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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 930868, 11 pages
A Problem Concerning Yamabe-Type Operators of Negative Admissible Metrics
1Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China
2Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
Received 14 February 2012; Accepted 29 February 2012
Academic Editor: Yonghong Yao
Copyright © 2012 Jin Liang and Huan Zhu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper is about a problem concerning nonlinear Yamabe-type operators of negative admissible metrics. We first give a result on 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.
Let be a compact closed, connected Riemannian manifold of dimension . In 2003, Gursky-Viaclovsky  introduced a modified Schouten tensor as follows: where and are the Ricci tensor and the scalar curvature of , respectively.
Define The Yamabe problem is to find a metric conformal to , such that where denotes the eigenvalue of with respect to the metric . This problem has attracted great interest since the work of Viaclovsky in  (cf., e.g., [2–7] and references therein).
Assume . Then the Yamabe problem in negative cone is still elliptic (see ).
Definition 1.1. A metric conformal to 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 is a positive function satisfying the monotone condition: For this equation, we have the following.
Theorem 1.2. Let be a compact, closed, connected Riemannian manifold of dimension and Suppose that are open convex symmetric cones with vertex at the origin, satisfying where Let satisfy(i) in , on , and on , where (ii) is concave on , and where is a positive constant. Moreover, assume that is a positive satisfying condition (1.9). Then there exists a solution to (1.7).
Theorem 1.3. Let be a compact, closed, connected Riemannian manifold of dimension and Let be those as in Theorem 1.2. Then there exist a function and a positive number , such that is a solution to the eigenvalue problem where for conformal metric and denotes the eigenvalue of with respect to metric .
Remark 1.4. (1) is unique in Theorem 1.3 under the sense that, if there is another solution satisfying (1.16), then
for some constant .
(2) is called the eigenvalue related to fully nonlinear Yamabe-type operators of negative admissible metrics, and is called an eigenfunction with respect to .
2. Proof of Theorem 1.2
To prove Theorem 1.2, firstly, let us give the following proposition.
Proof. Assume is a solution to (1.7) with . Denote
It is easy to verify that .
Write Then On the other hand, for some bound and constant , where by condition (ii).
Therefore, we know that is an elliptic operator, and By the maximum principle, we get . That is, Similarly, we can derive for solution with .
Thus, we have the following Gradient and Hessian estimates for solutions to (1.7).
Lemma 2.2. Let be a solution to (1.7) for some satisfying . Then
where depends only upon .
Moreover, where 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., ).
For each , let (here as in Section 1) which is defined on We consider the problem on , where Since , we have by condition (ii). Hence for , it follows from the maximum principle that is the unique solution.
In view of Proposition 2.1, we see that, for each , every solution to (2.15) with satisfies This, together with Lemma 2.2, shows that for each and solution to (2.15) with , the following estimate holds for some constant independent of .
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. ), we know that there exists a constant independent of such that where is a solution to (2.15) with .
Set and define by Then, by (2.19), we see that there is no solution to the equation So the degree of is well defined and independent of . As mentioned above, there is a unique solution at . Therefore Since the degree is homotopy invariant, we have Thus, we conclude that (1.7) has a solution in .
The proof of Theorem 1.2 is completed.
3. Proof of Theorem 1.3
Proof of Theorem 1.3. Take a look at the following equation:
We will prove that, for small , (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 small enough, we can find two constants , such that That is, condition (1.9) for 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 .
Set Since , we can define We claim is finite. Actually, If we assume that at , achieves its maximum, then , and so This means that
For any sequence with , let be the corresponding solution to (3.1) with .
First, we claim that Suppose this is not true, that is, for a positive constant . Then, by (3.1), at any maximum point of , for some constant depending only on . Then the apriori estimates imply that (by taking a subsequence) converges to a smooth function in , such that satisfies (3.1) for . 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.
Step 1. Let Then, following the above argument, and is a solution to (3.1). Assume attains its maximum at . Then at , Therefore, So That means that (3.13) holds.Step 2. Let Then, if (3.13) is not true, that is, for a positive constant , write Then we have as .
On the other hand, satisfies Since at any minimum point of , Consequently, at , we obtain Thus, it is easy to verify that is bounded from below as . This is a contradiction. So we see that (3.13) is true.
By a priori estimates results again, we deduce that converges to a smooth function in and satisfies (1.16) with .
Finally, let us prove the uniqueness.
Denote and for any smooth functions and , set Then we get for some bounded . Thus, if are two solutions to (1.16) for some and , respectively, then is positive definite. Therefore, for some constant by the maximum principle.
The authors acknowledge support from the NSF of China (11171210) and the Chinese Academy of Sciences.
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