# Separable infinite harmonic functions in cones

## TL;DR

This paper investigates the existence and uniqueness of separable infinite harmonic functions in cones, showing they can be expressed in a specific form and satisfy a nonlinear eigenvalue problem on the sphere.

## Contribution

It establishes the existence and uniqueness of such harmonic functions with a separable form in cones, linking the problem to a nonlinear eigenvalue problem on the sphere.

## Key findings

- Existence of separable infinite harmonic functions in cones.
- Unique determination of exponents for smooth domains.
- Reduction to a nonlinear eigenvalue problem on the sphere.

## Abstract

We study the existence of separable infinite harmonic functions in any cone of R N vanishing on its boundary under the form u(r, $\sigma$) = r --$\beta$ $\omega$($\sigma$). We prove that such solutions exist, the spherical part $\omega$ satisfies a nonlinear eigenvalue problem on a subdomain of the sphere S N --1 and that the exponents $\beta$ = $\beta$ + > 0 and $\beta$ = $\beta$ -- < 0 are uniquely determined if the domain is smooth.