1,720,968 research outputs found
Sign changing solutions of the Hardy-Sobolev-Maz'ya equation
No full textIn 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
Improved higher order Poincaré inequalities on the hyperbolic space via hardy-type remainder terms
No full textThe paper deals about Hardy-type inequalities associated with the following higher order Poincar\'e inequality: \left( \frac{N-1}{2} \right)^{2(k -l)} := \inf_{ u \in C^{\infty}_{0}(\mathbb{H}^{N} ) \setminus \{0\}} \frac{\int_{\hn} |\nabla_{\hn}^{k} u|^2 \ dv_{\hn}}{\int_{\hn} |\nabla_{\hn}^{l} u|^2 \ dv_{\hn} }\,, where are integers and \hn denotes the hyperbolic space. More precisely, we improve the Poincar\'e inequality associated with the above ratio by showing the existence of Hardy-type remainder terms. Furthermore, when and the existence of further remainder terms are provided and the sharpness of some constants is also discussed. As an application, we derive improved Rellich type inequalities on upper half space of the Euclidean space with non-standard remainder terms
Sign changing solutions of the Brezis-Nirenberg problem in the Hyperbolic space
No full textIn this article we will study the existence and nonexistence of sign changing solutions for the Brezis-Nirenberg type problem in the Hyperbolic space. We will also establish sharp asymptotic estimates for the solutions and the compactness properties of solutions
Sharp Poincaré–Hardy and Poincaré–Rellich inequalities on the hyperbolic space
Full text linkWe study Hardy-type inequalities associated to the quadratic form of the shifted Laplacian −ΔHN−(N−1)2/4 on the hyperbolic space HN, (N−1)2/4 being, as it is well-known, the bottom of the L2-spectrum of −ΔHN. We find the optimal constant in a resulting Poincaré–Hardy inequality, which includes a further remainder term which makes it sharp also locally: the resulting operator is in fact critical in the sense of [17]. A related improved Hardy inequality on more general manifolds, under suitable curvature assumption and allowing for the curvature to be possibly unbounded below, is also shown. It involves an explicit, curvature dependent and typically unbounded potential, and is again optimal in a suitable sense. Furthermore, with a different approach, we prove Rellich-type inequalities associated with the shifted Laplacian, which are again sharp in suitable senses
Nondegeneracy of positive solutions of semilinear elliptic problems in the hyperbolic space
Full text linkIn this article, we will study the nondegeneracy properties of positive finite energy solutions of the equation -Δu - λu = |u|p-1u in the hyperbolic space. We will show that the degeneracy occurs only in an N-dimensional subspace. We will prove that the positive solutions are nondegenerate in the case of geodesic balls
An optimal improvement for the Hardy inequality on the hyperbolic space and related manifolds
Full text linkWe prove optimal improvements of the Hardy inequality on the hyperbolic space. Here, optimal means that the resulting operator is critical in the sense of Devyver, Fraas, and Pinchover (2014), namely the associated inequality cannot be further improved. Such inequalities arise from more general, optimal ones valid for the operator where and is the bottom of the L^2 spectrum of the laplacian, a problem that had been studied in Berchio, Ganguly, and Grillo (2017) only for . A different, critical and new inequality on the hyperbolic space, locally of Hardy type is also shown. Such results have in fact greater generality since they are proved on general Cartan-Hadamard manifolds under curvature assumptions, possibly depending on the point. Existence/nonexistence of extremals for the related Hardy-Poincaré inequalities are also proved using concentration-compactness technique and a Liouville comparison theorem. As applications of our inequalities, we obtain an improved Rellich inequality and we derive a quantitative version of Heisenberg-Pauli-Weyl uncertainty principle for the operator
The fractional porous medium equation on the hyperbolic space
Full text linkWe consider a nonlinear degenerate parabolic equation of porous medium type, whose diffusion is driven by the (spectral) fractional Laplacian on the hyperbolic space. We provide existence results for solutions, in an appropriate weak sense, for data belonging either to the usual spaces or to larger (weighted) spaces determined either in terms of a ground state of , or of the (fractional) Green’s function. For such solutions, we also prove different kind of smoothing effects, in the form of quantitative estimates. To the best of our knowledge, this seems the first time in which the fractional porous medium equation has been treated on non-compact, geometrically non-trivial examples
Hardy-Rellich and second order Poincar\'e identities on the hyperbolic space via Bessel pairs
Full text linkWe 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.Comment: 22 Page
Classification of radial solutions to on Riemannian models
Full text linkWe provide a complete classification with respect to asymptotic behaviour,
stability and intersections properties of radial smooth solutions to the
equation on Riemannian model manifolds in dimension
. Our assumptions include Riemannian manifolds with sectional
curvatures bounded or unbounded from below. Intersection and stability
properties of radial solutions are influenced by the dimension in the sense
that two different kinds of behaviour occur when or ,
respectively. The crucial role of these dimensions in classifying solutions is
well-known in Euclidean space.Comment: 26 page
Improved Lp -Poincaré inequalities on the hyperbolic space
Full text linkWe investigate the possibility of improving the optimal Lp-Poincaré inequality on the hyperbolic space, where p>1. We prove several different, and independent, improved inequalities, one of which is a Poincaré–Hardy inequality, namely an improvement of the best Lp-Poincaré inequality in terms of a Hardy weight related to geodesic distance from a given pole. Certain Hardy–Maz’ya-type inequalities in the Euclidean half-space are also obtained
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