This paper is devoted to studies of non-negative, non-trivial (classical, punctured, or distributional) solutions to the higher order Hardy-Hénon equations \[ (-Δ)^m u = |x|^σu^p \] in $\mathbf R^n$ with $p > 1$. We show that the condition \[ n - 2m - \frac{2m+σ}{p-1} >0 \] is necessary for the existence of distributional solutions. For $n \geq 2m$ and $σ> -2m$, we prove that any distributional solution satisfies an integral equation and a weak super polyharmonic property. We establish some sufficient conditions for punctured or classical solution to be a distributional solution. As application, we show that if $n \geq 2m$ and $σ> -2m$, there is no non-negative, non-trivial, classical solution to the equation if \[ 1 < p < \frac{n+2m+2σ}{n-2m}. \] At last, we prove that for for $n > 2m$, $σ> -2m$ and $$p \geq \frac{n+2m+2σ}{n-2m},$$ there exist positive, radially symmetric, classical solutions to the equation.
In this paper some important inequalities are revisited. First, as motivation, we give another proof of the Hardy's inequality applying convenient vector fields as introduced by Mitidieri, see [6]. Then, we investigate a particular case of the Caffarelli-Kohn-Nirenberg's inequality. Finally, we study the Rellic's inequality.
In this paper, we consider the first order Hardy inequalities using simple equalities. This basic setting not only permits to derive quickly many well-known Hardy inequalities with optimal constants, but also supplies improved or new estimates in miscellaneous situations, such as multipolar potential, the exponential weight, hyperbolic space, Heisenberg group, the edge Laplacian, or the Grushin type operator.
We give a real-variable proof of the Hardy uncertainty principle. The method is based on energy estimates for evolutions with positive viscosity, convexity properties of free waves with Gaussian decay at two different times, elliptic $L^2$-estimates and the invertibility of the Fourier transform on $L^2(\Rn)$ and $\mathcal S'(\Rn)$.
We consider the problem of essential self-adjointness of the drift-diffusion operator $H=-\frac{1}ρ\nabla\cdot ρ\mathbb D\nabla +V$ on domains $Ω\subset \mathbb{R}^d$ with $\mathcal{C}^2$-boundary $\partial Ω$ and for large classes of coefficients $ρ,\; \mathbb{D}$ and $V$. We give criteria showing how the behavior as $x \rightarrow \partial Ω$ of these coefficients balances to ensure essential self-adjointness of $H$. On the way we prove a weak anisotropic Hardy inequality which is of independent interest.
In this article, we prove a weighted Hardy inequality for $1<p<+\infty$ and dimension $d\ge1$. If $p>d$, then we can deduce from our weighted Hardy inequality a Poincaré inequality. The proof of the weighted Hardy inequality is based on the method of vector fields firstly introduced by Mitidieri \cite{MR1769903}. By the same method, we show for $1<p<+\infty$ and $d\ge1$ that weighted Caffarelli-Kohn-Nirenberg inequalities hold true. As an application of our weighted Hardy inequality, we prove a non-existence result for a p-Kolmogorov parabolic equation.
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.
In a very general quasilinear setting, we show that the regularizing effect of a first order term causes the existence of energy solutions for problems involving the Hardy potential and $L^1$ data. In the same setting we study sharp (local and global) integral estimates for the second derivatives of the solutions.
This paper examines systems of poly-harmonic equations of the Hardy--Sobolev type and the closely related weighted systems of integral equations involving Riesz potentials. Namely, it is shown that the two systems are equivalent under some appropriate conditions. Then a sharp criterion for the existence and non-existence of positive solutions is determined for both differential and integral versions of a Hardy--Sobolev type system with variable coefficients. In the constant coefficient case, Liouville type theorems for positive radial solutions are also established using radial decay estimates…
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.
Let $ (M,g) $ be a smooth compact Riemannian manifold of dimension $ N \geq 3 $. Given $p_0 \in M$, $λ\in \mathcal{R}$ and $σ\in (0,2]$, we study existence and non existence of minimizers of the following quotient: \begin{equation}\label{Paper Equation} μ_{λ,σ}=\inf_{u \in H^1(M)\setminus \lbrace0\rbrace} \frac{\displaystyle\int_M |\nabla u|^2 dv_g -λ\int_M u^2 dv_g }{\biggl(\displaystyle\int_M ρ^{-σ} |u|^{2^*(σ)} dv_g\biggl)^{2/2^*(σ)}}, \end{equation} where $ρ(.):=dist(p_0,.)$ denoted the geodesic distance from $p \in M$ to $p_0$. In particular for $σ=2$, we provide sufficient and necessary …
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.
