G₂-Poisson equation on homogeneous spheres

This paper studies the Poisson equation for the G₂-Laplacian on 3-forms on the 7-sphere that are invariant under a transitive group action. We establish the existence and uniqueness of G-invariant solutions for G=SU(4),\: Spin(7),\: (Sp(2)× Sp(1))/Z₂. In the case G=Sp(2)× U(1)/Z₂, we show that the operator does not preserve the set of positive 3-forms. The paper also discusses the eigenvalue problem for the G₂-Laplacian. We classify G-invariant solutions for the above choices of G and determine which of these solutions are nearly parallel G₂-structures.