Isoparametric functions and nodal solutions of the Yamabe equation

Henry, Guillermo

doi:10.1007/s10455-019-09664-x

Cited by 7 publications

(9 citation statements)

References 74 publications

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“…Here the numbers n−n±+2 n−n±−2 ≥ p n are just the critical Sobolev exponents in dimensions n − n ± . Our Theorem improves the existence result stated by Henry in [27], giving an infinite number of distinct solutions instead of one. It also extends the multiplicity result in [22] to the subcritical and supercritical exponents.…”

Section: Introductionsupporting

confidence: 82%

“…[22,29]) yields the existence and uniqueness of the solutions to equation (2.4) with initial conditions w(0) = d ∈ R and w ′ (0) = 0, depending continuously on d. For d > 0, let w d := w(·, d) be the solution with initial values w d (0) = d and w ′ d (0) = 0. To assure the existence of an arbitrarily large number of zeroes, we use the following result, proven in [22,27].…”

Section: Double Shooting and The Proof Of Theorem 14mentioning

confidence: 99%

“…However, very little is known about the existence, multiplicity and blow-up of nodal solutions for the supercritical problem on the sphere (1.2). One of the few results known by the authors is given in [27], where the existence of at least one signchanging solution to the supercritical problem was settled. In this direction we will follow and generalize the ideas introduced in [22] to obtain an infinite number of nodal solutions to the supercritical and subcritical problem (1.2), having as level and critical sets isoparametric hypersurfaces and its focal submanifolds.…”

Section: Introductionmentioning

confidence: 99%

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Supercritical elliptic problems on the round sphere and nodal solutions to the Yamabe problem in projective spaces

Fernandez¹,

Palmas²,

Petean³

2020

Discrete &Amp; Continuous Dynamical Systems - A

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Given an isoparametric function f on the n-dimensional round sphere, we consider functions of the form u = w • f to reduce the semilinear elliptic problem −∆g 0 u + λu = λ |u| p−1 u on S n with λ > 0 and 1 < p, into a singular ODE in [0, π] of the form w ′′ + h(r)where h is an strictly decreasing function having exactly one zero in this interval and ℓ is a geometric constant. Using a double shooting method, together with a result for oscillating solutions to this kind of ODE, we obtain a sequence of sign-changing solutions to the first problem which are constant on the isoparametric hypersurfaces associated to f and blowing-up at one or two of the focal submanifolds generating the isoparametric family. Our methods apply also when p > n+2 n−2 , i.e., in the supercritical case. Moreover, using a reduction via harmonic morphisms, we prove existence and multiplicity of sign-changing solutions to the Yamabe problem on the complex and quaternionic space, having a finite disjoint union of isoparametric hipersurfaces as regular level sets.

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Section: Introductionsupporting

confidence: 82%

Section: Double Shooting and The Proof Of Theorem 14mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Supercritical elliptic problems on the round sphere and nodal solutions to the Yamabe problem in projective spaces

Fernandez¹,

Palmas²,

Petean³

2020

Discrete &Amp; Continuous Dynamical Systems - A

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“…To prove Theorem A we state and prove a Rellich-Kondrachov embedding theorem for the subspace of foliated Sobolev functions in Theorem H below, and a Principle of Symmetric criticality for the energy functional associated to problem (Y) in Theorem G. This is a generalization of Palais' Principle of Symmetric criticality [42], and also generalizes the work of Henry [33] in codimension one foliations. We point out that Theorem H has been proven independently by Alexandrino and Cavenaghi [3].…”

Section: Introductionmentioning

confidence: 92%

“…The study of the existence of sign-changing solutions has been recently carried out by several authors using very different techniques [4,13,12,21,25,33,48]. One of the approaches considered has been to find equivariant solutions with respect to a given compact Lie group action by isometries with positive dimensional orbits.…”

Section: Introductionmentioning

confidence: 99%

Yamabe problem in the presence of singular Riemannian Foliations

Corro¹,

Fernández²,

Perales³

2022

Preprint

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Using variational methods together with symmetries given by singular Riemannian foliations with positive dimensional leaves, we prove the existence of an infinite number of sign-changing solutions to Yamabe type problems, which are constant along the leaves of the foliation, and one positive solution of minimal energy among any other solution with these symmetries. In particular, we find sign-changing solutions to the Yamabe problem on the round sphere with new qualitative behavior when compared to previous results, that is, these solutions are constant along the leaves of a singular Riemannian foliation which is not induced neither by a group action nor by an isoparametric function. To prove the existence of these solutions, we prove a Sobolev embedding theorem for general singular Riemannian foliations, and a Principle of Symmetric Criticality for the associated energy functional to a Yamabe type problem.

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Yamabe problem in the presence of singular Riemannian Foliations

Corro

Fernandez²,

Perales³

2022

Calc. Var.

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We prove that the group of isometries preserving a metric foliation on a closed Alexandrov space X, or a singular Riemannian foliation on a manifold M is a closed subgroup of the isometry group of X in the case of a metric foliation, or of the isometry group of M for the case of a singular Riemannian foliation. We obtain a sharp upper bound for the dimension of these subgroups and show that, when equality holds, the foliations that realize this upper bound are induced by fiber bundles whose fibers are round spheres or projective spaces. Moreover, singular Riemannian foliations that realize the upper bound are induced by smooth fiber bundles whose fibers are round spheres or projective spaces.

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Isoparametric functions and nodal solutions of the Yamabe equation

Abstract: We prove existence results for nodal solutions of the Yamabe equation that are constant along the level sets of an isoparametric function.

Cited by 7 publications

References 74 publications

Supercritical elliptic problems on the round sphere and nodal solutions to the Yamabe problem in projective spaces

Supercritical elliptic problems on the round sphere and nodal solutions to the Yamabe problem in projective spaces

Yamabe problem in the presence of singular Riemannian Foliations

Yamabe problem in the presence of singular Riemannian Foliations

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