Existence and uniqueness of radial solutions of semi-linear equations on manifolds

We investigate the existence and uniqueness of solutions for second-order semi-linear partial differential equations defined on a Riemannian manifold $M$. By combining differential geometry and analysis techniques, we establish the existence and uniqueness of constant solutions through the orbits of a group action. Our approach transforms such problems into equivalent ones over a submanifold $\Sigma$ of dimension one, which is transversal to the group action. This reduction leads us to a one-dimensional setting, where we can apply different results from the theory of ordinary differential equations. Our framework is versatile and includes the setups of polar actions or exponential coordinates, with particular examples such as the sphere, surfaces of revolution, and others.