1 Introduction
In the analysis of Einstein field equations in general relativity the initial data set for the non-linear wave system plays an essential role. These initial data have to satisfy the Einstein constraint conditions that can be expressed in a geometric form as follows.
Let (M,g^) be a Riemannian manifold and K^ a symmetric 2-covariant tensor on M.
Then, (M,g^) is said to satisfy the Einstein constraint equations with non-gravitational energy density ρ^ and non-gravitational momentum density J^ if
(1.1){|K^|g^2-(trg^K^)2=Sg^-ρ^,divg^K^-∇trg^K^=J^.
Here Sg^ stands for the scalar curvature of the metric g^. To look for solutions of (1.1) using the conformal method introduced by Lichnerowicz in [15] means that we generate an initial data set (M,g^,K^,ρ^,J^) satisfying (1.1) by first choosing the following conformal data:
A Riemannian manifold (M,〈⋅,⋅〉).
A symmetric 2-covariant tensor σ required to be traceless and transverse with respect to 〈⋅,⋅〉, that is, for which tr〈⋅,⋅〉σ=0 and div〈⋅,⋅〉σ=0.
A scalar function τ (that will play the role of a non normalized mean curvature).
A non-negative scalar function ρ and a vector field J.
Then, one looks for a positive function u and a vector field W that solve the conformal constraint equations
(1.2){Δu-cmSu+cm|σ+ℒ̊W|2u-N-1-bmτ2uN-1+cmρu-N2=0,Δ𝕃W+m-1muN∇τ+J=0,
where Δ, S and |⋅| denote respectively the Laplace–Beltrami operator, the scalar curvature and the norm in the metric 〈⋅,⋅〉. The operator ℒ̊ is the traceless Lie derivative, that is, in a local orthonormal coframe,
(ℒ̊W)ij=Wij-Wji-2m(divW)δij
and Δ𝕃=div∘ℒ̊ is the vector laplacian.
The constants appearing in (1.2) are respectively given by
N=2mm-2,cm=m-24(m-1),bm=m-24m.
In particular, we note that N is the critical Sobolev exponent. If (u,W) is a solution of
(1.2) then the initial data set
g^=u4m-2〈⋅,⋅〉,ρ^=u-3N+22ρ,J^=u-NJ,K^ij=u-2m+2m-2(σ+ℒ̊W)ij+τmu4m-2δij
satisfies the Einstein constraints (1.1). For further informations on the physical content of system (1.2), we refer to the recent surveys [4, 9], and the references therein.
The aim of this paper is to study the existence, a priori bounds
and uniqueness of positive solutions of the Lichnerowicz-type equation
(1.3)Δu+a(x)u-b(x)uσ+c(x)uτ=0
with, at least, continuous coefficients, a(x), b(x), c(x), σ>1, τ<1, and with the sign restrictions
(1.4)b(x)≥0,c(x)≥0,
on a complete, non compact, connected manifold (M,〈⋅,⋅〉).
Equation (1.3) is of the same form as the scalar equation of system (1.2)
in the case ρ≡0. The coefficients b(x) and c(x) correspond, respectively, to
bmτ2 and cm|σ+ℒW|g2.
This latter fact justifies the sign condition (1.4).
In recent years this type of equations has been studied by many authors, see for instance [16, 8, 17, 10].
In the present work we significantly generalize many of the results obtained in the aforementioned papers.
We now introduce some notations and state our main existence results.
Let a(x)∈C0(M) and L=Δ+a(x).
If Ω is a non-empty open set, the first Dirichlet eigenvalue λ1L(Ω) is variationally characterized by means of the formula
(1.5)λ1L(Ω)=inf{∫Ω|∇φ|2-a(x)φ2:φ∈W01,2(Ω),∫Ωφ2=1}.
We recall that if Ω is bounded and ∂Ω is sufficiently regular, then the infimum is attained by the unique normalized eigenfunction v on Ω satisfying
(1.6){Δv+a(x)v+λ1L(Ω)v=0on Ω,v=0on ∂Ω,v>0on Ω.
We then define the first eigenvalue of L on M as
λ1L(M)=infΩλ1L(Ω),
where Ω runs over all bounded domains of M. Observe that, due to the monotonicity of λ1L with respect to the domain, that is,
(1.7)Ω1⊆Ω2 implies λ1L(Ω1)≥λ1L(Ω2),
we have
(1.8)λ1L(M)=limr→+∞λ1L(Br),
where, from now on, Br denotes the geodesic ball in the complete manifold (M,〈⋅,⋅〉) of radius r centered at a fixed origin o∈M.
Note that in case Ω2∖Ω1 has non-empty interior the inequality in (1.7) becomes strict.
We need to extend definition (1.5) to an arbitrary bounded subset B of M. We do this by setting
(1.9)λ1L(B)=supΩλ1L(Ω),
where the supremum is taken over all open bounded sets Ω with smooth boundary such that B⊆Ω . Observe that, by definition, if B=∅ then λ1L(B)=+∞.
We would like to remark that since the first Dirichlet eigenvalue for the Laplacian of a ball Br grows like r-2 as r→0+, we have that λ1L(Br)≥0 provided r is sufficiently small and one may think of λ1L(B)>0 as a condition expressing the fact that B is small in a spectral sense. Of course, this condition also depends on the behaviour of a(x) therefore small in a spectral sense does not necessarily mean, for instance, small in a Lebesgue measure sense.
This is clear if a(x)≤0 because λ1Δ(M)≥0 in any complete manifold (M,〈⋅,⋅〉), so that in this case λ1L(M)≥0 and thus λ1L(B)>0 on any bounded set B⊂M.
The main results of the paper are the following two existence theorems for positive solutions of equation (1.3).
Theorem ALet a(x), b(x), c(x)∈Cloc0,α(M) for some 0<α≤1. Assume (1.4) and suppose that b(x) is strictly positive outside some compact set. Let
(1.10)B0={x∈M:b(x)=0},
and suppose
(1.11)λ1L(B0)>0
with L=Δ+a(x). Assume that
(1.12)λ1L(M)<0.
Then, (1.3) has a maximal positive solution u∈C2(M).
In this case maximal means that if 0<w∈C2(M) is a second solution of (1.3), then w≤u.
In the same vein we have the following result, where the spectral condition (1.12) is substituted by a spectral smallness requirement on the zero set of the coefficient c(x) and a pointwise control on the coefficients. Recall that given the real function α(x), its positive and negative part are, respectively, defined by
α+(x)=max{α(x),0} and α-(x)=-min{α(x),0}.
Theorem BLet a(x), b(x), c(x)∈Cloc0,α(M) for some 0<α≤1.
Assume the validity of (1.4), and let
B0={x∈M:b(x)=0},C0={x∈M:c(x)=0}
and
(1.13)λ1Δ+a(B0)>0.
Suppose that there exist two bounded open sets Ω1, Ω2 such that C0⊂Ω1⊂⊂Ω2,
(1.14)supM∖Ω¯1a-(x)+b(x)c(x)<+∞
and
(1.15)λ1Δ-a(Ω2)>0.
