On solutions to Hardy-Sobolev equations on Riemannian manifolds

2025/06/25 by Henry, Guillermo, Petean, Jimmy · 1 citation
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.20089

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=\fracu^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.

Citations

• A note on sign-changing solutions to supercritical Yamabe-type equations
• Compact Sobolev embeddings on non-compact manifolds via orbit expansions of isometry groups
• The second best constant for the Hardy-Sobolev inequality on manifolds
• Global bifurcation techniques for Yamabe type equations on Riemannian\n manifolds
• Hardy-Sobolev inequality with higher dimensional singularity
• Hardy-Sobolev inequality with singularity a curve
• Hardy and Hardy-Sobolev inequalities on Riemannian manifolds
• Isoparametric functions on exotic spheres
• Geometry of isoparametric hypersurfaces in Riemannian manifolds
• Isoparametric functions and exotic spheres
• The Effect of Curvature on the Best Constatnt in the Hardy-Sobolev Inequalities

Cited by

• Compactness for the Hardy-Sobolev equation on manifolds

Related

• Hardy and Hardy-Sobolev inequalities on Riemannian manifolds
• Hardy-Sobolev inequality with higher dimensional singularity
• Influence of the curvature in the existence of solutions for a two Hardy-Sobolev critical exponents
• Multiple solutions for Schrödinger equations on Riemannian manifolds via ∇-theorems
• Multiple solutions of an elliptic Hardy-Sobolev equation with critical exponents on compact Riemannian manifolds