TY - ELEC
AU - Juan Carlos Fernandez
PY - 2019
TI - Supercritical problems on the round sphere and the Yamabe problem in projective spaces.
LA - eng
M3 - Moving Image
AB - Given an isoparametric function $f$ on the round sphere and
considering the space of functions $w\circ f$, we reduce the Yamabe-type problem
$$(1)\qquad -\Delta_{g_0}+\lambda u=\lambda |u|^{p-1}u\ \hbox{on}\ \mathbb S^n$$
with $\lambda>0$ and $p>1$, into a second order singular ODE of the form
$$w\rq{}\rq{}+{h(r)\over \sin r} w\rq{}+\lambda \left(|w|^{p-1}w-w\right)=0,$$
with boundary conditions $w\rq{}(0)=0$ and $w\rq{}(\pi)=0$, and where $h$ is a monotone function with exactly one zero
on $[0, \pi]$. Using a double shooting method, for any $k\in\mathbb N$, if $n_1\le n_2$
are the dimensions of the focal submanifolds determined by $f$ and if $p \in \left(1,\frac{n-n_1+2}{n-n_1-2}\right)$, this problem admits a nodal solution having at least $k$ zeroes.
This yields a solution to problem $(1)$ having as nodal set a disjoint union
of at least $k$ connected isoparametric hypersurfaces. As an application and
using that the Hopf fibrations are Riemannian submersions with minimal
fibers, we give a multiplicity result of nodal solutions to the Yamabe problem
on $\mathbb C P^m$ and on $\mathbb HP^m,$ the complex and quaternionic projective spaces
respectively, with $m $ odd.
This is a joint work with Jimmy Petean and Oscar Palmas.
N2 - Given an isoparametric function $f$ on the round sphere and
considering the space of functions $w\circ f$, we reduce the Yamabe-type problem
$$(1)\qquad -\Delta_{g_0}+\lambda u=\lambda |u|^{p-1}u\ \hbox{on}\ \mathbb S^n$$
with $\lambda>0$ and $p>1$, into a second order singular ODE of the form
$$w\rq{}\rq{}+{h(r)\over \sin r} w\rq{}+\lambda \left(|w|^{p-1}w-w\right)=0,$$
with boundary conditions $w\rq{}(0)=0$ and $w\rq{}(\pi)=0$, and where $h$ is a monotone function with exactly one zero
on $[0, \pi]$. Using a double shooting method, for any $k\in\mathbb N$, if $n_1\le n_2$
are the dimensions of the focal submanifolds determined by $f$ and if $p \in \left(1,\frac{n-n_1+2}{n-n_1-2}\right)$, this problem admits a nodal solution having at least $k$ zeroes.
This yields a solution to problem $(1)$ having as nodal set a disjoint union
of at least $k$ connected isoparametric hypersurfaces. As an application and
using that the Hopf fibrations are Riemannian submersions with minimal
fibers, we give a multiplicity result of nodal solutions to the Yamabe problem
on $\mathbb C P^m$ and on $\mathbb HP^m,$ the complex and quaternionic projective spaces
respectively, with $m $ odd.
This is a joint work with Jimmy Petean and Oscar Palmas.
UR - https://open.library.ubc.ca/collections/48630/items/1.0384929
ER - End of Reference