Then, (1.3) has a maximal positive solution u∈C2(M).
The proof of the theorems is the content of Section 2.
These existence results should be compared with those obtained at the end of the very recent paper [17].
The remaining two sections of the paper are devoted to uniqueness of solutions. In particular, the results of Section 3 should be interpreted as Liouville-type theorems and compared with those obtained in [16, 8, 10].
The main differences with previous work in the literature is that our geometric requirement on the manifold consist only in a mild volume growth assumption for geodesic balls and in the fact that we allow non constant coefficients a(x), b(x), c(x) in equation (1.3). In this last setting in general there are no trivial solutions at hand. Thus, to provide a complete analysis of the problem in this case, we need to find an a priori estimate and use it to detect a trivial solution of (1.3). In particular, Corollary 3.12 is our main Liouville-type theorem.
In Section 4 we analyze another uniqueness result, this time under a spectral assumption on the manifold, in the spirit of the very recent papers [5, 6].
2 Proof of Theorems A and B
The main result of the paper is in fact the following proposition, whose proof will be the content of the section. At the end we will prove Theorems A and B as corollaries of the proposition.
Proposition 2.1Let a(x), b(x), c(x)∈Cloc0,α(M) for some 0<α≤1.
Assume (1.4) and suppose that b(x) is strictly positive outside some compact set.
Furthermore, suppose
(2.1)λ1L(B0)>0
with L=Δ+a(x). If 0<u-∈C0(M)∩Wloc1,2(M) is a global subsolution of (1.3) on M, then (1.3) has a maximal positive solution u∈C2(M).
The proof of Proposition 2.1 is divided into several steps.
Note that in what follows we keep the notations of the proposition.
Lemma 2.2Let a¯(x), b(x), c(x)∈Cloc0,α(M) for some 0<α≤1, and let (1.4) hold. Suppose that B0 is bounded and
(2.2)λ1L¯(B0)>0,
where L¯=Δ+a¯(x). If Ω is a bounded open set such that B0⊂Ω, then there exists v+ solution of
(2.3){Δv++a¯(x)v+-b(x)v+σ+c(x)v+τ≤0on Ω,v+>0on Ω¯.
Proof.
Let D and D′ be bounded open domains such that
B0⊂⊂D′⊂⊂D⊂⊂Ω,
and λ1L¯(D)>0.
Let u1 be a positive solution of
{Δu1+a¯(x)u1+λ1L¯(D)u1=0on D,u1=0on ∂D.
Since b(x)>0 on M∖B0 and Ω¯∖D′⊂⊂M∖B0,
β=infΩ¯∖D′b(x)>0.
Define
α=supΩ¯∖D′a¯(x),δ=supΩ¯∖D′c(x),
and note that α, δ<+∞ since Ω is bounded. Let U be a positive constant. Then,
ΔU+a¯(x)U-b(x)Uσ+c(x)Uτ=U(a¯(x)-b(x)Uσ-1+c(x)Uτ-1)
≤U(α-βUσ-1+δUτ-1)
on Ω∖D′. We observe that the RHS of the above is non-positive for U sufficiently large, say
(2.4)U≥Λ0>0.
Next we choose a cut-off function ψ∈C0∞(M) such that 0≤ψ≤1, ψ≡1 on D′, and suppψ⊂D. Fix a positive constant γ and define
(2.5)u=γ(ψu1+(1-ψ)Λ0).
Since b(x)≥0 and λ1L¯(D)>0, on D′¯ we have
L¯u-b(x)uσ+c(x)uτ=L¯(γu1)-b(x)(γu1)σ+c(x)(γu1)τ
=-[λ1L¯(D)(γu1)+b(x)(γu1)σ-c(x)(γu1)τ]
=-(γu1)[λ1L¯(D)+b(x)(γu1)σ-1-c(x)(γu1)τ-1].
For the RHS of the above to be non-positive on D′¯ it is sufficient to have
(2.6)c(x)(γu1)τ-1≤λ1L¯(D) on D′¯.
To this end, we note that, since D′¯ is compact, u1>0 on D′¯, and τ<1. Thus, (2.6) is satisfied for
(2.7)γ≥Γ0=Γ0(u1)>0,
sufficiently large.
We now consider Ω∖D, since suppψ⊂D, it follows that u=γΛ0 there.
Thus, using Ω∖D⊂Ω∖D′, from (2.4) it follows that
Δu+a¯(x)u-b(x)uσ+c(x)uτ≤0 on Ω∖D
is satisfied if we choose γ≥1. Indeed, in this case
γΛ0≥Λ0.
It remains to analyze the situation on D∖D′¯. First of all we note that, by standard elliptic regularity theory, u1∈C2(D). Thus, since suppψ⊂D, it follows that u∈C2(Ω). In particular, this implies that there exists a positive constant C0 such that
(Δ+a¯(x))u≤γC0 on D∖D′¯.
Thus, on D∖D′¯ we have
Δu+a¯(x)u-b(x)uσ+c(x)uτ≤γC0-b(x)γσ(ψu1+(1-ψ)Λ0)σ+c(x)γτ(ψu1+(1-ψ)Λ0)τ.
Now there exists constants ε and E such that
infD∖D′¯b(x)(ψu1+(1-ψ)Λ0)σ=ε>0,
supD∖D′¯c(x)(ψu1+(1-ψ)Λ0)τ=E<+∞.
Therefore, on D∖D′¯,
Δu+a¯(x)u-b(x)uσ+c(x)uτ≤γ(C0-εγσ-1+Eγτ-1).
Since σ>1 and τ<1, it follows that there exists a positive constant Γ1 depending only on D and D′ such that
C0-εγσ-1+Eγτ-1≤0
for γ≥Γ1. Thus, by choosing
γ≥max{1,Γ0,Γ1},
u is the desired supersolution v+ of (2.3) on Ω.
∎
Definition 2.3We say that the property (Σ) holds on M (for equation (1.3)) if there exists Ro∈ℝ+ such that for all R≥Ro, there exists a solution φ∈C0(BR¯)∩Wloc1,2(BR) of
(2.8){Δφ+a(x)φ-b(x)φσ+c(x)φτ≥0on BR,φ≥0on BR¯.
When τ<0, in order to avoid singularities, in the equation above it is assumed that suppc(x)⊆suppφ. In Proposition 2.5 below we shall give some sufficient conditions for the validity of property (Σ).
Lemma 2.4Let a(x), b(x), c(x)∈Cloc0,α(M) for some 0<α≤1, and let (1.4) hold. Suppose that B0 is bounded and λ1L(B0)>0 with L=Δ+a(x). Furthermore assume that property (Σ) holds on M. Let Ω be a bounded domain such that B0⊂Ω. Then, for each n∈N, there exists a solution u of the problem
(2.9){Δu+a(x)u-b(x)uσ+c(x)uτ=0on Ω,u>0on Ω,u=non ∂Ω.
Proof.
