On solutions to Hardy-Sobolev equations on Riemannian manifolds

Let $(M,g)$ be a closed Riemannian manifold of dimension at least $3$. Let $S$ be the union of the focal submanifolds of an isoparametric function on $(M,g)$. In this article we address the existence of solutions of the Hardy-Sobolev type equation ${\mathrm{\Delta }}_{g}u+K(x)u=\frac{u^{q-1}}{{(d_S(x))}^s}$, where $d_S(x)$ is the distance from $x$ to $S$ and $q>2$. In particular, we will prove the existence of infinite sign-changing solutions to the equation.