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

A Problem Concerning Yamabe-Type Operators of Negative Admissible Metrics

Jin Liang1Β and Huan Zhu2

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 𝜎 π‘˜ 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 𝑛 β‰₯ 3 . In 2003, Gursky-Viaclovsky [1] introduced a modified Schouten tensor as follows: 𝐴 𝑑 𝑔 = 1 ξ‚΅ 𝑛 βˆ’ 2 R i c 𝑔 βˆ’ 𝑑 𝑅 𝑔 𝑔 ξ‚Ά 2 ( 𝑛 βˆ’ 1 ) , 𝑑 ≀ 1 , ( 1 . 1 ) where R i c 𝑔 and 𝑅 𝑔 are the Ricci tensor and the scalar curvature of 𝑔 , respectively.

Define 𝜎 π‘˜ (  πœ† ) = 1 ≀ 𝑖 1 ≀ β‹― ≀ 𝑖 π‘˜ ≀ 𝑛 πœ† 𝑖 1 β‹― πœ† 𝑖 π‘˜ ξ€· πœ† f o r πœ† = 1 , … , πœ† 𝑛 ξ€Έ ∈ ℝ 𝑛 , Ξ© + π‘˜ = ξ€½ ξ€· πœ† πœ† = 1 , … , πœ† 𝑛 ξ€Έ ∈ ℝ 𝑛 ; 𝜎 𝑗 ξ€Ύ . ( πœ† ) > 0 , 1 ≀ 𝑗 ≀ π‘˜ ( 1 . 2 ) The 𝜎 π‘˜ Yamabe problem is to find a metric Μƒ 𝑔 conformal to 𝑔 , such that 𝜎 π‘˜ ξ€· πœ† Μƒ 𝑔 ξ€· 𝐴 Μƒ 𝑔 ξ€Έ ξ€Έ = 1 , πœ† Μƒ 𝑔 ξ€· 𝐴 Μƒ 𝑔 ξ€Έ ∈ Ξ© + π‘˜ o n 𝑀 , ( 1 . 3 ) where πœ† Μƒ 𝑔 ( 𝐴 Μƒ 𝑔 ) denotes the eigenvalue of 𝐴 Μƒ 𝑔 with respect to the metric Μƒ 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 𝜎 π‘˜ Yamabe problem in negative cone 𝜎 π‘˜ ξ€· βˆ’ πœ† Μƒ 𝑔 ξ€· 𝐴 Μƒ 𝑔 ξ€Έ ξ€Έ = 1 , πœ† Μƒ 𝑔 ξ€· 𝐴 Μƒ 𝑔 ξ€Έ ∈ Ξ© βˆ’ π‘˜ o n 𝑀 , ( 1 . 4 ) is still elliptic (see [1]).

Definition 1.1. A metric Μƒ 𝑔 conformal to 𝑔 is called negative admissible if πœ† Μƒ 𝑔 ξ‚€ 𝐴 𝑑 Μƒ 𝑔  ∈ Ξ© βˆ’ π‘˜ o n 𝑀 . ( 1 . 5 )
Under the conformal relation Μƒ 𝑔 = 𝑒 2 𝑧 𝑔 , the transformation law for the modified Schouten tensor above is as follows: 𝐴 𝜏 Μƒ 𝑔 = 𝐴 𝜏 𝑔 βˆ’ βˆ‡ 2 𝑧 βˆ’ 1 βˆ’ 𝜏 𝑛 βˆ’ 2 ( Ξ” 𝑧 ) 𝑔 βˆ’ 2 βˆ’ 𝜏 2 | βˆ‡ 𝑧 | 2 𝑔 + 𝑑 𝑧 βŠ— 𝑑 𝑧 . ( 1 . 6 ) We consider the following nonlinear equation: 𝑃 ξ€· πœ† ( 𝑍 ) ∢ = 𝛽 𝑔 ξ€Έ ( 𝑍 ) = πœ‘ ( π‘₯ , z ) , πœ† 𝑔 ( 𝑍 ) ∈ Ξ© o n 𝑀 , ( 1 . 7 ) where 𝑍 = βˆ‡ 2 𝑧 + 1 βˆ’ 𝑑 𝑛 βˆ’ 2 ( Ξ” 𝑧 ) 𝑔 + 2 βˆ’ 𝑑 2 | βˆ‡ 𝑧 | 2 𝑔 βˆ’ 𝑑 𝑧 βŠ— 𝑑 𝑧 βˆ’ 𝐴 𝑑 𝑔 , ( 1 . 8 ) 𝛽 ∈ 𝐢 ∞ ( Ξ© + ) ∩ 𝐢 0 ( Ξ© + ) is a symmetric function and is homogeneous of degree one normalized, and πœ‘ is a positive 𝐢 ∞ function satisfying the monotone condition: t h e r e e x i s t s t w o c o n s t a n t s 𝛾 < 0 < πœ‘ ξ‚€ 𝛾 w i t h π‘₯ , 𝛾  ξ€· < 𝛽 βˆ’ πœ† 𝑔 ξ€· 𝐴 𝑑 𝑔 ξ€· ξ€Έ ξ€Έ < πœ‘ π‘₯ , 𝛾 ξ€Έ , βˆ€ π‘₯ ∈ 𝑀 . ( 1 . 9 ) For this equation, we have the following.

