A variational problem associated to a hyperbolic Caffarelli--Kohn--Nirenberg inequality

TL;DR

This paper proves a hyperbolic space version of the Caffarelli--Kohn--Nirenberg inequality and establishes existence and non-existence results for related elliptic equations using variational methods.

Contribution

It introduces a new hyperbolic space inequality and applies variational techniques to analyze solutions of associated semilinear elliptic equations.

Findings

Established a hyperbolic Caffarelli--Kohn--Nirenberg inequality.

Proved existence of positive radial solutions in the subcritical regime.

Showed non-existence of solutions in star-shaped domains for supercritical exponents.

Abstract

We prove a Caffarelli--Kohn--Nirenberg inequality in the hyperbolic space. For a semilinear elliptic equation involving the associated weighted Laplace--Beltrami operator, we establish variationally the existence of positive radial solutions in the subcritical regime. We also show a non-existence result in star-shaped domains when the exponent is supercritical.