By the definition of λ1L(B0) and the assumption of positivity, there exists an open domain D with smooth boundary such that B0⊂D⊂⊂Ω and λ1L(D)>0. Let ρ∈C∞(M) be a cut-off function such that 0≤ρ≤1, ρ≡1 on D, ρ≡0 on M∖Ω¯. Fix N≥max{supΩ¯|a(x)|+1,λ1Δ(M∖Ω¯)+1}.
Define
a¯(x)=ρ(x)a(x)+N(1-ρ(x)),
and consider the operator
L¯=Δ+a¯(x). Since a¯(x)=a(x) on D,
(2.10)λ1L¯(B0)=λ1L(B0)>0.
Furthermore, from N≥λ1Δ(M∖Ω¯)+1,
we deduce λ1L¯(M∖Ω¯)≤-1, and it follows that there exists sufficiently large R>0 such that
(2.11)Ω¯⊂BR and λ1L¯(BR)<0.
Fix ε>0. Then, λ1L¯(BR+ε)<0.
Let φ be the normalized eigenfunction of L¯ on BR+ε relative to the eigenvalue λ1L¯(BR+ε) (here, without loss of generality, that is, possibly substituting BR+ε with a slighly larger open set with smooth boundary, we are assuming that ∂BR+ε is smooth) so that
(2.12){L¯φ+λ1L¯(BR+ε)φ=0on BR+ε,φ≡0on ∂BR+ε,φ>0on BR+ε,∥φ∥L2(BR+ε)=1.
We fix γ>0, sufficiently small, such that
∫BR+ε|∇φ|2-a¯(x)+γ[b(x)-c(x)]φ2=λ1L¯(BR+ε)+∫BR+εγ[b(x)-c(x)]φ2<0.
This shows that the operator L~=Δ+a¯(x)-γ[b(x)-c(x)] satisfies λ1L~(BR+ε)<0. Let ψ be a positive eigenfunction corresponding to λ1L~(BR+ε). Then, ψ satisfies
{Δψ+a¯(x)ψ-γb(x)ψ+γc(x)ψ≥0on BR+ε,ψ≡0on ∂BR+ε,ψ>0on BR+ε.
Thus,
(2.13){Δψ+a¯(x)ψ-γb(x)ψ+γc(x)ψ≥0on BR,ψ>0on BR¯.
Let μ>0 and define v-=μψ on BR. Choosing
μ≤min{γ1σ-1(supBRψ)-1,γ1τ-1(supBRψ)-1},
we have
(2.14)γμ1-σψ1-σ≥1 and γμ1-τψ1-τ≤1 on BR.
Hence, using (2.13) and (2.14), we deduce
0≤Δv-+a¯(x)v--γb(x)v-σ(μ1-σψ1-σ)+γc(x)v-τ(μ1-τψ1-τ)
≤Δv-+a¯(x)v--b(x)v-σ+c(x)v-τ,
that is,
(2.15){Δv-+a¯(x)v--b(x)v-σ+c(x)v-τ≥0on BR,v->0on BR¯.
Because of the validity of (2.10), Lemma 2.2 yields the existence of v+ satisfying
(2.16){Δv++a¯(x)v+-b(x)v+σ+c(x)v+τ≤0on BR,v+>0on BR¯.
Note that if 0<γ≤1, γv- still satisfies (2.15). Hence, up to choosing a suitable γ, we can assume that
(2.17)supBR¯v-≤infBR¯v+ on BR¯.
Let
α+=inf∂BRv+,α-=sup∂BRv-,
and fix α∈[α-,α+].
Then, by the monotone iteration scheme, there exists a solution w of
{Δw+a¯(x)w-b(x)wσ+c(x)wτ=0on BR,w≡α>0on ∂BR,
with the additional property
0<v-≤w≤v+ on BR¯.
Therefore, since a¯(x)≥a(x) on BR¯ and w>0, we have
(2.18){Δw+a(x)w-b(x)wσ+c(x)wτ≤0on BR,w≡α>0on ∂BR,w>0on BR.
Fix any n∈ℕ. Let ζ∈ℝ be such that
ζ≥max{1,nsup∂Ωw},
and define w+=ζw. Then, because of (2.18), the fact that Ω⊂⊂BR, and the signs of b(x) and c(x), w+ satisfies
{Δw++a(x)w+-b(x)w+σ+c(x)w+τ≤0on Ω,w+≥non ∂Ω,w+>0on Ω.
We can suppose that the R chosen above is such that R≥Ro, where Ro is that of the property (Σ) in Definition 2.3.
This choice implies that there exists a solution ψ of
{Δψ+a(x)ψ-b(x)ψσ+c(x)ψτ≥0on BR,ψ≥0on BR¯.
If we define w-=βψ, where 0<β≤1, then reasoning as above we can find β so small that
{Δw-+a(x)w--b(x)w-σ+c(x)w-τ≥0on Ω,0≤w-≤w+on Ω,w-≤non ∂Ω.
Using the monotone iteration scheme we easily arrange a solution w of (2.9) between w- and w+. We note that the positivity of w is obvious in the case suppc(x)=M, while in the general case it is a consequence of the strong maximum principle (see [11, 12]).
∎
We note that the corresponding result for Yamabe-type equations, namely equations of the type (1.3) with c(x)≡0, can be proved without the additional assumption of property (Σ). Indeed, in such case this is automatically satisfied by the global subsolution w-=0. The next proposition provides some sufficient conditions for the validity of property (Σ) on M.
Proposition 2.5Assume the validity of one of the following;
For some Λ>0, we have
λ1Δ+a-b+Λc(M)<0.
Let C0={x∈M:c(x)=0} be bounded and such that
λ1Δ-a(C0)>0.
There exists a positive subsolution φ-∈C0(M)∩Wloc1,2(M) of (1.3).
Then, property (Σ) holds on M.
Proof.
If (i) holds true, then there exists sufficiently large Ro>0 such that
λ1Δ+a-b+Λc(BR)<0 for R≥Ro.
Accordingly, there exists a corresponding positive eigenfunction ψ on BR+ε, say ε>0 and small, for which
(2.19){Δψ+a(x)ψ-b(x)ψ+Λc(x)ψ≥0on BR,ψ>0on BR¯.
We let
(2.20)0<μ≤min{Λ1τ-1,(supBRψ)-1},
and define
φ=μψ.
Note that from (2.19),
0≤Δφ+a(x)φ-b(x)φσ(μψ)1-σ+Λc(x)φτ(μψ)1-τ.
Now, because of (2.20),
(μψ)1-σ≥1 and (μψ)1-τ≤1
on BR¯. Hence, the above inequality implies the validity of (2.8) with φ strictly positive on BR¯.
If (ii) holds true, then by Lemma 2.2, there exists Ro>0 sufficiently large such that C0⊂BRo and for R≥Ro there exists a solution ψ of
{Δψ-a(x)ψ-c(x)ψ2-τ+b(x)ψ2-σ≤0on BR,ψ>0on BR¯.
Thus, defining φ=1ψ, we have
Δφ=-φ2Δψ+2φ3|∇ψ|2≥-φ2Δψ,
which implies (2.8) always with φ>0 on BR¯. Case (iii) is obvious.