Theorem 1.2. Let ( 𝑀 , 𝑔 ) be a compact, closed, connected Riemannian manifold of dimension 𝑛 β‰₯ 3 and 𝐴 𝑑 𝑔 ∈ Ξ© βˆ’ , f o r 𝑑 < 1 . ( 1 . 1 0 ) Suppose that Ξ© + , Ξ© βˆ’ βŠ‚ 𝑅 𝑛 are open convex symmetric cones with vertex at the origin, satisfying Ξ© 𝑛 βŠ‚ Ξ© βŠ‚ Ξ© 1 , Ξ© βˆ’ = βˆ’ Ξ© + , ( 1 . 1 1 ) where Ξ© 1 ξƒ― ξ€· πœ† ∢ = πœ† = 1 , … , πœ† 𝑛 ξ€Έ ; 𝑛  𝑖 = 1 πœ† 𝑖 ξƒ° , Ξ© > 0 𝑛 ξ€½ ξ€· πœ† ∢ = πœ† = 1 , … , πœ† 𝑛 ξ€Έ ; πœ† 𝑖 ξ€Ύ . > 0 f o r 1 ≀ 𝑖 ≀ 𝑛 ( 1 . 1 2 ) Let 𝛽 satisfy(i) 𝛽 > 0 in Ξ© + , 𝛽 𝑖 ∢ = πœ• 𝛽 / πœ• πœ† 𝑖 > 0 on Ξ© + , and 𝛽 ( 𝑒 ) = 1 on Ξ© + , where 𝑒 = ( 1 , … , 1 ) . ( 1 . 1 3 ) (ii) 𝛽 is concave on Ξ© + , and 𝛽 ( πœ† ) ≀ 𝜚 𝜎 1 ( πœ† ) , βˆ€ πœ† ∈ Ξ© + , ( 1 . 1 4 ) 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 𝑛 β‰₯ 3 and 𝐴 𝑑 𝑔 ∈ Ξ© βˆ’ , f o r 𝑑 < 1 . ( 1 . 1 5 ) 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 𝑃 ξ€· πœ† ( π‘ˆ ) ∢ = 𝛽 𝑔 ξ€Έ ( π‘ˆ ) = Ξ› , ( 1 . 1 6 ) where π‘ˆ = βˆ’ 𝐴 𝑑 Μƒ 𝑔 = βˆ‡ 2 πœ™ + 1 βˆ’ 𝑑 𝑛 βˆ’ 2 ( Ξ” πœ™ ) 𝑔 + 2 βˆ’ 𝑑 2 | | | | βˆ‡ πœ™ 2 𝑔 βˆ’ 𝑑 πœ™ βŠ— 𝑑 πœ™ βˆ’ 𝐴 𝑑 𝑔 ( 1 . 1 7 ) for conformal metric Μƒ 𝑔 = 𝑒 2 πœ™ and πœ† 𝑔 ( 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 . 1 8 ) 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 𝐢 2 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 Μƒ 𝑧 = 𝑧 βˆ’ 𝛾 , 𝑧 𝑠 𝑍 = 𝑠 𝑧 + ( 1 βˆ’ 𝑠 ) 𝛾 , 𝑠 = βˆ‡ 2 𝑧 𝑠 + 1 βˆ’ 𝑑 ξ€· 𝑛 βˆ’ 2 Ξ” 𝑧 𝑠 ξ€Έ 𝑔 + 2 βˆ’ 𝑑 2 | βˆ‡ 𝑧 𝑠 | 2 𝑔 βˆ’ 𝑑 𝑧 𝑠 βŠ— 𝑑 𝑧 𝑠 βˆ’ 𝐴 𝑑 𝑔 . ( 2 . 3 ) It is easy to verify that 𝑍 𝑠 ∈ Ξ© + .
