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Journal of Applied Mathematics
VolumeΒ 2012Β (2012), Article IDΒ 930868, 11 pages
http://dx.doi.org/10.1155/2012/930868
Research Article

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: YonghongΒ Yao

Copyright Β© 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.

Abstract

This paper is about a problem concerning nonlinear Yamabe-type operators of negative admissible metrics. We first give a result on πœŽπ‘˜ 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.

1. Introduction

Let (𝑀,𝑔) be a compact closed, connected Riemannian manifold of dimension 𝑛β‰₯3. In 2003, Gursky-Viaclovsky [1] introduced a modified Schouten tensor as follows:𝐴𝑑𝑔=1ξ‚΅π‘›βˆ’2Ricπ‘”βˆ’π‘‘π‘…π‘”π‘”ξ‚Ά2(π‘›βˆ’1),𝑑≀1,(1.1) where Ric𝑔 and 𝑅𝑔 are the Ricci tensor and the scalar curvature of 𝑔, respectively.

DefineπœŽπ‘˜(ξ“πœ†)=1≀𝑖1β‰€β‹―β‰€π‘–π‘˜β‰€π‘›πœ†π‘–1β‹―πœ†π‘–π‘˜ξ€·πœ†forπœ†=1,…,πœ†π‘›ξ€Έβˆˆβ„π‘›,Ξ©+π‘˜=ξ€½ξ€·πœ†πœ†=1,…,πœ†π‘›ξ€Έβˆˆβ„π‘›;πœŽπ‘—ξ€Ύ.(πœ†)>0,1β‰€π‘—β‰€π‘˜(1.2) The πœŽπ‘˜ Yamabe problem is to find a metric ̃𝑔 conformal to 𝑔, such thatπœŽπ‘˜ξ€·πœ†Μƒπ‘”ξ€·π΄Μƒπ‘”ξ€Έξ€Έ=1,πœ†Μƒπ‘”ξ€·π΄Μƒπ‘”ξ€ΈβˆˆΞ©+π‘˜on𝑀,(1.3) where πœ†Μƒπ‘”(𝐴̃𝑔) denotes the eigenvalue of 𝐴̃𝑔 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 Ξ©βˆ’π‘˜=βˆ’Ξ©+π‘˜. Then the πœŽπ‘˜ Yamabe problem in negative coneπœŽπ‘˜ξ€·βˆ’πœ†Μƒπ‘”ξ€·π΄Μƒπ‘”ξ€Έξ€Έ=1,πœ†Μƒπ‘”ξ€·π΄Μƒπ‘”ξ€ΈβˆˆΞ©βˆ’π‘˜on𝑀,(1.4) is still elliptic (see [1]).

Definition 1.1. A metric ̃𝑔 conformal to 𝑔 is called negative admissible if πœ†Μƒπ‘”ξ‚€π΄π‘‘Μƒπ‘”ξ‚βˆˆΞ©βˆ’π‘˜on𝑀.(1.5)
Under the conformal relation ̃𝑔=𝑒2𝑧𝑔, the transformation law for the modified Schouten tensor above is as follows:π΄πœΜƒπ‘”=π΄πœπ‘”βˆ’βˆ‡2π‘§βˆ’1βˆ’πœπ‘›βˆ’2(Δ𝑧)π‘”βˆ’2βˆ’πœ2|βˆ‡π‘§|2𝑔+π‘‘π‘§βŠ—π‘‘π‘§.(1.6) We consider the following nonlinear equation: π‘ƒξ€·πœ†(𝑍)∢=𝛽𝑔(𝑍)=πœ‘(π‘₯,z),πœ†π‘”(𝑍)∈Ωon𝑀,(1.7) where 𝑍=βˆ‡2𝑧+1βˆ’π‘‘π‘›βˆ’2(Δ𝑧)𝑔+2βˆ’π‘‘2|βˆ‡π‘§|2π‘”βˆ’π‘‘π‘§βŠ—π‘‘π‘§βˆ’π΄π‘‘π‘”,(1.8)π›½βˆˆπΆβˆž(Ξ©+)∩𝐢0(Ξ©+) is a symmetric function and is homogeneous of degree one normalized, and πœ‘ is a positive 𝐢∞ function satisfying the monotone condition: thereexiststwoconstants𝛾<0<πœ‘ξ‚€π›Ύwithπ‘₯,𝛾<π›½βˆ’πœ†π‘”ξ€·π΄π‘‘π‘”ξ€·ξ€Έξ€Έ<πœ‘π‘₯,𝛾,βˆ€π‘₯βˆˆπ‘€.(1.9) For this equation, we have the following.