We establish the existence of ground states on Euclidean space for the Laplace operator involving the Hardy type potential. This gives rise to the existence of the principal eigenfunctions for the Laplace operator involving weighted Hardy potentials. We also obtain a higher integrability property for the principal eigenfunction. This is used to examine the behaviour of the principal eigenfunction around 0.
In this paper, we consider the critical order Hardy-Hénon equations \begin{equation*} (-Δ)^{\frac{n}{2}}u(x)=\frac{u^{p}(x)}{|x|^{a}}, \,\,\,\,\,\,\,\,\,\,\, x \, \in \,\, \mathbb{R}^{n}, \end{equation*} where $n\geq4$ is even, $-\infty<a<n$, and $1<p<+\infty$. We first prove a Liouville theorem (Theorem \ref{Thm0}), that is, the unique nonnegative solution to this equation is $u\equiv0$. Then as an immediate application, we derive a priori estimates and hence existence of positive solutions to critical order Lane-Emden equations in bounded domains (Theorem \ref{Thm1} and \ref{Thm2}). Our resu…
In this paper, we are concerned with the non-critical higher order Lane-Emden-Hardy equations \begin{equation*} (-Δ)^{m}u(x)=\frac{u^{p}(x)}{|x|^{a}} \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n} \end{equation*} with $n\geq3$, $1\leq m<\frac{n}{2}$, $0\leq a<2m$, $1<p<\frac{n+2m-2a}{n-2m}$ if $0\leq a<2$, and $1<p<\infty$ if $2\leq a<2m$. We prove Liouville theorems for nonnegative classical solutions to the above Lane-Emden-Hardy equations (Theorem \ref{Thm0}), that is, the unique nonnegative solution is $u\equiv0$. As an application, we derive a priori estimates and existence of p…
Well-posedness and global existence for Hardy--Hénon parabolic equations in Herz spaces
Laboratory of Functional Analysis and Geometry of Spaces, Faculty of Mathematics and Informatics, Department of Mathematics, M’sila University（穆西拉大学数学与信息学院功能分析与空间几何实验室）
In this paper, we obtain necessary conditions and sufficient conditions on the initial data for the local-in-time solvability of the Cauchy problem \[ \partial_t u +(-Δ)^\fracθ{2} u=|x|^{-γ} u^p ,\quad x\in{\bf R}^N, t>0, \qquad u(0)=μ\quad \mbox{in} \quad {\bf R}^N, \] where $N\ge 1$, $0<θ\le2$, $p>1$, $γ>0$ and $μ$ is a nonnegative Radon measure on ${\bf R}^N$. Using these conditions, we attempt to identify the optimal strength of the singularity of $μ$ for the existence of solutions to this problem.
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.
The paper concentrates on the application of the following Hardy inequality \begin{equation*} \int_Ω\ |ξ(x)|^p ω_{1 }(x)dx\le \int_Ω|\nabla ξ(x)|^pω_{2 }(x)dx, \end{equation*} to the proof of existence of weak solutions to degenerate parabolic problems of the type \begin{equation*} \left\{\begin{array}{ll} u_t-div(ω_2(x)|\nabla u|^{p-2} \nabla u )= λW(x) |u|^{p-2}u& x\inΩ, u(x,0)=f(x)& x\inΩ, u(x,t)=0& x\in\partialΩ,\ t>0,\\ \end{array}\right. \end{equation*} on an open subset $Ω\subseteq\mathbb{R}^n$, not necessarily bounded, where \[W(x)\leq \min\{m,ω_1(x)\},\qquad m\in\mathbb{R}_+.\]
In this paper we study existence and nonexistence of nonnegative distributional solutions for a class of semilinear fractional elliptic equations involving the Hardy potential.