Write 𝑄 [ 𝑧 ] = 𝑃 ( 𝑍 ) βˆ’ πœ‘ ( π‘₯ , 𝑧 ) . ( 2 . 4 ) Then 𝑄 [ 𝑧 ]  𝛾 βˆ’ 𝑄 ξ‚„ ξ€· = 0 βˆ’ 𝑃 βˆ’ 𝐴 𝑑 𝑔 ξ€Έ ξ‚€ + πœ‘ π‘₯ , 𝛾  . ( 2 . 5 ) On the other hand, 𝑄 [ 𝑧 ]  𝛾 βˆ’ 𝑄 ξ‚„ = ξ€œ 1 0 𝑑 𝑄 ξ€Ί 𝑧 𝑑 𝑠 𝑠 ξ€» = ξ€œ 𝑑 𝑠 1 0 𝑇 𝑖 𝑗 ξ€· 𝑍 𝑠 ξ€Έ 𝑑 𝑠 𝐷 𝑖 𝑗 Μƒ 𝑧 + 𝑏 𝑖 𝐷 𝑖 Μƒ 𝑧 + 𝑐 Μƒ 𝑧 = 𝐿 ( Μƒ 𝑧 ) ( 2 . 6 ) for some bound 𝑏 𝑖 and constant 𝑐 , where 𝑇 𝑖 𝑗 = 𝑃 𝑖 𝑗 + 1 βˆ’ 𝑑  𝑛 βˆ’ 2 𝑙 𝑃 𝑙 𝑙 𝛾 𝑖 𝑗 𝑃 β‰₯ 0 , 𝑖 𝑗 = πœ• 𝑃 πœ• 𝑍 𝑖 𝑗 β‰₯ 0 ( 2 . 7 ) by condition (ii).
Therefore, we know that 𝐿 is an elliptic operator, and 𝐿 ( Μƒ 𝑧 ) < 0 w i t h Μƒ 𝑧 β‰₯ 0 . ( 2 . 8 ) By the maximum principle, we get Μƒ 𝑧 > 0 . That is, 𝑧 > 𝛾 . ( 2 . 9 ) Similarly, we can derive 𝑧 < 𝛾 , ( 2 . 1 0 ) 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 𝐢 3 solution to (1.7) for some 𝑑 < 1 satisfying 𝛾 < 𝑧 < 𝛾 . Then β€– βˆ‡ 𝑧 β€– 𝐿 ∞ < 𝐢 1 , ( 2 . 1 1 ) where 𝐢 1 depends only upon 𝛾 , 𝛾 , 𝑔 , 𝑑 , πœ‘ .
Moreover, β€– β€– βˆ‡ 2 𝑧 β€– β€– 𝐿 ∞ < 𝐢 2 , ( 2 . 1 2 ) where 𝐢 2 depends only upon 𝛾 , 𝛾 , 𝑔 , 𝑑 , πœ‘ , 𝐢 1 .

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 𝛽 𝜏 ξ€· ( πœ† ) ∢ = 𝛽 𝜏 πœ† + ( 1 βˆ’ 𝜏 ) 𝜎 1 ξ€Έ ( πœ† ) 𝑒 , ( 2 . 1 3 ) (here 𝑒 = ( 1 , … , 1 ) as in Section 1) which is defined on Ξ© + 𝜏 = ξ€½ πœ† ∈ ℝ 𝑛 ; 𝜏 πœ† + ( 1 βˆ’ 𝜏 ) 𝜎 1 ( πœ† ) 𝑒 ∈ Ξ© + ξ€Ύ . ( 2 . 1 4 ) We consider the problem 𝑃 ξ€· 𝜏 𝑍 + ( 1 βˆ’ 𝜏 ) 𝜎 1 ξ€Έ ( 𝑍 ) 𝑒 = 𝜏 πœ‘ ( π‘₯ , 𝑧 ) + ( 1 βˆ’ 𝜏 ) 𝜎 1 ξ€· βˆ’ 𝐴 𝑑 𝑔 ξ€Έ 𝑒 2 𝑧 ( 2 . 1 5 ) on 𝑀 , where 𝑍 = βˆ‡ 2 𝑧 + 1 βˆ’ 𝑑 𝑛 βˆ’ 2 ( Ξ” 𝑧 ) 𝑔 + 2 βˆ’ 𝑑 2 | | | | βˆ‡ 𝑧 2 𝑔 βˆ’ 𝑑 𝑧 βŠ— 𝑑 𝑧 βˆ’ 𝐴 𝑑 𝑔 . ( 2 . 1 6 ) Since 𝐴 𝑑 𝑔 ∈ Ξ© βˆ’ , we have 𝜎 1 ξ€· βˆ’ 𝐴 𝑑 𝑔 ξ€Έ > 0 ( 2 . 1 7 ) 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 𝐢 2 solution 𝑧 𝜏 to (2.15) with 𝛾 ≀ 𝑧 𝜏 ≀ 𝛾 satisfies 𝛾 < 𝑧 𝜏 < 𝛾 . ( 2 . 1 8 ) 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 β€– 𝑧 𝜏 β€– 𝐢 2 < 𝐢 , ( 2 . 1 9 ) 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 that β€– 𝑧 𝜏 β€– 𝐢 4 , 𝛼 < 𝐾 , ( 2 . 2 0 ) where 𝑧 𝜏 is a 𝐢 2 solution to (2.15) with 𝛾 ≀ 𝑧 𝜏 ≀ 𝛾 .
