A Problem Concerning Yamabe-Type Operators of Negative Admissible Metrics

Definition 1.1. A metric $$ conformal to g is called negative admissible if

Under the conformal relation $$, the transformation law for the modified Schouten tensor above is as follows:

We consider the following nonlinear equation: where $$ is a symmetric function and is homogeneous of degree one normalized, and φ is a positive C^∞ function satisfying the monotone condition: For this equation, we have the following.

Theorem 1.2. Let (M, g) be a compact, closed, connected Riemannian manifold of dimension n ≥ 3 and

Suppose that Ω⁺, Ω⁻ ⊂ Rⁿ are open convex symmetric cones with vertex at the origin, satisfying where Let β satisfy

• (i)

  β > 0 in Ω⁺, β_i : = ∂β/∂λ_i > 0 on Ω⁺, and β(e) = 1 on Ω⁺, where

  $$ (1.13)

• (ii)

  β is concave on Ω⁺, and

  $$ (1.14)
  where ϱ 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

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 $$ and λ_g(U) denotes the eigenvalue of U with respect to metric g.

Remark 1.4. (1) (ϕ, Λ) is unique in Theorem 1.3 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.

Proposition 2.1. Suppose all the conditions in Theorem 1.2 are satisfied. Then every C² solution z to (1.7) with

satisfies

Proof. Assume z is a solution to (1.7) with $$. Denote

It is easy to verify that Z_s ∈ Ω⁺.

Write

Then On the other hand, for some bound bⁱ and constant c, where by condition (ii).

Therefore, we know that L is an elliptic operator, and

By the maximum principle, we get $$. That is, Similarly, we can derive for solution z with $$.

Lemma 2.2. Let z be a C³ solution to (1.7) for some t < 1 satisfying $$. Then

where C₁ depends only upon $$.

Moreover,

where C₂ depends only upon $$.

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]).

For each 0 ≤ τ ≤ 1, let

(here e = (1, …, 1) as in Section 1) which is defined on We consider the problem on M, where Since $$, we have by condition (ii). Hence for τ = 0, it follows from the maximum principle that z = 0 is the unique solution.

In view of Proposition 2.1, we see that, for each τ ∈ [0,1], every C² solution z^τ to (2.15) with $$ satisfies

This, together with Lemma 2.2, shows that for each τ ∈ [0,1] and solution z^τ to (2.15) with $$, 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 C² solution to (2.15) with $$.

Set

and define T_τ : C^4,α → C^2,α by Then, by (2.19), we see that there is no solution to the equation 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 Since the degree is homotopy invariant, we have Thus, we conclude that (1.7) has a solution in S₁.

The proof of Theorem 1.2 is completed.

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

We will prove that, for small λ > 0, (3.1) has a unique smooth solution.

Since $$, 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 $$, such that 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

Since E ≠ ∅, we can define We claim Λ is finite. Actually, If we assume that at x₀, u achieves its maximum, then ∇²u ≤ 0, and so This means that

For any sequence λ_i ⊂ E with λ_i → Λ, let $$ be the corresponding solution to (3.1) with λ = λ_i.

First, we claim that

Suppose this is not true, that is, for a positive constant C₀. Then, by (3.1), at any maximum point x₀ of $$, for some constant C depending only on $$. Then the apriori estimates imply that $$ (by taking a subsequence) converges to a smooth function u₀ in C^∞, such that u₀ satisfies (3.1) for λ = λ₀. Since the linearized operator of (3.1) is invertible, by the standard implicit function theorem, we have a solution to (3.1) for This is a contradiction. Hence (3.9) holds.

Next, we prove that

We divided our proof into two steps.