Stability and Multiple Solutions to Einstein-scalar Field Lichnerowicz Equation on Manifolds

In this paper, we study the stability and multiple solutions to Einstein-scalar field Lichnerowicz equation on compact Riemannian manifolds. In particular, in dimension no more than 5, we can find a different way (comparing with the previous result of Hebey-PacardPollack) by showing that there are at least two positive solutions or a unique positive solution according to the coercivity property of a quadratic form defined by the minimal solution obtained by the monotone method. When the coercive condition fails, we prove a uniqueness result. A positive solution of the Lichnerowicz equation is also found in a complete non-compact Riemannian manifold. Mathematics Subject Classification 2000: 35Jxx