Set 𝑆 𝜏  𝛾 ∢ = < 𝑧 𝜏 < 𝛾  ∩ ξ€½ β€– 𝑧 𝜏 β€– 𝐢 4 , 𝛼 ξ€Ύ ∩ ξ€½ < 𝐾 𝑍 ∈ Ξ© + 𝜏 ξ€Ύ , ( 2 . 2 1 ) and define 𝑇 𝜏 ∢ 𝐢 4 , 𝛼 β†’ 𝐢 2 , 𝛼 by 𝑇 𝜏 ξ€· ( 𝑧 ) = 𝑃 𝜏 𝑍 + ( 1 βˆ’ 𝜏 ) 𝜎 1 ξ€Έ ( 𝑍 ) 𝑒 βˆ’ 𝜏 πœ‘ ( π‘₯ , 𝑧 ) βˆ’ ( 1 βˆ’ 𝜏 ) 𝜎 1 ξ€· βˆ’ 𝐴 𝑑 𝑔 ξ€Έ 𝑒 2 𝑧 . ( 2 . 2 2 ) Then, by (2.19), we see that there is no solution to the equation 𝑇 𝜏 ( 𝑧 ) = 0 o n πœ• 𝑆 𝜏 . ( 2 . 2 3 ) 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 ξ€· 𝑇 d e g 0 , 𝑆 0 ξ€Έ , 0 β‰  0 . ( 2 . 2 4 ) Since the degree is homotopy invariant, we have ξ€· 𝑇 d e g 1 , 𝑆 1 ξ€Έ , 0 β‰  0 . ( 2 . 2 5 ) Thus, we conclude that (1.7) has a solution in 𝑆 1 .
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‚€ βˆ‡ 𝑃 ( 𝑒 ) = 𝑃 2 𝑒 + 1 βˆ’ 𝑑 𝑛 βˆ’ 2 ( Ξ” 𝑒 ) 𝑔 + 2 βˆ’ 𝑑 2 | βˆ‡ 𝑒 | 2 𝑔 βˆ’ 𝑑 𝑒 βŠ— 𝑑 𝑒 βˆ’ 𝐴 𝑑 𝑔  βˆ’ 𝑒 𝑒 = πœ† . ( 3 . 1 ) We will prove that, for small πœ† > 0 , (3.1) has a unique smooth solution.
Since πœ•  𝑃 / πœ• 𝑒 < 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 𝐴 𝑑 𝑔 ∈ Ξ© βˆ’ ( 3 . 2 ) that, for πœ† > 0 small enough, we can find two constants 𝛾 < 0 < 𝛾 , such that 𝑒 𝛾 ξ€· + πœ† < 𝑃 βˆ’ 𝐴 𝑑 𝑔 ξ€Έ < 𝑒 𝛾 + 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 .
