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Supercritical problems on the round sphere and the Yamabe problem in projective spaces. Fernandez, Juan Carlos


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.

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