Theorem 1.2. Let (𝑀,𝑔) be a compact, closed, connected Riemannian manifold of dimension 𝑛β‰₯3 and π΄π‘‘π‘”βˆˆΞ©βˆ’,for𝑑<1.(1.10) Suppose that Ξ©+,Ξ©βˆ’βŠ‚π‘…π‘› are open convex symmetric cones with vertex at the origin, satisfying Ξ©π‘›βŠ‚Ξ©βŠ‚Ξ©1,Ξ©βˆ’=βˆ’Ξ©+,(1.11) where Ξ©1ξƒ―ξ€·πœ†βˆΆ=πœ†=1,…,πœ†π‘›ξ€Έ;𝑛𝑖=1πœ†π‘–ξƒ°,Ξ©>0π‘›ξ€½ξ€·πœ†βˆΆ=πœ†=1,…,πœ†π‘›ξ€Έ;πœ†π‘–ξ€Ύ.>0for1≀𝑖≀𝑛(1.12) Let 𝛽 satisfy(i)𝛽>0 in Ξ©+, π›½π‘–βˆΆ=πœ•π›½/πœ•πœ†π‘–>0 on Ξ©+, and 𝛽(𝑒)=1 on Ξ©+, where 𝑒=(1,…,1).(1.13)(ii)𝛽 is concave on Ξ©+, and 𝛽(πœ†)β‰€πœšπœŽ1(πœ†),βˆ€πœ†βˆˆΞ©+,(1.14) where 𝜚 is a positive constant. Moreover, assume that πœ‘(π‘₯,𝑧) is a positive 𝐢∞ 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 𝑛β‰₯3 and π΄π‘‘π‘”βˆˆΞ©βˆ’,for𝑑<1.(1.15) Let (𝛽,Ξ©+) be those as in Theorem 1.2. Then there exist a function πœ™ and a positive number πœ†, such that πœ™ is a solution to the eigenvalue problem π‘ƒξ€·πœ†(π‘ˆ)∢=𝛽𝑔(π‘ˆ)=Ξ›,(1.16) where π‘ˆ=βˆ’π΄π‘‘Μƒπ‘”=βˆ‡2πœ™+1βˆ’π‘‘π‘›βˆ’2(Ξ”πœ™)𝑔+2βˆ’π‘‘2||||βˆ‡πœ™2π‘”βˆ’π‘‘πœ™βŠ—π‘‘πœ™βˆ’π΄π‘‘π‘”(1.17) for conformal metric ̃𝑔=𝑒2πœ™ and πœ†π‘”(π‘ˆ) denotes the eigenvalue of π‘ˆ with respect to metric 𝑔.

Remark 1.4. (1) (πœ™,Ξ›) is unique in Theorem 1.3 under the sense that, if there is another solution (πœ™ξ…ž,Ξ›ξ…ž) satisfying (1.16), then Ξ›=Ξ›ξ…ž,πœ™=πœ™ξ…ž+𝑐(1.18) for some constant 𝑐.
(2) Ξ› is called the eigenvalue related to fully nonlinear Yamabe-type operators of negative admissible metrics, and πœ™ is called an eigenfunction with respect to Ξ›.

2. 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 𝐢2 solution 𝑧 to (1.7) with 𝛾≀𝑧≀𝛾(2.1) satisfies 𝛾<𝑧<𝛾.(2.2)

Proof. Assume 𝑧 is a solution to (1.7) with 𝛾≀𝑧. Denote ̃𝑧=π‘§βˆ’π›Ύ,𝑧𝑠𝑍=𝑠𝑧+(1βˆ’π‘ )𝛾,𝑠=βˆ‡2𝑧𝑠+1βˆ’π‘‘ξ€·π‘›βˆ’2Δ𝑧𝑠𝑔+2βˆ’π‘‘2|βˆ‡π‘§π‘ |2π‘”βˆ’π‘‘π‘§π‘ βŠ—π‘‘π‘§π‘ βˆ’π΄π‘‘π‘”.(2.3) It is easy to verify that π‘π‘ βˆˆΞ©+.
Write𝑄[𝑧]=𝑃(𝑍)βˆ’πœ‘(π‘₯,𝑧).(2.4) Then 𝑄[𝑧]ξ‚ƒπ›Ύβˆ’π‘„ξ‚„ξ€·=0βˆ’π‘ƒβˆ’π΄π‘‘π‘”ξ€Έξ‚€+πœ‘π‘₯,𝛾.(2.5) On the other hand, 𝑄[𝑧]ξ‚ƒπ›Ύβˆ’π‘„ξ‚„=ξ€œ10𝑑𝑄𝑧𝑑𝑠𝑠=ξ€œπ‘‘π‘ 10𝑇𝑖𝑗𝑍𝑠𝑑𝑠𝐷𝑖𝑗̃𝑧+𝑏𝑖𝐷𝑖̃𝑧+𝑐̃𝑧=𝐿(̃𝑧)(2.6) for some bound 𝑏𝑖 and constant 𝑐, where 𝑇𝑖𝑗=𝑃𝑖𝑗+1βˆ’π‘‘ξ“π‘›βˆ’2𝑙𝑃𝑙𝑙𝛾𝑖𝑗𝑃β‰₯0,𝑖𝑗=πœ•π‘ƒπœ•π‘π‘–π‘—β‰₯0(2.7) by condition (ii).
Therefore, we know that 𝐿 is an elliptic operator, and𝐿(̃𝑧)<0with̃𝑧β‰₯0.(2.8) By the maximum principle, we get ̃𝑧>0. That is, 𝑧>𝛾.(2.9) Similarly, we can derive 𝑧<𝛾,(2.10) for solution 𝑧 with 𝑧≀𝛾.

