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

TL;DR

This paper establishes the existence of positive solutions for Lichnerowicz-type equations with sign-changing nonlinearities on complete manifolds with boundary, relevant to Einstein-scalar field equations in General Relativity.

Contribution

It provides a new existence theorem for nonlinear boundary value problems on manifolds, connecting geometric analysis with physical models in General Relativity.

Findings

Proved existence of positive solutions under certain conditions.

Applied to Einstein-scalar field equations in the conformal method.

Addresses nonlinear Neumann boundary conditions.

Abstract

We prove an existence theorem for positive solutions to Lichnerowicz-type equations on complete manifolds with boundary 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.