∎
Remark 2.6It is obvious that the existence of a positive global subsolution
φ-∈C0(M)∩Wloc1,2(M) of (1.3) implies the validity of the Σ-property. What is more interesting is that the validity of condition (ii) in Proposition 2.5 yields the existence of a global subsolution φ-∈C0(M)∩Wloc1,2(M).
In the sequel we shall need the following a priori estimate. Here BT(q) denotes the geodesic ball of radius T centered at q.
Lemma 2.7Let a(x), b(x), c(x)∈C0(M), σ>1, τ<1, 0<T~<T, and Ω⊂⊂BT~(q)⊂M. Assume b(x)>0 on BT(q)¯. Then, there exists an absolute constant C>0 such that any positive solution u∈C2(BT(q)¯) of
(2.21)Δu+a(x)u-b(x)uσ+c(x)uτ≥0
satisfies
(2.22)supΩu≤C.
Proof.
We let ρ(x)=dist(x,q) and, on the compact ball BT(q)¯ we consider the continous function
F(x)=[T2-ρ(x)2]2σ-1u(x),
where u(x) is any nonnegative C2 solution of (2.21). Note that F(x)|∂BT(q)=0, thus, unless u≡0 and in this case there is nothing to prove, F has a positive absolute maximum at some point x¯∈BT(q). In particular, u(x¯)>0.
Now, proceeding as in the proof of [21, Lemma 2.6], we conclude that, at x¯,
buσ-1≤8(σ+1)(σ-1)2ρ2(T2-ρ2)2+4σ-1m+(m-1)AρT2-ρ2+a++cuτ-1
for some constant A≥0, independent of u.
We now state an elementary lemma postponing its proof.
Lemma 2.8Let α, β∈[0,+∞) and μ, ν∈(0,+∞).
If t∈R+ satisfies
tμ≤α+βtν,
then
(2.23)t≤(α+βμμ+ν)1μ.
Since σ>1 and τ<1, from the lemma we conclude that, at x¯,
u≤b-1σ-1(8(σ+1)(σ-1)2ρ2(T2-ρ2)2+4σ-1m+(m-1)AρT2-ρ2+a++cσ-1σ-τ)1σ-1.
Now the proof proceeds exactly as in [21, Lemma 2.6] by substituting the a+ there with a++cσ-1σ-τ.
∎
Proof of Lemma 2.8.
If tμ≤α, then we are done, since μ>0 and β≥0.
In other case set s=tμ,. Then, s>α and thus
s≤α+βsνμ<α+β(s-α)νμ.
Setting r=s-α ,we conclude that
rμ+νμ<β,
and (2.23) follows.
∎
The next simple comparison result reveals something quite useful.
Lemma 2.9Let Ω⊆M be a bounded open set. Assume that fi:M×R→R for i=1,2 are measurable functions such that for all x∈M
(2.24)f2(x,s)s≥f1(x,t)t
for s≤t.
Let u,v∈C0(Ω¯)∩C2(Ω) be solutions on Ω, respectively, of
(2.25){Δu+f1(x,u)≥0,Δv+f2(x,v)≤0,
with u≥0, v>0. If u≤v on ∂Ω, then u≤v on Ω.
Proof.
Set ψ(x)=u(x)v(x)∈C0(Ω¯)∩C2(Ω). From (2.25) a standard computation yields
(2.26)Δψ≥uv2f2(x,v)-1vf1(x,u)-2〈∇ψ,∇logv〉.
Now, if we assume by contradiction that u>v somewhere in Ω, then there exists ε>0 such that
Ωε={x∈Ω:ψ(x)>1+ε}≠∅
and ∂Ωε⊂Ω.
Thus, it follows from (2.26), that the following inequality holds true on Ωε:
Δψ+2〈∇ψ,∇logv〉≥ψ[f2(x,v)v-f1(x,u)u]≥0.
Moreover, ψ≡1+ε on ∂Ωε, and thus by the maximum principle, ψ≤1+ε on Ωε contradicting the definition of Ωε.
∎
Remark 2.10We note that the hypohteses on fi of Lemma 2.9 are satisfied, for instance, if f1=f2:M×ℝ→ℝ is a measurable function such that for all x∈M,
s↦fi(x,s)s
is a non increasing function and that the lemma can also be stated for f1=f2:M×ℝ+→ℝ with u, v>0. In particular, this is the case for the Lichnerowicz-type nonlinearities considered in this paper, namely
f(x,s)=a(x)s-b(x)sσ+c(x)sτ
with b(x), c(x) non negative and σ>1, τ<1. Indeed, for any fixed x∈M,
the function
gx(s)=f(x,s)s=a(x)-b(x)sσ-1+c(x)sτ-1
is smooth on ℝ+ and its derivative is given by
gx′(s)=-(σ-1)b(x)sσ-2+c(x)(τ-1)sτ-2,
which is non positive by our assumptions on b(x), c(x), σ and τ.
A reasoning similar to that in the proof of Lemma 2.9 will be used at the end of the argument in the proof of the next lemma.
Lemma 2.11In the assumptions of Lemma 2.4 there exists a positive solution u of the problem
(2.27){Δu+a(x)u-b(x)uσ+c(x)uτ=0on Ω,u=+∞on ∂Ω.
Proof.
For n∈ℕ, let un>0 on Ω be the solution of (2.9) obtained in Lemma 2.4, so that
(2.28){Δun+a(x)un-b(x)unσ+c(x)unτ=0on Ω,un>0on Ω,un=non ∂Ω.
First of all we claim that
(2.29)un≤un+1.
Indeed, un=n<n+1=un+1 on ∂Ω. We then apply Lemma 2.9 with the choice f1=f2=f and, recalling Remark 2.10, we obtain the validity of (2.29).
If we show the convergence of the monotone sequence un to a function u, which is a solution of (2.27), then we are done. Indeed, u will certainly be positive.
To this end, by standard regularity theory, it is enough to show that the sequence {un} is uniformly bounded on any compact subset K of Ω. If K⊂Ω∖B0, then we can find a finite covering of balls {Bi} for K such that b(x)>0 on each Bi.
Applying Lemma 2.7, we deduce the existence of a constant C1>0 such that
(2.30)un(x)≤C1 for all x∈K and for all n∈ℕ.
It remains to find an upper bound on a neighborhood of B0. Towards this end, for η>0, we let
Nη={x∈M:d(x,B0)<η},
where η is small enough that Nη¯⊂Ω. Furthermore, by the definition of λ1L(B0) and the fact that λ1L(B0)>0, we can also assume to have chosen η so small that
λ1L(Nη)>0.
Now ∂Nη/2 is closed and bounded (because B0 is so), hence compact by the completeness of M. This implies the existence of a constant C2 such that
un(x)≤C2 for all x∈∂Nη/2 and for all n∈ℕ.
This follows from Lemma 2.7 by considering a finite covering of ∂Nη/2 with balls of radii less than η/2.
Next we let φ be a positive eigenfunction corresponding to λ1L(Nη). Then, since infNη/2φ>0, it follows that there exists a constant μo>0 such that
μφ(x)>C2 for all x∈∂Nη/2 and for all μ≥μo.
