A Problem Concerning Yamabe-Type Operators of Negative Admissible Metrics

This paper is about a problem concerning nonlinear Yamabe-type operators of negative admissible metrics. We ﬁrst give a result on σ k Yamabe problem of negative admissible metrics by virtue of the degree theory in nonlinear functional analysis and the maximum principle and then establish an existence and uniqueness theorem for the solutions to the problem.


Introduction
Let M, g be a compact closed, connected Riemannian manifold of dimension n ≥ 3.In 2003, Gursky-Viaclovsky 1 introduced a modified Schouten tensor as follows: where Ric g and R g are the Ricci tensor and the scalar curvature of g, respectively.Define

1.2
The σ k Yamabe problem is to find a metric g conformal to g, such that where λ g A g denotes the eigenvalue of A g with respect to the metric g.This problem has attracted great interest since the work of Viaclovsky in 2 cf., e.g., 2-7 and references therein . Assume is still elliptic see 1 .
Definition 1.1.A metric g conformal to g is called negative admissible if Under the conformal relation g e 2z g, the transformation law for the modified Schouten tensor above is as follows: We consider the following nonlinear equation: where where

1.13
ii β is concave on Ω , and where is a positive constant.
Moreover, assume that ϕ x, z is a positive C ∞ satisfying condition 1.9 .Then there exists a solution to 1.7 .

Theorem 1.3. Let M, g be a compact, closed, connected Riemannian manifold of dimension n ≥ 3 and
A t g ∈ Ω − , for t < 1.

1.15
Let β, Ω be those as in Theorem 1.2.Then there exist a function φ and a positive number λ, such that φ is a solution to the eigenvalue problem where for conformal metric g e 2φ and λ g U denotes the eigenvalue of U with respect to metric g.
3 under the sense that, if there is another solution φ , Λ satisfying 1.16 , then for some constant c. 2 Λ is called the eigenvalue related to fully nonlinear Yamabe-type operators of negative admissible metrics, and φ is called an eigenfunction with respect to Λ.

Proof of Theorem 1.2
To prove Theorem 1.2, firstly, let us give the following proposition.Proposition 2.1.Suppose all the conditions in Theorem 1.2 are satisfied.Then every C 2 solution z to 1.7 with Proof.Assume z is a solution to 1.7 with γ ≤ z.Denote

2.3
It is easy to verify that Z s ∈ Ω .Write Then On the other hand, for some bound b i and constant c, where by condition ii .
Therefore, we know that L is an elliptic operator, and By the maximum principle, we get z > 0. That is, z > γ. 2.9 Similarly, we can derive for solution z with z ≤ γ.
Thus, we have the following Gradient and Hessian estimates for solutions to 1.7 .
Proof of Theorem 1.2.We now prove Theorem 1.2 using a priori estimates in Lemma 2.2, the maximum principle in Proposition 2.1, and the degree theory in nonlinear functional analysis cf., e.g., 8 .

2.14
We consider the problem

2.16
Since A t g ∈ Ω − , we have .17 by condition ii .Hence for τ 0, it follows from the maximum principle that z 0 is the unique solution.

2.18
This, together with Lemma 2.2, shows that for each τ ∈ 0, 1 and solution z τ to 2.15 with γ ≤ z τ ≤ γ, the following estimate holds for some constant C independent of τ.This estimate yields uniform ellipticity, and by virtue of the concavity condition ii , the well-known theory of Evans-Krylov, and the standard Schauder estimate cf. 9 , we know that there exists a constant K independent of τ such that where z τ is a C 2 solution to 2.15 with γ ≤ z τ ≤ γ.Set and define T τ : C 4,α → C 2,α by

2.22
Then, by 2.19 , we see that there is no solution to the equation

2.23
So the degree of T τ is well defined and independent of τ.As mentioned above, there is a unique solution at τ 0. Therefore deg T 0 , S 0 , 0 / 0.

2.24
Since the degree is homotopy invariant, we have Thus, we conclude that 1.7 has a solution in S 1 .
The proof of Theorem 1.2 is completed.

Proof of Theorem 1.3
Proof of Theorem 1.3.Take a look at the following equation:

3.1
We will prove that, for small λ > 0, 3.1 has a unique smooth solution.
Since ∂ P/∂u < 0, the uniqueness of the solution to 3.1 follows from the maximum principle.
Next, we show the existence of the solution to 3.1 by using Theorem 1.2.It follows from that, for λ > 0 small enough, we can find two constants γ < 0 < γ, such that e γ λ < P −A t g < e γ λ.

3.3
That is, condition 1.9 for ϕ x, z in Theorem 1.2 is satisfied.Therefore, by the result in Theorem 1.2, the existence of unique solution to 3.1 is established for small λ > 0. Set E : {λ > 0; 3.1 has a solution}.

3.5
We claim Λ is finite.Actually,

3.6
If we assume that at x 0 , u achieves its maximum, then ∇ 2 u ≤ 0, and so

3.7
This means that For any sequence λ i ⊂ E with λ i → Λ, let u λ i be the corresponding solution to 3.1 with λ λ i .
First, we claim that inf Suppose this is not true, that is, for some constant C depending only on P −A t g .Then the apriori estimates imply that u λ i by taking a subsequence converges to a smooth function u 0 in C ∞ , such that u 0 satisfies 3.1 for λ λ 0 .Since the linearized operator of 3.1 is invertible, by the standard implicit function theorem, we have a solution to 3.1 for λ λ 0 δ with δ > 0 small enough.
3.12 This is a contradiction.Hence 3.9 holds.Next, we prove that max M u λ i −→ −∞ as i −→ ∞.

3.13
We divided our proof into two steps.
Step 1.Let Λ P −A t g .

3.18
That means that 3.13 holds. Step constant C 0 .Then, by 3.1 , at any maximum point x 0 of u λ i , | 2 − dz λ i ⊗ dz λ i − A t g e max M u λ i e z λ i λ i .3.23Since at any minimum point z 0 of z λ i , ∇ 2 z λ i ≥ 0, ∇z λ i 0.3.24