necessary and sufficient conditions for the existence of weak solutions to $$ -Δu- \fracμ{δ^2}u= u^p +τ\quad \text{in } Ω, \quad \quad u=ν\quad\text{on } \partial Ω, $$ where $μ\in \mathbb{R}$, $p>0$, $τ$ and $ν$ are measures on $Ω$ and $\partial Ω$ respectively. We then establish existence results for the system $$ \left\{ \begin{aligned} &-Δu- \fracμ{δ^2}u = ε\, v^p +τ\quad \text{in } Ω, \\ &-Δv- \fracμ{δ^2}v = ε\, u^{\tilde p}+\tilde τ\quad \text{in } Ω…
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…
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 …
Poisson problem with Hardy operator on lattice graph $ \mathbb{Z}^d$ and the application to semilinear equations
Center for Mathematics and Interdisciplinary Sciences, Fudan University（复旦大学数学与交叉科学中心）；Shanghai Institute for Mathematics and Interdisciplinary Sciences（上海数学与交叉科学研究院）；School of Mathematical Sciences, LMNS, Fudan University（复旦大学数学科学学院）；Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）；Research Institute of Intelligent Complex Systems, Fudan University（复旦大学复杂系统研究所）
本文研究晶格图 $\mathbb{Z}^d$ 上带 Hardy 算子的 Poisson 问题，通过 Green 核估计和加权可和性条件刻画其可解性，并完整给出半线性椭圆不等式正解不存在性的充要条件。
We study a class of fractional semilinear elliptic equations and formulate the corresponding Calderón problem. We determine the nonlinearity from the exterior partial measurements of the Dirichlet-to-Neumann map by using first order linearization and the Runge approximation property.
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 look for solutions of a semilinear Maxwell type equation, in even dimension, greater than four. These solutions are critical points of a functional which is strongly degenerate because of the presence of the exterior derivative. We prove that, assuming a suitable convexity condition on the nonlinearity, the equation possesses infinitely many finite energy solutions.
This paper is concerned with the forward and inverse problems for the fractional semilinear elliptic equation $(-Δ)^s u +a(x,u)=0$ for $0<s<1$. For the forward problem, we proved the problem is well-posed and has a unique solution for small exterior data. The inverse problems we consider here consists of two cases. First we demonstrate that an unknown coefficient $a(x,u)$ can be uniquely determined from the knowledge of exterior measurements, known as the Dirichlet-to-Neumann map. Second, despite the presence of an unknown obstacle in the media, we show that the obstacle and the coefficient ca…
We consider the initial boundary value problem in exterior domain for semilinear wave equations with power-type nonlinearity |u| p. We will establish blow-up results when p is less than or equal to Strauss' exponent which is the same one for the whole space case R n .
We consider the $L^p$ Hardy inequality involving the distance to the boundary of a domain in the $n$-dimensional Euclidean space with nonempty compact boundary. We extend the validity of known existence and non-existence results, as well as the appropriate tight decay estimates for the corresponding minimizers, from the case of domains of class $C^2$ to the case of domains of class $C^{1,γ}$ with $γ\in (0,1]$. We consider both bounded and exterior domains. The upper and lower estimates for the minimizers in the case of exterior domains and the corresponding related non-existence result seem to…
The purpose in this paper is to determine the global behavior of solutions to the initial-boundary value problems for energy-subcritical and critical semilinear heat equations by initial data with lower energy than the mountain pass level in energy spaces associated with self-adjoint operators satisfying Gaussian upper bounds. Our self-adjoint operators include the Dirichlet Laplacian on an open set, Robin Laplacian on an exterior domain, and Schrödinger operators, etc.
We consider a semilinear elliptic problem with a nonlinear term which is the product of a power and the Riesz potential of a power. This family of equations includes the Choquard or nonlinear Schroedinger--Newton equation. We show that for some values of the parameters the equation does not have nontrivial nonnegative supersolutions in exterior domains. The same techniques yield optimal decay rates when supersolutions exists.
We prove the existence of a family of slow decay positive solutions of a supercritical elliptic equation with Hardy potential in the entire space and study stability and oscillation properties of these solutions. We also establish the existence of a continuum of stable slow decay positive solutions for the relevant exterior Dirichlet problem.
We study semilinear problems in general bounded open sets for non-local operators with exterior and boundary conditions. The operators are more general than the fractional Laplacian. We also give results in case of bounded $C^{1,1}$ open sets.
We study the Cauchy problem for the semilinear heat equation with the singular potential, called the Hardy-Sobolev parabolic equation, in the energy space. The aim of this paper is to determine a necessary and sufficient condition on initial data below or at the ground state, under which the behavior of solutions is completely dichotomized. More precisely, the solution exists globally in time and its energy decays to zero in time, or it blows up in finite or infinite time. The result on the dichotomy for the corresponding Dirichlet problem is also shown as a by-product via comparison principle…
We investigate the initial-value problem for the semilinear plate equation containing localized strong damping, localized weak damping and nonlocal nonlinearity. We prove that if nonnegative damping coefficients are strictly positive almost everywhere in the exterior of some ball and the sum of these coefficients is positive a.e. in $% \mathbb{R} ^{n}$, then the semigroup generated by the considered problem possesses a global attractor in $H^{2}\left( \mathbb{R} ^{n}\right) \times L^{2}\left( \mathbb{R} ^{n}\right) $. We also establish boundedness of this attractor in $ H^{3}\left( \mathbb{R} …
In this paper, we consider exterior problem of the critical semilinear wave equation in three space dimensions with variable coefficients and prove global existence of smooth solutions. Similar to the constant coefficients case, we show that the energy cannot concentrate at any point $(t,x)\in(0,\infty)\timesΩ$. For that purpose, following Ibrahim and Majdoub \cite{Ibrahim}, we use a geometric multiplier close to the well-known Morawetz multiplier used in the constant coefficients case. Then we use comparison theorem from Riemannian Geometry to estimate the error terms. Finally, using Strichar…
