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We prove an existence theorem for positive solutions to Lichnerowicz-typeequations on complete manifolds with boundary and nonlinear Neumann conditions.This kind of nonlinear problems arise quite naturally in the study of solutionsfor the Einstein-scalar field equations of General Relativity in the frameworkof the so called Conformal Method.

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