$G_2$-Poisson equation on homogeneous spheres

This paper studies the Poisson equation for the $G_2$-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 $G=SU(4),\: Spin(7),\: (Sp(2)\times Sp(1))/\mathbb{Z}_2$. In the case $G=Sp(2)\times U(1)/\mathbb{Z}_2$, we show that the operator does not preserve the set of positive 3-forms. The paper also discusses the eigenvalue problem for the $G_2$-Laplacian. We classify $G$-invariant solutions for the above choices of $G$ and determine which of these solutions are nearly parallel $G_2$-structures.