This talk is about a joint work, still in progress, with Friedrich Tomi (Heidelberg University, Germany) where one investigates the existence of solutions which are invariant by a Lie subgroup of the isometry group of a Riemannian manifold \(M\); acting freely and properly on \(M\) , to the Dirichlet problem of a certain class of partial differential equations on \(M\): Typical examples of this class are the \(p-\)Laplacian PDE and the minimal surface equation. This approach may reduce the study of the Dirichlet problem in unbounded to bounded domains and also allows to prove the existence of solutions on domains which are not necessarily mean convex in the case of the minimal surface equation for certain boundary data.
