On solutions to Hardy-Sobolev equations on Riemannian manifolds

TL;DR

This paper investigates the existence of solutions, including infinitely many sign-changing ones, for Hardy-Sobolev equations on Riemannian manifolds with singularities along focal submanifolds.

Contribution

It establishes the existence of infinitely many solutions, including sign-changing solutions, for Hardy-Sobolev equations on Riemannian manifolds with singularities.

Findings

Existence of solutions to Hardy-Sobolev equations on manifolds.

Construction of infinitely many sign-changing solutions.

Extension of classical results to singular geometric settings.

Abstract

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 $\Delta_g u+K(x)u=\frac{u^{q-1}}{\left(d_{S}(x)\right)^s}$, 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.