Thus, we have the following Gradient and Hessian estimates for solutions to (1.7).

Lemma 2.2. Let 𝑧 be a 𝐢3 solution to (1.7) for some 𝑑<1 satisfying 𝛾<𝑧<𝛾. Then β€–βˆ‡π‘§β€–πΏβˆž<𝐢1,(2.11) where 𝐢1 depends only upon 𝛾,𝛾,𝑔,𝑑,πœ‘.
Moreover,β€–β€–βˆ‡2π‘§β€–β€–πΏβˆž<𝐢2,(2.12) where 𝐢2 depends only upon 𝛾,𝛾,𝑔,𝑑,πœ‘,𝐢1.

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 0β‰€πœβ‰€1, letπ›½πœξ€·(πœ†)∢=π›½πœπœ†+(1βˆ’πœ)𝜎1ξ€Έ(πœ†)𝑒,(2.13) (here 𝑒=(1,…,1) as in Section 1) which is defined on Ξ©+𝜏=ξ€½πœ†βˆˆβ„π‘›;πœπœ†+(1βˆ’πœ)𝜎1(πœ†)π‘’βˆˆΞ©+ξ€Ύ.(2.14) We consider the problem π‘ƒξ€·πœπ‘+(1βˆ’πœ)𝜎1ξ€Έ(𝑍)𝑒=πœπœ‘(π‘₯,𝑧)+(1βˆ’πœ)𝜎1ξ€·βˆ’π΄π‘‘π‘”ξ€Έπ‘’2𝑧(2.15) on 𝑀, where 𝑍=βˆ‡2𝑧+1βˆ’π‘‘π‘›βˆ’2(Δ𝑧)𝑔+2βˆ’π‘‘2||||βˆ‡π‘§2π‘”βˆ’π‘‘π‘§βŠ—π‘‘π‘§βˆ’π΄π‘‘π‘”.(2.16) Since π΄π‘‘π‘”βˆˆΞ©βˆ’, we have 𝜎1ξ€·βˆ’π΄π‘‘π‘”ξ€Έ>0(2.17) 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 𝐢2 solution π‘§πœ to (2.15) with π›Ύβ‰€π‘§πœβ‰€π›Ύ satisfies𝛾<π‘§πœ<𝛾.(2.18) This, together with Lemma 2.2, shows that for each 𝜏∈[0,1] and solution π‘§πœ to (2.15) with π›Ύβ‰€π‘§πœβ‰€π›Ύ, the following estimate holds β€–π‘§πœβ€–πΆ2<𝐢,(2.19) for some constant 𝐢 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 thatβ€–π‘§πœβ€–πΆ4,𝛼<𝐾,(2.20) where π‘§πœ is a 𝐢2 solution to (2.15) with π›Ύβ‰€π‘§πœβ‰€π›Ύ.
Setπ‘†πœξ‚†π›ΎβˆΆ=<π‘§πœ<π›Ύξ‚‡βˆ©ξ€½β€–π‘§πœβ€–πΆ4,π›Όξ€Ύβˆ©ξ€½<πΎπ‘βˆˆΞ©+πœξ€Ύ,(2.21) and define π‘‡πœβˆΆπΆ4,𝛼→𝐢2,𝛼 by π‘‡πœξ€·(𝑧)=π‘ƒπœπ‘+(1βˆ’πœ)𝜎1ξ€Έ(𝑍)π‘’βˆ’πœπœ‘(π‘₯,𝑧)βˆ’(1βˆ’πœ)𝜎1ξ€·βˆ’π΄π‘‘π‘”ξ€Έπ‘’2𝑧.(2.22) Then, by (2.19), we see that there is no solution to the equation π‘‡πœ(𝑧)=0onπœ•π‘†πœ.(2.23) So the degree of π‘‡πœ is well defined and independent of 𝜏. As mentioned above, there is a unique solution at 𝜏=0. Therefore 𝑇deg0,𝑆0ξ€Έ,0β‰ 0.(2.24) Since the degree is homotopy invariant, we have 𝑇deg1,𝑆1ξ€Έ,0β‰ 0.(2.25) Thus, we conclude that (1.7) has a solution in 𝑆1.
