# A Problem Concerning Yamabe-Type Operators of Negative Admissible Metrics

Authors: Jin Liang; Huan Zhu
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper is about a problem concerning nonlinear Yamabe-type operators of negative admissible metrics. We first give a result onσk 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.


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## Body

## 1. 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:(1.1)Agt=1n-2(Ricg-tRg2(n-1)g),t≤1,
where Ricg and Rg are the Ricci tensor and the scalar curvature of g, respectively.Define(1.2)σk(λ)=∑1≤i1≤⋯≤ik≤nλi1⋯λikforλ=(λ1,…,λn)∈Rn,Ωk+={λ=(λ1,…,λn)∈Rn;σj(λ)>0,1≤j≤k}.
The σk Yamabe problem is to find a metric g̃ conformal to g, such that(1.3)σk(λg̃(Ag̃))=1,λg̃(Ag̃)∈Ωk+on⁡M,
where λg̃(Ag̃) denotes the eigenvalue of Ag̃ with respect to the metric g̃. This problem has attracted great interest since the work of Viaclovsky in [2] (cf., e.g., [2–7] and references therein).AssumeΩk-=-Ωk+. Then the σk Yamabe problem in negative cone(1.4)σk(-λg̃(Ag̃))=1,λg̃(Ag̃)∈Ωk-on⁡M,
is still elliptic (see [1]).Definition 1.1.
A metricg̃ conformal to g is called negative admissible if
(1.5)λg̃(Ag̃t)∈Ωk-on⁡M.
Under the conformal relationg̃=e2zg, the transformation law for the modified Schouten tensor above is as follows:(1.6)Ag̃τ=Agτ-∇2z-1-τn-2(Δz)g-2-τ2|∇z|2g+dz⊗dz.
We consider the following nonlinear equation:
(1.7)P(Z):=β(λg(Z))=φ(x,z),λg(Z)∈Ωon⁡M,
where
(1.8)Z=∇2z+1-tn-2(Δz)g+2-t2|∇z|2g-dz⊗dz-Agt,β∈C∞(Ω+)∩C0(Ω+¯) is a symmetric function and is homogeneous of degree one normalized, and φ is a positive C∞ function satisfying the monotone condition:
(1.9)thereexiststwoconstantsγ̲<0<γ¯withφ(x,γ̲)<β(-λg(Agt))<φ(x,γ¯),∀x∈M.
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
(1.10)Agt∈Ω-,fort<1.
Suppose that  Ω+,Ω-⊂Rn are open convex symmetric cones with vertex at the origin, satisfying
(1.11)Ωn⊂Ω⊂Ω1,Ω-=-Ω+,
where
(1.12)Ω1:={λ=(λ1,…,λn);∑i=1nλi>0},Ωn:={λ=(λ1,…,λn);λi>0for1≤i≤n}.
Let β satisfy(i)
β>0 in Ω+, βi:=∂β/∂λi>0 on Ω+, and β(e)=1 on Ω+, where
(1.13)e=(1,…,1).(ii)
β is concave on Ω+, and
(1.14)β(λ)≤ϱσ1(λ),∀λ∈Ω+,
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
(1.15)Agt∈Ω-,fort<1.
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
(1.16)P(U):=β(λg(U))=Λ,
where
(1.17)U=-Ag̃t=∇2ϕ+1-tn-2(Δϕ)g+2-t2|∇ϕ|2g-dϕ⊗dϕ-Agt
for conformal metric g̃=e2ϕ and λg(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
(1.18)Λ=Λ′,ϕ=ϕ′+c
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.


