Effective multiplicity for the Einstein-scalar field Lichnerowicz equation

TL;DR

This paper proves the stability of the Einstein-scalar field Lichnerowicz equation under certain perturbations and demonstrates the existence of multiple solutions in various cases, highlighting the equation's multiplicity properties.

Contribution

It establishes stability results and the existence of multiple solutions for the Einstein-scalar field Lichnerowicz equation in specific dimensions and conditions.

Findings

Proves stability under subcritical perturbations in dimensions 3, 4, 5

Shows existence of a second solution in several cases

Identifies a unique solution at a critical value

Abstract

We prove the stability of the Einstein-scalar field Lichnerowicz equation under subcritical perturbations of the critical nonlinearity in dimensions 3, 4 and 5. As a consequence, we obtain the existence of a second solution to the equation in several cases. In particular, in the positive case, including the CMC positive cosmological constant case, we show that each time a solution exists, the equation produces a second solution with the exception of one critical value for which the solution is unique.