We are concerned with the existence of ground states for nonlinear Choquard equations involving a critical nonlinearity in the sense of Hardy-Littlewood-Sobolev. Our result complements previous results by Moroz and Van Schaftingen where the subcritical case was considered. Then, we focus on the existence of semi-classical states and by using a truncation argument approach, we establish the existence and concentration of single peak solutions concentrating around minima of the Schrödinger potential, as the Planck constant goes to zero. The result is robust in the sense that the nonlinearity is …
In order to obtain solutions to problem $$ {{array}{c} -Δu=\dfrac{A+h(x)} {|x|^2}u+k(x)u^{2^*-1}, x\in {\mathbb R}^N, u>0 \hbox{in}{\mathbb R}^N, {and}u\in {\mathcal D}^{1,2}({\mathbb R}^N), {array}. $$ $h$ and $k$ must be chosen taking into account not only the size of some norm but the shape. Moreover, if $h(x)\equiv 0$, to reach multiplicity of solution, some hypotheses about the local behaviour of $k$ close to the points of maximum are needed.
On a compact Riemannian manifold, we prove the existence of multiple solutions for an elliptic equation with critical Sobolev growth and critical Hardy potential.
We investigate the existence of extremals for Hardy-Sobolev inequalities involving the Dirichlet fractional Laplacian of order s, 0<s<1, on half-spaces.
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…
\ In this paper, the following biharmonic elliptic problem \begin{eqnarray*} \begin{cases} Δ^2u-λ\frac{|u|^{q-2}u}{|x|^s}=|u|^{2^{**}-2}u+ f(x,u), &x\inΩ,\\ u=\dfrac{\partial u}{\partial n}=0, &x\in\partialΩ\end{cases} \end{eqnarray*} is considered. The main feature of the equation is that it involves a Hardy term and a nonlinearity with critical Sobolev exponent. By combining a careful analysis of the fibering maps of the energy functional associated with the problem with the Mountain Pass Lemma, it is shown, for some positive parameter $λ$ depending on $s$ and $q$, that the problem admits at…
In this paper we deal with an inhomogeneous parabolic Dirichlet problem involving the 1-Laplacian operator. We show the existence of a unique solution when data belong to $L^1(0,T;L^2(Ω))$ for every $T>0$. As a consequence, global existence and uniqueness for data in $L^1_{loc}(0,+\infty;L^2(Ω))$ is obtained. Our analysis retrieves the results of \cite{SW} in a correct and complete way.
This paper deals with the Cauchy problem for the Hardy-Hénon equation (and its fractional analogue). Local well-posedness for initial data in the class of continuous functions with slow decay at infinity is investigated. Small data (in critical weak-Lebesgue space) global well-posedness is obtained in $C_{b}([0,\infty);L^{q_c,\infty}(\RR^{n}))$. As a direct consequence, global existence for data in strong critical Lebesgue $L^{q_c}(\RR^{n})$ follows under a smallness condition while uniqueness is unconditional. Besides, we prove the existence of self-similar solutions and examine the long time…
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…
The present work is concerned with existence of positive solutions for a class of fractional equation involving a Kirchhoff term and singular potential.
Mixed local-nonlocal quasilinear problems with mixed interpolated Hardy potential
Ghent University（根特大学）；National Institute of Technology Calicut（卡利卡特国立技术学院）；Indian Institute of Technology (BHU)（印度理工学院（贝拿勒斯））；Queen Marry University of London（伦敦玛丽女王大学）
We prove multipolar Hardy inequalities on complete Riemannian manifolds, providing various curved counterparts of some Euclidean multipolar inequalities due to Cazacu and Zuazua [Improved multipolar Hardy inequalities, 2013]. We notice that our inequalities deeply depend on the curvature, providing (quantitative) information about the deflection from the flat case. By using these inequalities together with variational methods and group-theoretical arguments, we also establish non-existence, existence and multiplicity results for certain Schrödinger-type problems involving the Laplace-Beltrami …
This paper deals with a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponent \begin{equation*} \begin{cases} -Δu+(V_1(x)+λ_1)u=μ_1(|x|^{-4}*u^{2})u+β(|x|^{-4}*v^{2})u, \ \ &x\in R^N, -Δv+(V_2(x)+λ_2)v=μ_2(|x|^{-4}*v^{2})v+β(|x|^{-4}*u^{2})v, \ \ &x\in R^N, \end{cases} \end{equation*} where $N\geq 5$, $λ_1$, $λ_2\geq 0$ with $λ_1+λ_2\neq 0$, $V_1(x), V_{2}(x)\in L^{\frac{N}{2}}(R^N)$ are nonnegative functions and $μ_1$, $μ_2$, $β$ are positive constants. Such system arises from mathematical models in Bose-Einstein condensates theory and nonlinear optics. By variational methods…
This paper studies the existence of extremal problems for the Hardy-Littlewood-Sobolev inequalities on compact manifolds without boundary via Concentration-Compactness principle.