On Nη/2 we have
(2.31)Δ(μφ)+a(x)(μφ)=-λ1L(Nη)(μφ)<0.
We now choose μ≥μo sufficiently large such that
μτ-1(infNη/2φ)τ-1(supNη/2c(x))<λ1L(Nη),
which is possible since τ<1 and infNη/2φ>0.
Then, for each ε>0,
(2.32)μτ-1(infNη/2φ)τ-1(supNη/2c(x))(1+ε)τ-1<λ1L(Nη).
We let ψ=uμφ on Nη/2, where u is any of the functions of the sequence {un}. The same computations as in the proof of Lemma 2.9 using (2.28) and (2.31) yields
(2.33)Δψ+2〈∇ψ,∇log(μφ)〉≥(λ1L(Nη)+b(x)uσ-1-c(x)uτ-1)ψ.
Note that, according to our choice of μ,
μφ>C2>u on ∂Nη/2.
We claim that ψ≤1 on ∂Nη/2. By contradiction suppose the contrary. Then, for some ε1>0,
Ωε1={x∈Nη/2:ψ(x)>1+ε1}≠∅
and Ωε1⊂⊂Nη/2.
On Ωε1,
u>(1+ε1)μφ.
Therefore, since τ<1,
uτ-1≤(1+ε1)τ-1(μφ)τ-1.
Then, inserting this into (2.33), together with (2.32), we deduce
Δψ+2〈∇ψ,∇log(μφ)〉≥(λ1L(Nη)-(supNη/2c(x))(1+ε1)τ-1μτ-1φτ-1)ψ≥0.
By the maximum principle it follows that ψ attains its maximum on ∂Ωε1, but there ψ(x)=1+ε1, contradicting the assumption Ωε1≠∅.
Thus, ψ≤1 on Nη/2, that is, u≤μψ on Nη/2. Hence, for all n∈ℕ,
un≤max{C2,supNη/2μφ}.
This completes the proof of the lemma.
∎
We are now ready to prove Proposition 2.1. The proof, which is the same as the proof of [18, Theorem 6.5], follows a standard argument and it is reported here for the sake of completeness.
Proof of Proposition 2.1.
First of all we note that, by part (iii) of Proposition 2.5, the existence of the global positive subsolution u- implies that the Σ-property holds on M. We fix an exhausting sequence {Dk} of open precompact sets with smooth boundaries such that
B0⊂Dk⊂Dk¯⊂Dk+1,
and for each k we denote by uk∞ the solution of the problem
{Δu+a(x)u-b(x)uσ+c(x)uτ=0on Dk,u=+∞on ∂Dk,
which exists by Lemma 2.11.
It follows from Lemma 2.9 that
(2.34)u-≤uk+1∞≤uk∞ on Dk¯.
Thus, {uk∞} converges monotonically to a function u
which solves
(1.3),
and satisfies,
because of (2.34), u≥u->0. Let now u1>0 be a second solution of (1.3) on M. By Lemma 2.9, u1≤uk∞ on Dk for all k, and therefore u1≤u.
Thus, u is a maximal positive solution.
∎
We can now prove Theorem A using an existence result for solutions of Yamabe-type equations contained in [18].
Proof of Theorem A.
By Proposition 2.1 it follows that to prove the theorem is sufficient to show that assumption (1.12) implies the existence of a global subsolution u- of (1.3). To this end, we consider the following Yamabe-type equation:
(2.35)Δv+a(x)v-b(x)vσ=0 on M
with σ, a(x), and b(x) as in Theorem A. Then, by the sign assumptions (1.4) it follows that a global subsolution v of (2.35) is also a global subsolution of (1.3).
Now we recall [18, Theorem 6.7], which provides a positive solution v of (2.35) under assumptions (1.11) and (1.12). ∎
We conclude the section with the proof of Theorem B. The technique is the same as that used in Theorem A: Provide a global subsolution and then apply Proposition 2.1. In this case the subsolution is obtained by pasting a subsolution defined inside a compact set and another one defined in the complement of a compact set.
Proof of Theorem B.
Reasoning as in Lemma 2.2, assumption (1.13) implies the existence of a solution ψ∈C2(Ω2) of the following problem:
{Δψ+a(x)ψ-b(x)ψσ+c(x)ψτ≥0on Ω2,ψ>0on Ω2,ψ=0on ∂Ω2,
thus u1=ψ is a subsolution of (1.3) in Ω2.
In particular, since ∂Ω1⊂Ω2, if we set ν=min∂Ω1ψ, then we have that ν>0.
Now we note that (1.14) implies that there exists μ∈ℝ+ such that
supM∖Ω¯1a-(x)+b(x)c(x)=μ.
Let us define μ*=min{1,μ1τ-1,ν2}. Then, on M∖Ω¯1 we have that
Δμ*+a(x)μ*-b(x)μ*σ+c(x)μ*τ=a(x)μ*-b(x)μ*σ+c(x)μ*τ
≥-a-(x)μ*-b(x)μ*+c(x)μ*τ
=c(x)μ*[μ*τ-1-a-(x)+b(x)c(x)]
≥c(x)μ*[μ*τ-1-μ]
≥0.
Thus, u2=μ* is a subsolution of (1.3) in M∖Ω¯1. Set
u-={u1on Ω¯1,max{u1,u2}on Ω2∖Ω¯1,u2on M∖Ω2.
We claim that u- is the required global subsolution. To prove the claim, we start by noting that the fact that 0<μ*<ν2 implies 0<u-∈C0(M)∩Wloc1,2(M).
For the same reason, it is clear that there exists ε>0 such that u- is a subsolution of (1.3) on (Ω¯1)ε∪(M∖Ω2)ε, where
(U)ε=⋃x∈UBε(x)
for any set U⊂M (Bε(x) denotes the geodesic ball of radius ε centered in x). Thus, we are left to show that u- is a subsolution of (1.3) on Ω2∖Ω¯1, this is a rather standard fact but we sketch the proof here for the sake of completeness. First of all we set
f(x,v)=a(x)v-b(x)vσ+c(x)vτ.
Then, we note that for any test function φ∈W01,2(Ω2∖Ω¯1), φ≥0, we have that
∫Ω2∖Ω¯1〈∇u1,∇φ〉-φf(x,u1)≤0
and f(x,u2)≥0 on Ω2∖Ω¯1.
Now, for any φ∈W01,2(Ω2∖Ω¯1) and w∈W1,2(Ω2∖Ω¯1), consider
H(w,φ)=∫Ω2∖Ω¯1〈∇w,∇φ〉-φf(x,u-).
It is clear that H(⋅,φ):W1,2(Ω2∖Ω¯1)→ℝ is a continous functional for any φ.
We want to show that H(u-,φ)≤0 for any test function φ≥0 on Ω2∖Ω¯1.
For ε>0, let
uε=12(u1+u2+(u1-u2)2+ε2).
Then, uε is smooth with
∇uε=12(1+u1-u2(u1-u2)2+ε2)∇u1.
