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

Abstract

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.

Yamabe problem in the presence of singular Riemannian Foliations

“…Theorem A and Corollaries B and C directly expand the results obtained in [12,25,24]. In [12] solutions to (Y) were found for foliations arising from closed subgroups of isometries acting on a given Riemannian manifold, meanwhile [25,24] provided solutions for foliations arising from isoparametric functions (codimension one foliations).…”

“…Theorem A and Corollaries B and C directly expand the results obtained in [12,25,24]. In [12] solutions to (Y) were found for foliations arising from closed subgroups of isometries acting on a given Riemannian manifold, meanwhile [25,24] provided solutions for foliations arising from isoparametric functions (codimension one foliations). We point out that by the work of Radeschi [43] and of Farrell and Wu [23], the notion of a singular Riemannian foliation is more general than the one of a group action and a Riemannian submersion; that is, there are singular Riemannian foliations of arbitrary dimension which cannot arise from a group action nor from a Riemannian submersion (see Subsection 3.2 for a description of Radeschi's examples).…”

“…In the second part we give examples of singular Riemannian foliations. In particular, we present examples for which previous methods to find solutions to (Y) such as those [12,25,24,3] do not apply.…”

“…This construction was generalized by Radeschi in [43] and we present some details here. We recall that solutions to (Y) such as [12,25,24] do not apply to the foliations in [43] while our main theorems do apply.…”