Symmetries, Hopf fibrations and supercritical elliptic problems

Clapp, Mónica; Pistoia, Angela

doi:10.1090/conm/656/13100

Cited by 14 publications

(6 citation statements)

References 28 publications

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“…When M = Ω is a bounded domain of R n+1 with smooth boundary, there has been recent progress in handling supercritical exponent problems like (1.1). A fruitful approach consists in reducing the supercritical problem to a more general elliptic critical or subcritical problem, either by considering rotational symmetries or by means of maps preserving the Laplace operator or by a combination of both, see [17] and the references therein. In case of closed Riemannian manifolds, these reduction methods also apply and have been combined with the Lyapunov-Schmidt reduction method in order to obtain sequences of positive and sign-changing solutions to similar supercritical problems, such that they blow-up or concentrate at minimal submanifolds of M [16,21,25,32,39].…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

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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“…Lemma 3.2. The functional E K defined in (15) satisfies the mountain pass geometry and (PS) compactness condition.…”

Section: The Specific Settingmentioning

confidence: 99%

“…We now return to the Dirichlet problems. There have been many supercritical works that deal with domains that have certain symmetry, for instance, see [15,16,17,18,19,20,21].…”

Section: Introductionmentioning

confidence: 99%

Supercritical elliptic problems on nonradial domains via a nonsmooth variational approach

Cowan¹,

Moameni²

2021

Preprint

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In this paper we are interested in positive classical solutions ofwhere Ω is a bounded annular domain (not necessarily an annulus) in R N (N ≥ 3) and a(x) is a nonnegative smooth function. We show the existence of a classical positive solution for all p > 2 when the problem enjoys certain mild symmetry conditions. As a consequence of our results, we shall show that (1) has a nonradial solution when a(x) = 1 and Ω is an annulus with certain assumptions on the radii. Our approach is based on a new variational principle that allows one to deal with supercritical problems variationally by limiting the corresponding functional on a proper convex subset instead of the whole space.

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“…In this case (1.1) reduces to a subcritical equation on the base. Supercritical equations are more difficult to study and have also been of great interest in analysis, see for instance [8,9,15].…”

Section: Introductionmentioning

confidence: 99%

Nodal solutions of Yamabe-type equations on positive Ricci curvature manifolds

Julio-Batalla¹,

Petean²

2020

Preprint

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We consider a closed cohomogeneity one Riemannian manifold (M n , g) of dimension n ≥ 3. If the Ricci curvature of M is positive, we prove the existence of infinite nodal solutions for equations of the form −∆ g u + λu = λu q with λ > 0, q > 1. In particular for a positive Einstein manifold which is of cohomogeneity one or fibers over a cohomogeniety one Einstein manifold we prove the existence of infinite nodal solutions for the Yamabe equation, with a prescribed number of connected components of its nodal domain.

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Symmetries, Hopf fibrations and supercritical elliptic problems

Cited by 14 publications

References 28 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

Supercritical elliptic problems on nonradial domains via a nonsmooth variational approach

Nodal solutions of Yamabe-type equations on positive Ricci curvature manifolds

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