Moreover, by the definition of u-, we have uε→W1,2u- as ε→0. If φ∈W01,2(Ω2∖Ω¯1) and ε>0, then
φε=12(1+u1-u2(u1-u2)2+ε2)φ
belongs to W01,2(Ω2∖Ω¯1) and its gradient is given by
∇φε=12(1+u1-u2(u1-u2)2+ε2)∇φ+ε22((u1-u2)2+ε2)3φ∇u1.
The following computation uses the properties of uε, φε, and the fact that u1 and u2 are subsolutions:
H(uε,φ)=∫Ω2∖Ω¯112(1+u1-u2(u1-u2)2+ε2)〈∇u1,∇φ〉-φf(x,u-)
=∫Ω2∖Ω¯1〈∇u1,∇φε〉-ε2φ|∇u1|22((u1-u2)2+ε2)3-φf(x,u-)
≤∫Ω2∖Ω¯1〈∇u1,∇φε〉-φf(x,u-)
≤∫Ω2∖Ω¯1φεf(x,u1)-φf(x,u-).
Now, since
φε→L2{φif u1>u2,0if u1≤u2.
From the continuity of H(⋅,φ), we conclude that
H(u-,φ)=limε→0H(uε,φ)≤-∫{u1≤u2}∩Ω2∖Ω¯1φf(x,u2)≤0
for any test function φ.
∎
3 Uniqueness results and “a priori” estimates
The aim of this section is to prove uniqueness of positive solutions of equation (1.3) on M or outside a relatively compact open set Ω. To avoid technicalities we suppose u∈C2(M) or u∈C2(M∖Ω¯) but this assumption can be relaxed as it will become clear from the arguments we are going to present.
Also the positivity of u can be relaxed as it will be remarked below. We begin by proving a further comparison result with the aid of the open form of the q-Weak Maximum Principle (q-WMP) introduced in [3], see also [2]. For the present purposes we let L be a linear operator of the form
(3.1)Lu=Δu-〈X,∇u〉
for some vector field X on M. Let q(x)∈C0(M) be such that q(x)>0 on M.
Definition 3.1We say that the q-Weak Maximum Principle holds on M for the operator L in (3.1) if for each u∈C2(M) with u*=supMu and for each γ∈ℝ with γ<u*, we have
(3.2)infΩγ(q(x)Lu)≤0,
where
Ωγ={x∈M:u(x)>γ}.
The next result is contained in [3, Theorem 3.5].
Theorem 3.2The q-WMP holds on M for the operator L if and only if the open q-WMP holds on M for L, that is, for each f∈C0(R), for each open set Ω∈M with ∂Ω≠∅ and for each v∈C0(Ω¯)∩C2(Ω) satisfying
q(x)Lv≥f(v) on Ω,
supΩv<+∞,
we have that either
supΩv=sup∂Ωv 𝑜𝑟 f(supΩv)≤0.
The following is a sufficient condition for the validity of the q-WMP (see [21, Theorem 4.1]).
Theorem 3.3Let (M,〈⋅,⋅〉) be complete and q(x)≤Cr(x)μ for some constants C>0, 2>μ≥0, and r(x)≫1. Assume that
(3.3)lim infr→+∞logvol(Br)r2-μ<+∞.
Then, the q-WMP holds on M for the operator Δ.
We observe that when q is constant or more generally bounded between two positive constants then the q-WMP is equivalent to stochastic completeness of the manifold (M,〈⋅,⋅〉) this underlines the fact that the q-WMP does not require completeness of the metric and that Theorem 3.3 indeed gives a sufficient condition.
The following result generalizes [25, Theorem 3.1] and [1, Theorem 5.1]. The proof follows the same ideas of the aforementioned results and it is tailored for the case of Lichnerowicz-type equations. Since the presence of the possibly negative exponent τ generates some subtleties, we prefer, for the ease of the reader, to give a detailed proof of the result.
Theorem 3.4Let a(x),b(x),c(x)∈C0(M) and σ,τ∈R be such that σ>1 and τ<1. Let Ω be a relatively compact open set in M. Assume that
(3.4){(i)b(x)>0 on M∖Ω,(ii)c(x)≥0 on M∖Ω,(iii)supMa-(x)b(x)<+∞,(iv)supMc(x)b(x)<+∞,
where a- denotes the negative part of a. Let u,v∈C0(M∖Ω)∩C2(M∖Ω¯) be positive solutions of
(3.5){Lu+a(x)u-b(x)uσ+c(x)uτ≥0,Lv+a(x)v-b(x)vσ+c(x)vτ≤0,
on M∖Ω¯, satisfying
(3.6)lim infx→+∞v(x)>0,
(3.7)lim supx→+∞u(x)<+∞
and
(3.8)0<u(x)≤v(x) on ∂Ω.
Then,
(3.9)u(x)≤v(x) on M∖Ω,
provided that the 1/b-WMP holds on M∖Ω¯ for L.
Remark 3.5As it will be observed in the proof of the theorem, in case 0≤τ<1 assumption (3.4) (iv) can be dropped.
Proof.
Without loss of generality we can suppose that M∖Ω¯ is connected. From the positivity of v, (3.6), (3.7), and (3.8), there exist positive constants C1, C2 such that
(3.10)v(x)≥C1,u(x)≤C2 on M∖Ω¯.
We set ζ=supM∖Ω¯(uv). From the assumptions on v, u, and (3.10) it follows that ζ satisfies
(3.11)0<ζ<+∞.
Note that if ζ≤1, then u≤v on M∖Ω¯. Thus, assume by contradiction that ζ>1, and define
φ=u-ζv.
Then, φ≤0 on M∖Ω¯. It is not hard to realize, using (3.11) and the definition of ζ, that
(3.12)supM∖Ω¯φ=0.
We now use (3.5) to compute
(3.13)Lφ≥-a(x)φ+b(x)[uσ-(ζv)σ]-c(x)[uτ-(ζv)τ]+b(x)ζv[(ζv)σ-1-vσ-1]+c(x)ζv[vτ-1-(ζv)τ-1].
We let
h(x)={σuσ-1(x)if u(x)=ζv(x),σu(x)-ζv(x)∫ζv(x)u(x)tσ-1𝑑tif u(x)<ζv(x),
and, similarly, for τ≠0,
j(x)={-τuτ-1(x)if u(x)=ζv(x),τζv(x)-u(x)∫ζv(x)u(x)tτ-1𝑑tif u(x)<ζv(x).
In case τ=0 choose j(x)≡0. Observe that h and j are continous on M∖Ω¯ and h is non-negative. Using h and j, and observing that -a(x)φ≥a-(x)φ, from (3.13) we obtain
(3.14)Lφ≥[a-(x)+b(x)h(x)+c(x)j(x)]φ+b(x)ζv[(ζv)σ-1-vσ-1]+c(x)ζv[vτ-1-(ζv)τ-1].
Let
Ω-1={x∈M∖Ω¯:φ(x)>-1}.
Since u is bounded above on M∖Ω¯, there exists a constant C>0 such that
(3.15)v(x)=1ζ(u(x)-φ(x))≤1ζ(C+1) on Ω-1.
