Global bifurcation techniques for Yamabe type equations on Riemannian manifolds

Abstract

We consider a closed Riemannian manifold $(M^n,g)$ of dimension $n\ge 3$ and study positive solutions of the equation $-{\Delta }_{g}u+\lambda u=\lambda u^q$, with $\lambda >0$, $q>1$. If $M$ supports a proper isoparametric function with focal varieties $M_1$, $M_2$ of dimension $d_1\ge d_2$ we show that for any $q<\frac{n-d_2+2}{n-d_2-2}$ the number of positive solutions of the equation $-{\Delta }_{g}u+\lambda u=\lambda u^q$ tends to $\infty$ as $\lambda \to +\infty$. We apply this result to prove multiplicity results for positive solutions of critical and supercritical equations. In particular we prove multiplicity results for the Yamabe equation on Riemannian manifolds.