On solutions to Hardy-Sobolev equations on Riemannian manifolds

TL;DR

The paper analyzes Hardy-Sobolev type equations on closed Riemannian manifolds with singularities supported on the union $S$ of focal submanifolds of a proper isoparametric function. By restricting to $f$-invariant functions, the PDE reduces to an ordinary differential equation along the distance to $M_0$, enabling precise regularity results and a variational construction of solutions. Under coercivity of $\Delta_g+K$ and subcritical range $q<2^*_f(s)$, it proves the existence of a positive $f$-invariant minimizer with strong regularity, and it establishes a uniform $C^0$-bound to rule out blow-up. Utilizing the Mountain Pass theorem, the authors obtain infinitely many $f$-invariant solutions, including sign-changing ones, and show that their associated energy levels become unbounded. Overall, the work extends Hardy-Sobolev theory to manifolds with isoparametric foliations, providing a symmetry-based reduction and robust existence/regularity results for an entire family of solutions.

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

Theorems & Definitions (36)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Corollary 1.6
• Remark 1.7
• Remark 1.8
• Remark 2.1
• proof