## 2. Proof of Theorem1.2

To prove Theorem1.2, firstly, let us give the following proposition.Proposition 2.1.
Suppose all the conditions in Theorem1.2 are satisfied. Then every C2 solution z to (1.7) with
(2.1)γ̲≤z≤γ¯
satisfies
(2.2)γ̲<z<γ¯.Proof.
Assumez is a solution to (1.7) with γ̲≤z. Denote
(2.3)z̃=z-γ̲,zs=sz+(1-s)γ,Zs=∇2zs+1-tn-2(Δzs)g+2-t2|∇zs|2g-dzs⊗dzs-Agt.
It is easy to verify that Zs∈Ω+.
Write(2.4)Q[z]=P(Z)-φ(x,z).
Then
(2.5)Q[z]-Q[γ̲]=0-P(-Agt)+φ(x,γ̲).
On the other hand,
(2.6)Q[z]-Q[γ̲]=∫01ddsQ[zs]ds=∫01Tij(Zs)dsDijz̃+biDiz̃+cz̃=L(z̃)
for some bound bi and constant c, where
(2.7)Tij=Pij+1-tn-2∑lPllγij≥0,Pij=∂P∂Zij≥0
by condition (ii).
Therefore, we know thatL is an elliptic operator, and(2.8)L(z̃)<0withz̃≥0.
By the maximum principle, we get z̃>0. That is,
(2.9)z>γ̲.
Similarly, we can derive
(2.10)z<γ¯,
for solution z with z≤γ¯.Thus, we have the following Gradient and Hessian estimates for solutions to (1.7).Lemma 2.2.
Letz be a C3 solution to (1.7) for some t<1 satisfying γ̲<z<γ¯. Then
(2.11)‖∇z‖L∞<C1,
where C1 depends only upon γ̲,γ¯,g,t,φ.
Moreover,(2.12)‖∇2z‖L∞<C2,
where C2 depends only upon γ̲,γ¯,g,t,φ,C1.Proof of Theorem1.2.
We now prove Theorem1.2 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 each0≤τ≤1, let(2.13)βτ(λ):=β(τλ+(1-τ)σ1(λ)e),
(here e=(1,…,1) as in Section 1) which is defined on
(2.14)Ωτ+={λ∈Rn;τλ+(1-τ)σ1(λ)e∈Ω+}.
We consider the problem
(2.15)P(τZ+(1-τ)σ1(Z)e)=τφ(x,z)+(1-τ)σ1(-Agt)e2z
on M, where
(2.16)Z=∇2z+1-tn-2(Δz)g+2-t2|∇z|2g-dz⊗dz-Agt.
Since Agt∈Ω-, we have
(2.17)σ1(-Agt)>0
by condition (ii). Hence for τ=0, it follows from the maximum principle that z=0 is the unique solution.
In view of Proposition2.1, we see that, for each τ∈[0,1], every C2 solution zτ to (2.15) with γ̲≤zτ≤γ¯ satisfies(2.18)γ̲<zτ<γ¯.
This, together with Lemma 2.2, shows that for each τ∈[0,1] and solution zτ to (2.15) with γ̲≤zτ≤γ¯, the following estimate holds
(2.19)‖zτ‖C2<C,
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(2.20)‖zτ‖C4,α<K,
where zτ is a C2 solution to (2.15) with γ̲≤zτ≤γ¯.
Set(2.21)Sτ:={γ̲<zτ<γ¯}∩{∥zτ∥C4,α<K}∩{Z∈Ωτ+},
and define Tτ:C4,α→C2,α by
(2.22)Tτ(z)=P(τZ+(1-τ)σ1(Z)e)-τφ(x,z)-(1-τ)σ1(-Agt)e2z.
Then, by (2.19), we see that there is no solution to the equation
(2.23)Tτ(z)=0on⁡∂Sτ.
So the degree of Tτ is well defined and independent of τ. As mentioned above, there is a unique solution at τ=0. Therefore
(2.24)deg⁡(T0,S0,0)≠0.
Since the degree is homotopy invariant, we have
(2.25)deg⁡(T1,S1,0)≠0.
Thus, we conclude that (1.7) has a solution in S1.
The proof of Theorem1.2 is completed.