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: ξ‚ξ‚€βˆ‡π‘ƒ(𝑒)=𝑃2𝑒+1βˆ’π‘‘π‘›βˆ’2(Δ𝑒)𝑔+2βˆ’π‘‘2|βˆ‡π‘’|2π‘”βˆ’π‘‘π‘’βŠ—π‘‘π‘’βˆ’π΄π‘‘π‘”ξ‚βˆ’π‘’π‘’=πœ†.(3.1) We will prove that, for small πœ†>0, (3.1) has a unique smooth solution.
Since πœ•ξ‚π‘ƒ/πœ•π‘’<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 fromπ΄π‘‘π‘”βˆˆΞ©βˆ’(3.2) that, for πœ†>0 small enough, we can find two constants 𝛾<0<𝛾, such that 𝑒𝛾+πœ†<π‘ƒβˆ’π΄π‘‘π‘”ξ€Έ<𝑒𝛾+πœ†.(3.3) That is, condition (1.9) for πœ‘(π‘₯,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 πœ†>0.
Set𝐸∢={πœ†>0;(3.1)hasasolution}.(3.4) Since πΈβ‰ βˆ…, we can define Ξ›=supπœ†βˆˆπΈπœ†.(3.5) We claim Ξ› is finite. Actually, ξ‚€βˆ‡πœ†<𝑃2𝑒+1βˆ’π‘‘π‘›βˆ’2(Δ𝑒)𝑔+2βˆ’π‘‘2|βˆ‡π‘’|2π‘”βˆ’π‘‘π‘’βŠ—π‘‘π‘’βˆ’π΄π‘‘π‘”ξ‚.(3.6) If we assume that at π‘₯0, 𝑒 achieves its maximum, then βˆ‡2𝑒≀0, and so ξ‚€βˆ‡πœ†<𝑃2𝑒+1βˆ’π‘‘π‘›βˆ’2(Δ𝑒)π‘”βˆ’π΄π‘‘π‘”ξ‚ξ€·β‰€π‘ƒβˆ’π΄π‘‘π‘”ξ€Έ.(3.7) This means that ξ€·Ξ›β‰€π‘ƒβˆ’π΄π‘‘π‘”ξ€Έ.(3.8)
For any sequence πœ†π‘–βŠ‚πΈ with πœ†π‘–β†’Ξ›, let π‘’πœ†π‘– be the corresponding solution to (3.1) with πœ†=πœ†π‘–.
First, we claim thatinfπ‘€π‘’πœ†π‘–βŸΆβˆ’βˆžasπ‘–βŸΆβˆž.(3.9) Suppose this is not true, that is, infπ‘€π‘’πœ†π‘–β‰₯βˆ’πΆ0(3.10) for a positive constant 𝐢0. Then, by (3.1), at any maximum point π‘₯0 of π‘’πœ†π‘–, maxπ‘€π‘’πœ†π‘–β‰€πΆ(3.11) for some constant 𝐢 depending only on 𝑃(βˆ’π΄π‘‘π‘”). Then the apriori estimates imply that π‘’πœ†π‘– (by taking a subsequence) converges to a smooth function 𝑒0 in 𝐢∞, such that 𝑒0 satisfies (3.1) for πœ†=πœ†0. Since the linearized operator of (3.1) is invertible, by the standard implicit function theorem, we have a solution to (3.1) for πœ†=πœ†0+𝛿with𝛿>0smallenough.(3.12) This is a contradiction. Hence (3.9) holds.
Next, we prove thatmaxπ‘€π‘’πœ†π‘–βŸΆβˆ’βˆžasπ‘–βŸΆβˆž.(3.13)
We divided our proof into two steps.
