A couple of days ago, I got an acceptance for publication in Advances in Mathematics journal that makes me feel so exciting because of the prestige of the journal. This is part of my PhD thesis in NUS under the supervision of professor Xu. Besides, this is joint work with him.
The work looks like simple, I mean, we just try to solve the following PDE
where 
is the Laplace-Beltrami operator, 
is the critical Sobolev exponent, 
is a compact manifold without boundary of dimension 
, and 
, 
, 
are smooth functions. In our work, the above PDE is numbered as (1.2). I don’t want to mention the physical background of the equation, in a few words, this equation is motivated by the Hamiltonian constraint equations of General Relativity through the so-called conformal method. Apparently, the important and frequently studied prescribing scalar curvature equation is just a particular case.
In this work, we focus on the negative Yamabe-scalar field invariant, that is, 
. Our result basically consists of two theorems.
In the first result, we consider the case that may change its sign, we prove
Theorem 1.1. Let 
be a smooth compact Riemannian manifold without the boundary of dimension . Assume that and are smooth functions on such that 
, 
, 
, and 
where 
is given in (2.1) below. Let us also suppose that the integral of 
satisfies
where 
is the negative part of . Then there exists a number 
to be specified such that if
Equation (1.2) possesses at least two smooth positive solutions.
In the next result, we consider the case that 
. In this case, we are able to get a complete characterization of the existence of solutions. More precisely, we prove
Theorem 1.2. Let be a smooth compact Riemannian manifold without boundary of dimension . Let be a constant, and be smooth functions on with in , but not strictly negative. Then Equation (1.2) possesses one positive solution if and only if 
.
As one can see, the above theorem does not allow to be strictly negative. Fortunately, our approach can cover this case too. This is the last remark in the paper as we prove the following: if 
then Equation (1.2) always possesses one positive solution, I mean, without any condition on except the condition .
It is important to note that in the case , the solution is always unique by using the monotone trick.