## 3. Proof of Theorem1.3

Proof of Theorem1.3.
Take a look at the following equation:(3.1)P̃(u)=P(∇2u+1-tn-2(Δu)g+2-t2|∇u|2g-du⊗du-Agt)-eu=λ.
We will prove that, for small λ>0, (3.1) has a unique smooth solution.
Since∂P̃/∂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(3.2)Agt∈Ω-
that, for λ>0 small enough, we can find two constants γ̲<0<γ¯, such that
(3.3)eγ̲+λ<P(-Agt)<eγ¯+λ.
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(3.4)E:={λ>0;(3.1)hasasolution}.
Since E≠∅, we can define
(3.5)Λ=sup⁡λ∈Eλ.
We claim Λ is finite. Actually,
(3.6)λ<P(∇2u+1-tn-2(Δu)g+2-t2|∇u|2g-du⊗du-Agt).
If we assume that at x0, u achieves its maximum, then ∇2u≤0, and so
(3.7)λ<P(∇2u+1-tn-2(Δu)g-Agt)≤P(-Agt).
This means that
(3.8)Λ≤P(-Agt).
For any sequenceλi⊂E with λi→Λ, let uλi be the corresponding solution to (3.1) with λ=λi.
First, we claim that(3.9)inf⁡M⁡uλi⟶-∞asi⟶∞.
Suppose this is not true, that is,
(3.10)inf⁡M⁡uλi≥-C0
for a positive constant C0. Then, by (3.1), at any maximum point x0 of uλi,
(3.11)max⁡M⁡uλi≤C
for some constant C depending only on P(-Agt). Then the apriori estimates imply that uλi (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
(3.12)λ=λ0+δwithδ>0small enough.
This is a contradiction. Hence (3.9) holds.
Next, we prove that(3.13)max⁡M⁡uλi⟶-∞asi⟶∞.
We divided our proof into two steps.
Step 1.
Let(3.14)Λ=P(-Agt).
Then, following the above argument,
(3.15)uλi→ϕ0inC∞,
and (Λ,u0) is a solution to (3.1). Assume u0 attains its maximum at y0. Then at y0,
(3.16)∇2u0≤0,∇u0=0.
Therefore,
(3.17)eu0(y0)≤P(-Agt)-Λ=0.
So
(3.18)u0(y0)=-∞.
That means that (3.13) holds.Step 2.
Let(3.19)P(-Agt)-Λ=ϖ>0.
Then, if (3.13) is not true, that is,
(3.20)max⁡M⁡uλi≥-C0
for a positive constant C0, write
(3.21)zλi:=uλi-max⁡M⁡uλi.
Then we have
(3.22)max⁡M⁡zλi⟶0,inf⁡M⁡zλi⟶-∞,
as i→∞.
On the other hand,zλi satisfies(3.23)P(∇2zλi+1-tn-2(Δzλi)g+2-t2|∇zλi|2-dzλi⊗dzλi-Agt)=emax⁡Muλiezλi+λi.
Since at any minimum point z0 of zλi,
(3.24)∇2zλi≥0,∇zλi=0.
Consequently, at z0, we obtain
(3.25)emax⁡M⁡uλiezλi≥P(-Agt)-Λ>0.
Thus, it is easy to verify that zλi is bounded from below as i→∞. This is a contradiction. So we see that (3.13) is true.
By a priori estimates results again, we deduce thatzλi converges to a smooth function z in C∞ and z satisfies (1.16) with λ=Λ.
Finally, let us prove the uniqueness.
Denote(3.26)Z:=∇2z+1-tn-2(Δz)g+2-t2|∇z|2g-dz⊗dz-Agt,
and for any smooth functions z0 and z1, set
(3.27)v=z1-z0,zs=sz1+(1-s)z0,Zs=∇2zs+1-tn-2(Δzs)g+2-t2|∇zs|2g-dzs⊗dzs-Agt.
Then we get
(3.28)P(Z1)-P(Z0)=∫01ddsP(Zs)=∫01[Pij+1-tn-2∑lPllγij](Zs)dsvij+blvl
for some bounded bl. Thus, if
(3.29)z0=ϕ,z1=ϕ′
are two solutions to (1.16) for some λ and λ′, respectively, then aij is positive definite. Therefore,
(3.30)ϕ=ϕ′+c
for some constant c by the maximum principle.



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*Source: 930868-2012-05-10.xml*