Step 1. Let ξ€·Ξ›=π‘ƒβˆ’π΄π‘‘π‘”ξ€Έ.(3.14) Then, following the above argument, π‘’πœ†π‘–β†’πœ™0inC∞,(3.15) and (Ξ›,𝑒0) is a solution to (3.1). Assume 𝑒0 attains its maximum at 𝑦0. Then at 𝑦0, βˆ‡2𝑒0≀0,βˆ‡π‘’0=0.(3.16) Therefore, 𝑒𝑒0(𝑦0)ξ€·β‰€π‘ƒβˆ’π΄π‘‘π‘”ξ€Έβˆ’Ξ›=0.(3.17) So 𝑒0𝑦0ξ€Έ=βˆ’βˆž.(3.18) That means that (3.13) holds.Step 2. Let π‘ƒξ€·βˆ’π΄π‘‘π‘”ξ€Έβˆ’Ξ›=πœ›>0.(3.19) Then, if (3.13) is not true, that is, maxπ‘€π‘’πœ†π‘–β‰₯βˆ’πΆ0(3.20) for a positive constant 𝐢0, write π‘§πœ†π‘–βˆΆ=π‘’πœ†π‘–βˆ’maxπ‘€π‘’πœ†π‘–.(3.21) Then we have maxπ‘€π‘§πœ†π‘–βŸΆ0,infπ‘€π‘§πœ†π‘–βŸΆβˆ’βˆž,(3.22) as π‘–β†’βˆž.
On the other hand, π‘§πœ†π‘– satisfiesπ‘ƒξ‚€βˆ‡2π‘§πœ†π‘–+1βˆ’π‘‘ξ€·π‘›βˆ’2Ξ”π‘§πœ†π‘–ξ€Έπ‘”+2βˆ’π‘‘2|βˆ‡π‘§πœ†π‘–|2βˆ’π‘‘π‘§πœ†π‘–βŠ—π‘‘π‘§πœ†π‘–βˆ’π΄π‘‘π‘”ξ‚=𝑒maxπ‘€π‘’πœ†π‘–π‘’π‘§πœ†π‘–+πœ†π‘–.(3.23) Since at any minimum point 𝑧0 of π‘§πœ†π‘–, βˆ‡2π‘§πœ†π‘–β‰₯0,βˆ‡π‘§πœ†π‘–=0.(3.24) Consequently, at 𝑧0, we obtain 𝑒maxπ‘€π‘’πœ†π‘–π‘’π‘§πœ†π‘–ξ€·β‰₯π‘ƒβˆ’π΄π‘‘π‘”ξ€Έβˆ’Ξ›>0.(3.25) Thus, it is easy to verify that π‘§πœ†π‘– is bounded from below as π‘–β†’βˆž. This is a contradiction. So we see that (3.13) is true.
By a priori estimates results again, we deduce that π‘§πœ†π‘– converges to a smooth function 𝑧 in 𝐢∞ and 𝑧 satisfies (1.16) with πœ†=Ξ›.
Finally, let us prove the uniqueness.
Denoteπ‘βˆΆ=βˆ‡2𝑧+1βˆ’π‘‘π‘›βˆ’2(Δ𝑧)𝑔+2βˆ’π‘‘2|βˆ‡π‘§|2π‘”βˆ’π‘‘π‘§βŠ—π‘‘π‘§βˆ’π΄π‘‘π‘”,(3.26) and for any smooth functions 𝑧0 and 𝑧1, set 𝑣=𝑧1βˆ’π‘§0,𝑧𝑠=𝑠𝑧1+(1βˆ’π‘ )𝑧0,𝑍𝑠=βˆ‡2𝑧𝑠+1βˆ’π‘‘ξ€·π‘›βˆ’2Δ𝑧𝑠𝑔+2βˆ’π‘‘2|βˆ‡π‘§π‘ |2π‘”βˆ’π‘‘π‘§π‘ βŠ—π‘‘π‘§π‘ βˆ’π΄π‘‘π‘”.(3.27) Then we get 𝑃𝑍1ξ€Έξ€·π‘βˆ’π‘ƒ0ξ€Έ=ξ€œ10𝑑𝑃𝑍𝑑𝑠𝑠=ξ€œ10𝑃𝑖𝑗+1βˆ’π‘‘ξ“π‘›βˆ’2𝑙𝑃𝑙𝑙𝛾𝑖𝑗𝑍𝑠𝑑𝑠𝑣𝑖𝑗+𝑏𝑙𝑣𝑙(3.28) for some bounded 𝑏𝑙. Thus, if 𝑧0=πœ™,𝑧1=πœ™ξ…ž(3.29) are two solutions to (1.16) for some πœ† and πœ†ξ…ž, respectively, then π‘Žπ‘–π‘— is positive definite. Therefore, πœ™=πœ™ξ…ž+𝑐(3.30) for some constant 𝑐 by the maximum principle.

Acknowledgment

The authors acknowledge support from the NSF of China (11171210) and the Chinese Academy of Sciences.

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