By means of variational methods we establish existence and multiplicity of solutions for a class of nonlinear nonlocal problems involving the fractional p-Laplacian and a combined Sobolev and Hardy nonlinearity at subcritical and critical growth.
Existence and uniqueness of a specific self-similar solution is established for the following reaction-diffusion equation with Hardy singular potential $$ \partial_tu=Δu^m+|x|^{-2}u^p, \qquad (x,t)\in \real^N\times(0,\infty), $$ in the range of exponents $1\leq p<m$ and dimension $N\geq3$. The self-similar solution is unbounded at $x=0$ and has a logarithmic vertical asymptote, but it remains bounded at any $x\neq0$ and $t\in(0,\infty)$ and it is a weak solution in $L^1$ sense, which moreover satisfies $u(t)\in L^p(\real^N)$ for any $t>0$ and $p\in[1,\infty)$. As an application of this self-si…
We study linear and non-linear equations related to the fractional Hardy--Sobolev inequality. We prove nondegeneracy of ground state solutions to the basic equation and investigate existence and qualitative properties, including symmetry of solutions to some perturbed equations.
Let $Ω\subset \mathbb{R}^n$ be a smooth bounded domain having zero in its interior $0 \in Ω.$ We fix $0 < α\le 2$ and $0 \le s <α.$ We investigate a sufficient condition for the existence of a positive solution for the following perturbed problem associated with the Hardy-Schrödinger operator $ L_{γ,α,}: = ({-}{ Δ})^{\fracα{2}}- \fracγ{|x|^α}$ on $Ω:$ \begin{equation*} \left\{\begin{array}{rl} \displaystyle ({-}{ Δ})^{\fracα{2}}u- γ\frac{u}{|x|^α} - λu= {\frac{u^{2_α^*(s)-1}}{|x|^s}}+ h(x) u^{q-1} & \text{in } Ω\\ u=0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,…
We give existence and regularity results for solutions of some nonlinear elliptic problems. The equations we deal with are modeled on a problem which involves in its principal part an anisotropic operator, a Hardy-type potential, and a lower-order term depending on the gradient.
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.
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…
In this article, we study the following nonlinear doubly nonlocal problem involving the fractional Laplacian in the sense of Hardy-Littlewood-Sobolev inequality \begin{equation*} \left\{\begin{aligned} (-Δ)^s u & = au+bv+\frac{2p}{p+q}\int_Ω\frac{|v(y)|^q}{|x-y|^μ}dy|u|^{p-2}u+2ξ_1\int_Ω\frac{|u(y)|^{2^*_μ}}{|x-y|^μ}dy|u|^{2^*_μ-2}u,&& \text{in } Ω;\\ (-Δ)^s v & = bu+cv+\frac{2q}{p+q}\int_Ω\frac{|u(y)|^p}{|x-y|^μ}dy|v|^{q-2}v+2ξ_2\int_Ω\frac{|v(y)|^{2^*_μ}}{|x-y|^μ}dy|v|^{2^*_μ-2}v,&& \text{in } Ω;\\ u &=v=0,\text{ in } \R^N\setminusΩ, \end{aligned}\right. \end{equation*} where $Ω$ is a smooth…
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…
In this article we will study the existence, multiplicity and Morse index of sign changing solutions for the Hardy-Sobolev-Maz'ya(HSM) equation in bounded domain and involving critical growth. We obtain infinitely many sign changing solutions for HSM equation. We also establish an estimate on the Morse index for the sign changing solutions.
We introduce tow assumptions weaker than the classical Ambrosetti-Rabinowitz and the subcritical polynomial growth conditions to obtain the Palais-Smale Condition. Therefore, we improve the Ambrosetti- Rabinowitz existence theorems. Also, we prove some existence results under a weaken subcritical polynomial growth conditions where the nonlinearity does not satisfy the super-quadratic condition and even it does not have any limits at infinity.
In this paper, we study the existence of three solutions for a Kirchhoff equation involving the nonlocal fractional p-Laplacian considering Sobolev and Hardy nonlinearities at subcritical and critical growths. The proof is based on Mountain Pass Theorem, Ekeland's variational principle and constrained minimization in Nehari sets.