Using the definitions of h and j, then from the mean value theorem for integrals we deduce
h(x)=σyhσ-1,j(x)=-τyjτ-1
for some yh=yh(x) and yj=yj(x) in the range [u(x),ζv(x)].
Since u(x) and v(x) are bounded above on Ω-1,
(3.16)max{h(x),j(x})≤C on Ω-1
for some constant C>0.
Next we recall that b(x)>0 on M∖Ω¯ and rewrite (3.14) in the form
1b(x)Lφ≥[a-(x)b(x)+h(x)+c(x)b(x)j+(x)]φ+ζv[(ζv)σ-1-vσ-1]+c(x)b(x)ζv[vτ-1-(ζv)τ-1].
Since φ≤0, (3.4) and (3.16) imply
[a-(x)b(x)+h(x)+c(x)b(x)j+(x)]φ≥Cφ
for some constant C>0 on Ω-1. For further use, we observe here that when 0≤τ<1, j+(x)≡0, so that in this case assumption (3.4) (iv) is not needed to obtain the last inequality. Thus,
1b(x)Lφ≥Cφ+ζv[(ζv)σ-1-vσ-1]+c(x)b(x)ζv[vτ-1-(ζv)τ-1] on Ω-1.
Recalling the elementary inequalities
(3.17){as-bs≥sbs-1(a-b)for s<0 and s>1,as-bs≥sas-1(a-b)for 0≤s≤1,
a, b∈ℝ+, coming from the mean value theorem for integrals (see [13, Theorem 41]), we conclude
1b(x)Lφ≥Cφ+(σ-1)ζmin{1,σ-1}(ζ-1)vσ+(1-τ)c(x)b(x)ζ-1ζ1-τvτ on Ω-1.
Now we use the fact that τ<1, v is bounded from below by a positive constant, (3.4) (i), (ii) and (iv), to get (again if 0≤τ<1, then we do not need (3.4) (iv))
1b(x)Lφ≥Cφ+B on Ω-1
for some positive constants B, C. Finally, we choose 0<ε<1 sufficiently small such that
Cφ>-12B on Ω-ε={x∈M∖Ω¯:φ(x)>-ε}⊂Ω-1.
Therefore,
(3.18)1b(x)Lφ≥12B>0 on Ω-ε.
Furthermore, note that
φ(x)≤min{-ε,(1-ζ)min∂Ωv}<0 on Ω-ε.
As a consequence sup∂Ω-εφ<0 while supΩ-εφ=0. By Theorem 3.2, (3.18) and the above fact, we obtain the required contradiction, proving
that ζ≤1.
∎
As an immediate consequence of Theorem 3.4, we obtain the following uniqueness result.
Corollary 3.6In the assumptions of Theorem 3.4, the equation
Lu+a(x)u-b(x)uσ+c(x)uτ=0 on M∖Ω¯
admits at most a unique solution u∈C0(M∖Ω)∩C2(M∖Ω¯) with assigned boundary data on ∂Ω and satisfying
(3.19)C1≤u(x)≤C2 on M∖Ω¯
for some positive constants C1, C2, provided that the 1/b-WMP holds on M for the operator L.
We observe that the two main assumptions in Corollary 3.6 are the validity of the 1/b-WMP on M for L and the validity of the bounds (3.19). In case L=Δ in Theorem 3.3 we have given a sufficient condition for the validity of the 1/b-WMP, for b>0 everywhere it remains to analyze (3.19). To this end, we recall the following result companion of Theorem 3.3 and whose proof can be easily adapted from [21] and [20].
Theorem 3.7Let (M,〈⋅,⋅〉) be a complete Riemannian manifold and a(x), b(x)∈C0(M). Moreover, assume that ∥a+∥∞<+∞, b(x)>0 on M, and
(3.20)b(x)≥Cr(x)μ
outside a compact set K for some constants C>0 and μ<2.
Assume also (3.3) and
supMa+(x)b(x)<+∞.
Let u∈C2(M) be a non-negative solution of
Δu+a(x)u-b(x)uσ≥0 on Ωγ,
where σ>1 and
Ωγ={x∈M:u(x)>γ}
for some γ<u*≤+∞. Then, u*<+∞. Furthermore, having set
Hγ=supΩγa+(x)b(x),
we have
(3.21)u*≤Hγ1σ-1.
Proposition 3.8Let (M,〈⋅,⋅〉) be a complete Riemannian manifold. Let a(x), b(x), c(x)∈C0(M), and assume that ∥a++c+∥∞<+∞ and b(x)>0 on M
satisfying (3.20) for some μ<2 outside a compact set. Suppose the validity of (3.3) and
(3.22)supMa+(x)+c+(x)b(x)<+∞.
Let σ>1, τ<1, and let u∈C2(M) be a positive solution of
(3.23)Δu+a(x)u-b(x)uσ+c(x)uτ≥0 on Ωγ={x∈M:u(x)>γ}
for some γ<u*≤+∞. Then, u*<+∞ and indeed
(3.24)u*≤max{γ*,Hγ*1σ-1},
where γ*=max{1,γ} and
Hγ*=supΩγ*a+(x)+c+(x)b(x).
Proof.
First we show that u*<+∞. We can assume u*>1. If γ<1 we let γ~ be such that 1≤γ~<u* and note that Ωγ~⊂Ωγ. It follows that (3.23) holds on Ωγ~. Thus, without loss of generality, we can assume γ≥1. Since uτ≤u on Ωγ, from (3.23) we have
Δu+a+(x)u-b(x)uσ+c+(x)u≥Δu+a+(x)u-b(x)uσ+c+(x)uτ
≥Δu+a(x)u-b(x)uσ+c(x)uτ≥0
on Ωγ; in other words
Δu+[a+(x)+c+(x)]u-b(x)uσ≥0 on Ωγ.
Applying Theorem 3.7, we deduce u*<+∞. To prove (3.24) first let γ≥1 so that γ*=γ, Ωγ*=Ωγ, Hγ*=Hγ. Then, (3.24) follows directly from (3.21) of Theorem 3.7.
Suppose now γ<1. Then, γ*=1 and Ωγ*=Ω1⊂Ωγ.
If Ω1=∅, then u*≤γ*. If Ω1≠∅, then (3.23) holds on Ω1 and applying again Theorem 3.7 we deduce the validity of (3.24).
∎
Now, as in case (ii) of Proposition 2.5, we are going to exploit the symmetry of equation (1.3) to obtain a bilateral a priori estimate. This is the content of the next crucial theorem.
Theorem 3.9Let (M,〈⋅,⋅〉) be a complete Riemannian manifold. Let a(x), b(x), c(x)∈C0(M) such that ∥a++c∥∞<+∞ and ∥a-+b∥∞<+∞. Moreover, assume that b(x)>0 and c(x)>0 on M, and that both satisfy (3.20) for some μ<2. Suppose the validity of (3.3) and of the following:
(3.25)supMa+(x)+c(x)b(x)=H<+∞,
(3.26)supMa-(x)+b(x)c(x)=K<+∞.
