Nonexistence of Radial Optimal Functions for the Sobolev Inequality on Cartan-Hadamard Manifolds

Kawakami, Tatsuki; Muratori, Matteo

doi:10.1007/978-3-030-73363-6_9

Cited by 5 publications

(2 citation statements)

References 31 publications

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“…Recently, similar rigidity results regarding interpolation inequalities were proved in the papers [27,28], either upon requiring or not the validity of the Cartan-Hadamard conjecture (see also [15]). As for the Sobolev inequality with p = 2, it was shown in [24,Theorem 1.1] that no radial optimal function can exist unless M n ≡ R n , actually without assuming the Cartan-Hadamard conjecture. However, this result will now follow as a particular case of Theorem 1.3 below.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Some rigidity results for Sobolev inequalities and related PDEs on Cartan-Hadamard manifolds

Muratori,

Soave

2021

Preprint

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The Cartan-Hadamard conjecture states that, on every n-dimensional Cartan-Hadamard manifold M n , the isoperimetric inequality holds with Euclidean optimal constant, and any set attaining equality is necessarily isometric to a Euclidean ball. This conjecture was settled, with positive answer, for n ≤ 4. It was also shown that its validity in dimension n ensures that every p-Sobolev inequality (1 < p < n) holds on M n with Euclidean optimal constant. In this paper we address the problem of classifying all Cartan-Hadamard manifolds supporting an optimal function for the Sobolev inequality. We prove that, under the validity of the n-dimensional Cartan-Hadamard conjecture, the only such manifold is R n , and therefore any optimizer is an Aubin-Talenti profile (up to isometries). In particular, this is the case in dimension n ≤ 4.Optimal functions for the Sobolev inequality are weak solutions to the critical p-Laplace equation. Thus, in the second part of the paper, we address the classification of radial solutions (not necessarily optimizers) to such a PDE. Actually, we consider the more general critical or supercritical equationwhere q ≥ p * − 1. We show that if there exists a radial finite-energy solution, then M n is necessarily isometric to R n , q = p * − 1 and u is an Aubin-Talenti profile. Furthermore, on model manifolds, we describe the asymptotic behavior of radial solutions not lying in the energy space Ẇ 1,p (M n ), studying separately the p-stochastically complete and incomplete cases.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Some rigidity results for Sobolev inequalities and related PDEs on Cartan-Hadamard manifolds

Muratori,

Soave

2021

Preprint

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“…Concerning the critical or supercritical regimes, the situation is somehow more rigid, in the following sense. If u is a radial solution to (1.1) on a Cartan-Hadamard model manifold M n , with p + 1 ≥ 2 * and u D 1,2 (R n ) < +∞, then M n is necessarily isometric to R n and u is therefore an Aubin-Talenti function [20,Theorem 1.3] (see also [2,Theorem 2.4] for a related result obtained under additional assumptions on M n , and [16] for a previous rigidity result concerning solutions that minimize the Sobolev quotient). Furthermore, the asymptotic behavior of radial solutions is strongly affected by the global geometric properties of the underlying manifold: if M n is stochastically complete, then all radial solutions tend to 0 at infinity; otherwise, if it is stochastically incomplete, each solution converges to a strictly positive constant at infinity [20,Theorem 1.4].…”

Section: Introductionmentioning

confidence: 99%

Radial solutions of the Lane-Emden system on Cartan-Hadamard manifolds: asymptotics and rigidity

Muratori¹,

Soave²

2023

Preprint

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We investigate existence and qualitative properties of globally defined and positive radial solutions of the Lane-Emden system, posed on a Cartan-Hadamard model manifold M n . We prove that, for critical or supercritical exponents, there exists at least a one-parameter family of such solutions. Depending on the stochastic completeness or incompleteness of M n , we show that the existence region stays one dimensional in the former case, whereas it becomes two dimensional in the latter. Then, we study the asymptotics at infinity of solutions, which again exhibit a dichotomous behavior between the stochastically complete (where both components are forced to vanish) and incomplete cases. Finally, we prove a rigidity result for finite-energy solutions, showing that they exist if and only if M n is isometric to R n .

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Some Functional Properties on Cartan–Hadamard Manifolds of Very Negative Curvature

Marini,

Veronelli

2024

J Geom Anal

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Nonexistence of Radial Optimal Functions for the Sobolev Inequality on Cartan-Hadamard Manifolds

Cited by 5 publications

References 31 publications

Some rigidity results for Sobolev inequalities and related PDEs on Cartan-Hadamard manifolds

Some rigidity results for Sobolev inequalities and related PDEs on Cartan-Hadamard manifolds

Radial solutions of the Lane-Emden system on Cartan-Hadamard manifolds: asymptotics and rigidity

Some Functional Properties on Cartan–Hadamard Manifolds of Very Negative Curvature

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