Set 𝐸 ∢ = { πœ† > 0 ; ( 3 . 1 ) h a s a s o l u t i o n } . ( 3 . 4 ) Since 𝐸 β‰  βˆ… , we can define Ξ› = s u p πœ† ∈ 𝐸 πœ† . ( 3 . 5 ) We claim Ξ› is finite. Actually, ξ‚€ βˆ‡ πœ† < 𝑃 2 𝑒 + 1 βˆ’ 𝑑 𝑛 βˆ’ 2 ( Ξ” 𝑒 ) 𝑔 + 2 βˆ’ 𝑑 2 | βˆ‡ 𝑒 | 2 𝑔 βˆ’ 𝑑 𝑒 βŠ— 𝑑 𝑒 βˆ’ 𝐴 𝑑 𝑔  . ( 3 . 6 ) If we assume that at π‘₯ 0 , 𝑒 achieves its maximum, then βˆ‡ 2 𝑒 ≀ 0 , and so ξ‚€ βˆ‡ πœ† < 𝑃 2 𝑒 + 1 βˆ’ 𝑑 𝑛 βˆ’ 2 ( Ξ” 𝑒 ) 𝑔 βˆ’ 𝐴 𝑑 𝑔  ξ€· ≀ 𝑃 βˆ’ 𝐴 𝑑 𝑔 ξ€Έ . ( 3 . 7 ) This means that ξ€· Ξ› ≀ 𝑃 βˆ’ 𝐴 𝑑 𝑔 ξ€Έ . ( 3 . 8 )
For any sequence πœ† 𝑖 βŠ‚ 𝐸 with πœ† 𝑖 β†’ Ξ› , let 𝑒 πœ† 𝑖 be the corresponding solution to (3.1) with πœ† = πœ† 𝑖 .
First, we claim that i n f 𝑀 𝑒 πœ† 𝑖 ⟢ βˆ’ ∞ a s 𝑖 ⟢ ∞ . ( 3 . 9 ) Suppose this is not true, that is, i n f 𝑀 𝑒 πœ† 𝑖 β‰₯ βˆ’ 𝐢 0 ( 3 . 1 0 ) for a positive constant 𝐢 0 . Then, by (3.1), at any maximum point π‘₯ 0 of 𝑒 πœ† 𝑖 , m a x 𝑀 𝑒 πœ† 𝑖 ≀ 𝐢 ( 3 . 1 1 ) for some constant 𝐢 depending only on 𝑃 ( βˆ’ 𝐴 𝑑 𝑔 ) . Then the apriori estimates imply that 𝑒 πœ† 𝑖 (by taking a subsequence) converges to a smooth function 𝑒 0 in 𝐢 ∞ , such that 𝑒 0 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 + 𝛿 w i t h 𝛿 > 0 s m a l l e n o u g h . ( 3 . 1 2 ) This is a contradiction. Hence (3.9) holds.
Next, we prove that m a x 𝑀 𝑒 πœ† 𝑖 ⟢ βˆ’ ∞ a s 𝑖 ⟢ ∞ . ( 3 . 1 3 )
We divided our proof into two steps.
Step 1. Let ξ€· Ξ› = 𝑃 βˆ’ 𝐴 𝑑 𝑔 ξ€Έ . ( 3 . 1 4 ) Then, following the above argument, 𝑒 πœ† 𝑖 β†’ πœ™ 0 i n C ∞ , ( 3 . 1 5 ) and ( Ξ› , 𝑒 0 ) is a solution to (3.1). Assume 𝑒 0 attains its maximum at 𝑦 0 . Then at 𝑦 0 , βˆ‡ 2 𝑒 0 ≀ 0 , βˆ‡ 𝑒 0 = 0 . ( 3 . 1 6 ) Therefore, 𝑒 𝑒 0 ( 𝑦 0 ) ξ€· ≀ 𝑃 βˆ’ 𝐴 𝑑 𝑔 ξ€Έ βˆ’ Ξ› = 0 . ( 3 . 1 7 ) So 𝑒 0 ξ€· 𝑦 0 ξ€Έ = βˆ’ ∞ . ( 3 . 1 8 ) That means that (3.13) holds.Step 2. Let 𝑃 ξ€· βˆ’ 𝐴 𝑑 𝑔 ξ€Έ βˆ’ Ξ› = πœ› > 0 . ( 3 . 1 9 ) Then, if (3.13) is not true, that is, m a x 𝑀 𝑒 πœ† 𝑖 β‰₯ βˆ’ 𝐢 0 ( 3 . 2 0 ) for a positive constant 𝐢 0 , write 𝑧 πœ† 𝑖 ∢ = 𝑒 πœ† 𝑖 βˆ’ m a x 𝑀 𝑒 πœ† 𝑖 . ( 3 . 2 1 ) Then we have m a x 𝑀 𝑧 πœ† 𝑖 ⟢ 0 , i n f 𝑀 𝑧 πœ† 𝑖 ⟢ βˆ’ ∞ , ( 3 . 2 2 ) as 𝑖 β†’ ∞ .
