Symmetry reduces the fractional Q-curvature optimal partition problem on the sphere to one dimension, yielding a minimizer made of disjoint spherical shells.
Local bifurcation diagrams and degenerate solutions of Yamabe-type equations
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abstract
We study positive solutions of the equation $-\Delta_g u + \lambda u = \lambda u^q$, with $\lambda >0$, $q>1$ on the round sphere $\mathbb{S}^n$ . We reduce the equation to an ordinary differential equation by considering isoparametric functions and apply bifurcation theory. We study when the corresponding bifurcation points are transcritical. We apply this result to show the existence of degenerate solutions to the equation and to study multiplicity results for conformal constant scalar curvature metrics.
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Fractional $Q$-curvature on the sphere and optimal partitions
Symmetry reduces the fractional Q-curvature optimal partition problem on the sphere to one dimension, yielding a minimizer made of disjoint spherical shells.