Lichnerowicz-type equations with sign-changing nonlinearities on complete manifolds with boundary

We prove an existence theorem for positive solutions to Lichnerowicz-type equations on complete manifolds with boundary (M,∂M,〈,〉) and nonlinear Neumann conditions. This kind of nonlinear problems arise quite naturally in the study of solutions for the Einstein-scalar field equations of General Relativity in the framework of the so called Conformal Method.