On the other hand, 𝑧 πœ† 𝑖 satisfies 𝑃 ξ‚€ βˆ‡ 2 𝑧 πœ† 𝑖 + 1 βˆ’ 𝑑 ξ€· 𝑛 βˆ’ 2 Ξ” 𝑧 πœ† 𝑖 ξ€Έ 𝑔 + 2 βˆ’ 𝑑 2 | βˆ‡ 𝑧 πœ† 𝑖 | 2 βˆ’ 𝑑 𝑧 πœ† 𝑖 βŠ— 𝑑 𝑧 πœ† 𝑖 βˆ’ 𝐴 𝑑 𝑔  = 𝑒 m a x 𝑀 𝑒 πœ† 𝑖 𝑒 𝑧 πœ† 𝑖 + πœ† 𝑖 . ( 3 . 2 3 ) Since at any minimum point 𝑧 0 of 𝑧 πœ† 𝑖 , βˆ‡ 2 𝑧 πœ† 𝑖 β‰₯ 0 , βˆ‡ 𝑧 πœ† 𝑖 = 0 . ( 3 . 2 4 ) Consequently, at 𝑧 0 , we obtain 𝑒 m a x 𝑀 𝑒 πœ† 𝑖 𝑒 𝑧 πœ† 𝑖 ξ€· β‰₯ 𝑃 βˆ’ 𝐴 𝑑 𝑔 ξ€Έ βˆ’ Ξ› > 0 . ( 3 . 2 5 ) Thus, it is easy to verify that 𝑧 πœ† 𝑖 is bounded from below as 𝑖 β†’ ∞ . This is a contradiction. So we see that (3.13) is true.
By a priori estimates results again, we deduce that 𝑧 πœ† 𝑖 converges to a smooth function 𝑧 in 𝐢 ∞ and 𝑧 satisfies (1.16) with πœ† = Ξ› .
Finally, let us prove the uniqueness.
Denote 𝑍 ∢ = βˆ‡ 2 𝑧 + 1 βˆ’ 𝑑 𝑛 βˆ’ 2 ( Ξ” 𝑧 ) 𝑔 + 2 βˆ’ 𝑑 2 | βˆ‡ 𝑧 | 2 𝑔 βˆ’ 𝑑 𝑧 βŠ— 𝑑 𝑧 βˆ’ 𝐴 𝑑 𝑔 , ( 3 . 2 6 ) and for any smooth functions 𝑧 0 and 𝑧 1 , set 𝑣 = 𝑧 1 βˆ’ 𝑧 0 , 𝑧 𝑠 = 𝑠 𝑧 1 + ( 1 βˆ’ 𝑠 ) 𝑧 0 , 𝑍 𝑠 = βˆ‡ 2 𝑧 𝑠 + 1 βˆ’ 𝑑 ξ€· 𝑛 βˆ’ 2 Ξ” 𝑧 𝑠 ξ€Έ 𝑔 + 2 βˆ’ 𝑑 2 | βˆ‡ 𝑧 𝑠 | 2 𝑔 βˆ’ 𝑑 𝑧 𝑠 βŠ— 𝑑 𝑧 𝑠 βˆ’ 𝐴 𝑑 𝑔 . ( 3 . 2 7 ) Then we get 𝑃 ξ€· 𝑍 1 ξ€Έ ξ€· 𝑍 βˆ’ 𝑃 0 ξ€Έ = ξ€œ 1 0 𝑑 𝑃 ξ€· 𝑍 𝑑 𝑠 𝑠 ξ€Έ = ξ€œ 1 0  𝑃 𝑖 𝑗 + 1 βˆ’ 𝑑  𝑛 βˆ’ 2 𝑙 𝑃 𝑙 𝑙 𝛾 𝑖 𝑗 ξƒ­ ξ€· 𝑍 𝑠 ξ€Έ 𝑑 𝑠 𝑣 𝑖 𝑗 + 𝑏 𝑙 𝑣 𝑙 ( 3 . 2 8 ) for some bounded 𝑏 𝑙 . Thus, if 𝑧 0 = πœ™ , 𝑧 1 = πœ™ ξ…ž ( 3 . 2 9 ) are two solutions to (1.16) for some πœ† and πœ† ξ…ž , respectively, then π‘Ž 𝑖 𝑗 is positive definite. Therefore, πœ™ = πœ™ ξ…ž + 𝑐 ( 3 . 3 0 ) 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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