GJMS-type operators on a compact Riemannian manifold: best constants and Coron-type solutions

We study the Hardy identities and inequalities on Cartan-Hadamard manifolds using the notion of a Bessel pair. These Hardy identities offer significantly more information on the existence/nonexistence of the extremal functions of the Hardy inequalities. These Hardy inequalities are in the spirit of Brezis-Vázquez in the Euclidean spaces. As direct consequences, we establish several Hardy type inequalities that provide substantial improvements as well as simple understandings to many known Hardy inequalities and Hardy-Poincaré-Sobolev type inequalities on hyperbolic spaces in the literature. 

/pdf/hardys-identities-and-inequalities-on-cartan-hadamard-3fraaen3.pdf

Hardy’s Identities and Inequalities on Cartan-Hadamard Manifolds

We prove a family of Hardy-Rellich and Poincar\'e identities and inequalities on the hyperbolic space having, as particular cases, improved Hardy-Rellich, Rellich and second order Poincar\'e inequalities. All remainder terms provided considerably improve those already known in literature, and all identities hold with same constants for radial operators also. Furthermore, as applications of the main results, second order versions of the uncertainty principle on the hyperbolic space are derived. 

/pdf/hardy-rellich-and-second-order-poincare-identities-on-the-2qqbddeh.pdf

Hardy–Rellich and second order Poincaré identities on the hyperbolic space via Bessel pairs

In this paper we consider Cartan-Hadamard manifolds (i.e. simply connected of non-positive sectional curvature) whose negative Ricci curvature grows polynomially at infinity. We show that a number of functional properties, which typically hold when the curvature is bounded, remain true in this setting. These include the characterization of Sobolev spaces on manifolds, the so-called Calderon-Zygmund inequalities and the $L^p$-positivity preserving property, i.e. $u\in L^p\ \&\ (-\Delta + 1)u\ge 0 \Rightarrow u\ge 0$. The main tool is a new class of first and second order Hardy-type inequalities on Cartan-Hadamard manifolds with a polynomial upper bound on the curvature. 
In the last part of the manuscript we prove the $L^p$-positivity preserving property, $p\in[1,+\infty]$, on manifolds with subquadratic negative part of the Ricci curvature. This generalizes an idea of B. Guneysu and gives a new proof of a well-known condition for the stochastic completeness due to P. Hsu.

Some functional properties on Cartan-Hadamard manifolds of very negative curvature

Let $\Omega$ be a smooth bounded domain in R^n (n ≥ 3) such that 0 ∈ ∂Ω We consider issues of non-existence, existence, and multiplicity of variational solutions for the borderline Dirichlet problem, −∆u − γ u |x|^{-2} − h(x)u = |u|^{2*(s)−2} u |x|^{-s} in Ω, u = 0 on ∂Ω \ {0}, (E) where 0 < s < 2, 2*(s) := 2(n−s) n−2 , γ ∈ R and h ∈ C^0 (Ω) We use sharp blow-up analysis on-possibly high energy-solutions of corresponding subcritical problems to establish, for example, that if γ < n^2/4 − 1 and the principal curvatures of ∂Ω at 0 are non-positive but not all of them vanishing, then Equation (E) has an infinite number of high energy (possibly sign-changing) solutions This complements results of the first and third authors, who showed in [20] that if γ ≤ n^2/4 − 1/4 and the mean curvature of ∂Ω at 0 is negative, then (E) has a positive least energy solution On the other hand, the sharp blow-up analysis also allows us to show that if the mean curvature at 0 is nonzero and the mass, when defined, is also nonzero, then there is a surprising stability of regimes where there are no variational positive solutions under C^1-perturbations of the potential h In particular, and in sharp contrast with the non-singular case (ie, when γ = s = 0), we prove non-existence of such solutions for (E) in any dimension, whenever Ω is star-shaped and h is close to 0, which include situations not covered by the classical Pohozaev obstruction

Multiplicity and stability of the Pohozaev obstruction for Hardy-Schrödinger equations with boundary singularity

We consider the Hardy-Schr\"odinger operator $ -\Delta_{\mathbb{B}^n}-\gamma{V_2}$ on the Poincar\'e ball model of the Hyperbolic space ${\mathbb{B}^n}$ ($n \geq 3$). Here $V_2$ is a well chosen radially symmetric potential, which behaves like the Hardy potential around its singularity at $0$, i.e., $V_2(r)\sim \frac{1}{r^2}$. Just like in the Euclidean setting, the operator $ -\Delta_{\mathbb{B}^n}-\gamma{V_2}$ is positive definite whenever $\gamma <\frac{(n-2)^2}{4}$, in which case we exhibit explicit solutions for the equation $$-\Delta_{\mathbb{B}^n}u-\gamma{V_2}u=V_{2^*(s)}u^{2^*(s)-1}\quad{\text{ in }}\mathbb{B}^n,$$ where $0\leq s <2$, $2^*(s)=\frac{2(n-s)}{n-2}$, and $V_{2^*(s)}$ is a weight that behaves like $\frac{1}{r^s}$ around $0$. The same equation, on bounded domains $\Omega$ of ${\mathbb{B}^n}$ containing $0$ but not touching the hyperbolic boundary, has positive solutions if $0 < \gamma \leq \frac{(n-2)^{2}}{4}-\frac{1}{4}$. However, if $\frac{(n-2)^{2}}{4}-\frac{1}{4}< \gamma < \frac{(n-2)^{2}}{4}$, the existence of solutions requires the positivity of the "hyperbolic Hardy mass" $m_{_{\mathbb{B}^n}}(\Omega)$ of the domain, a notion that we introduce and analyse therein.

Mass and Extremals Associated with the Hardy-Schr\"odinger Operator on Hyperbolic Space

Mass and Extremals Associated with the Hardy-Schrödinger Operator on Hyperbolic Space

We study the Hardy identities and inequalities on Cartan-Hadamard manifolds using the notion of a Bessel pair. These Hardy identities offer significantly more information on the existence/nonexistence of the extremal functions of the Hardy inequalities. These Hardy inequalities are in the spirit of Brezis-Vazquez in the Euclidean spaces. As direct consequences, we establish several Hardy type inequalities that provide substantial improvements as well as simple understandings to many known Hardy inequalities and Hardy-Poincare-Sobolev type inequalities on hyperbolic spaces in the literature.

Hardy's Identities and Inequalities on Cartan-Hadamard Manifolds

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