Let σ>1, τ<1. Then, any positive C2 solution of
(3.27)Δu+a(x)u-b(x)uσ+c(x)uτ=0 on M
satisfies
(3.28)𝒦≤u(x)≤ℋ on M,
where
(3.29)𝒦=min{1,K1τ-1},ℋ=max{1,H1σ-1}.
Proof.
Suppose Ω1={x∈M:u(x)>1}≠∅, then the validity of (3.27) implies
Δu+a(x)u-b(x)uσ+c(x)uτ≥0 on Ω1.
Thus, the estimate from above in (3.28) follows from Proposition 3.8.
For the case where Ω1=∅, the same estimate is trivially true because of the definition (3.29) of ℋ. For the estimate from below, we consider the function v=1u∈C2(M), since u>0 on M. Since Δv=-v2Δu+2v3|∇u|2, using (3.27) we have
Δv+a~(x)v-b~(x)vσ~+c~(x)vτ~≥0 on M,
where we have set a~(x)=-a(x), b~(x)=c(x), c~(x)=b(x), σ~=2-τ>1, and τ~=2-σ<1. Now, since
a-(x)+b+(x)c(x)=a~+(x)+c~+(x)b~(x),
we can reason as above and deduce
v≤max{1,K1σ~-1}=max{1,K11-τ},
and the lower bound in (3.28) follows immediately from the definition of v.
∎
We note that the existence of solutions for equation (1.3) can be easily obtained under the hypoteses of Theorem 3.9 by direct application of the monotone iteration scheme of Amann (see, for instance, [26] or [18]). Indeed, in this case it is relatively easy to find an ordered pair of global sub and supersolutions. The key point is that their existence is a consequence of the a priori estimate. This is the content of the following lemma.
Lemma 3.10Let (M,〈⋅,⋅〉) be a complete Riemannian manifold. Let a(x), b(x), c(x), σ, τ, H, K, H, and K be as in Theorem 3.9. Then, u+≡H and u-≡K are respectively a global supersolution and a global subsolution of (3.27). Moreover, u-≤u+.
Proof.
First of all we note that since ℋ≥1 and τ<1, then it follows that ℋτ-1≤1. This implies that
Δu++a(x)u+-b(x)(u+)σ+c(x)(u+)τ=ℋ[a(x)-b(x)ℋσ-1+c(x)ℋτ-1]
≤ℋb(x)[a+(x)+c(x)b(x)-ℋσ-1]
≤0,
where in the last passage we have used (3.25) and the fact that ℋ≥H1σ-1, thus u+ is a global supersolution. The proof of the fact that u- is a subsolution is analogous and u-≤u+ follows from the definitions of ℋ and 𝒦.
∎
From this we immediately deduce the next existence result (see also [23] for a similar result).
Theorem 3.11Let (M,〈⋅,⋅〉) be a complete Riemannian manifold. Let a(x), b(x), c(x), σ, τ be as in Theorem 3.9 and assume that a(x), b(x), c(x)∈C0,α(M) for some α>0. Then, (1.3) has a positive solution u∈C2(M).
Proof.
Let {Ωk}k∈ℕ be a family of bounded open sets with smooth boundaries such that
Ωk⊂⊂Ωk+1 and ⋃k∈ℕΩk=M.
For each k∈ℕ, consider the Dirichlet problem
(3.30){Δv+a(x)v-b(x)vσ+c(x)vτ=0on Ωk,v=u+on ∂Ωk,
where u+=ℋ is the global supersolution of Lemma 3.10. Since u+ and u- of Lemma 3.10 are, respectively, a supersolution and a subsolution of (3.30) for any k∈ℕ, it follows from the monotone iteration scheme (see, for instance, [26, Theorem 2.1]) that for any k there exists a solution vk∈C2,α(Ωk) of (3.30) such that u-≤vk≤u+. From Lemma 2.9, it follows that
u-≤vi≤vj≤u+ on Ωk
for all i,j∈ℕ such that i≥j≥k. Thus, from the Schauder interior estimates and the compactness of the embedding C2,α(Ωk)⊂C2(Ωk), it follows that the vk converge uniformly on compact sets to a solution u∈C2(M) of (1.3). Moreover, u(x)≥u->0.
∎
Putting together Theorem 3.3, Corollary 3.6 with Ω=∅, Theorem 3.9, and Theorem 3.11, we have the following corollary.
Corollary 3.12In the assumptions of Theorem 3.9 with 0≤μ<2, the equation
Δu+a(x)u-b(x)uσ+c(x)uτ=0 on M
admits a unique positive solution u∈C2(M).
The next corollary deals with the easier case where a(x), b(x) and c(x) are of the form ζf(x), where ζ∈ℝ and 0<f(x)∈C0(M)∩L∞(M). It generalizes [16, Theorem 2] and [17, Theorem 7]. Furthermore, it should be compared with [10, Theorem 3.15 and Example 3.18].
Corollary 3.13Let (M,〈⋅,⋅〉) be a complete Riemannian manifold. Let α, β, γ∈R such that β, γ>0. Let σ>1, τ<0, 0<f(x)∈C0(M)∩L∞(M) satisfying (3.20) outside a compact set for some μ<2, and assume the validity of (3.3). Then, the unique positive solution of
Δu+f(x)(αu-βuσ+γuτ)=0 on M
is given by u≡λ, where λ∈R+ satisfies p(λ)=0, with
p(t)=α+βtσ-1-γtτ-1.
We conclude the section with a second uniqueness result whose proof is based on that of [7, Theorem 4.1], see also [18, Theorem 5.1].
Theorem 3.14Let (M,〈⋅,⋅〉) be a complete manifold, a(x),b(x),c(x)∈C0(M), σ>1, τ<1, and assume (1.4), that is,
b(x)≥0,c(x)≥0
and
(3.31)b(x)+c(x)≢0 on M.
Let u, v∈C2(M) be positive solutions of
Δu+a(x)u-b(x)uσ+c(x)uτ=0 on M,
such that
(3.32){∫∂Br(u-v)2}-1∉L1(+∞).
Then, u≡v on M.
Remark 3.15Note that condition (3.32) is implied by u-v∈L2(M) or even by the weaker assumption
∫Br(u-v)2=o(r2) as r→+∞.
See, for instance, [24, Proposition 1.3].
Proof.
The proof follows, mutatis mutandis, that reported in [18, Theorem 5.1] up to equation (5.7) that now becomes
(3.33)∫Mb(x)(v2-u2)(vσ-1-uσ-1)
+∫Mc(x)(v2-u2)(uτ-1-vτ-1)+∫M{|∇v-vu∇u|2+|∇u-uv∇v|2}=0.
Because of (1.4) we deduce
|∇u-uv∇v|≡0 on M,
so that u=Av for some constant A>0. Substituting into (3.33) yields
(1-A2)(1-Aσ-1)∫Mb(x)vσ+1=0
and (1-A2)(Aτ-1-1)∫Mc(x)vτ+1=0.
Since v>0, (3.31) implies A=1, that is, u=v on M.
∎
Remark 3.16The exponent 2 in (3.32) is sharp, see the discussion after [18, Theorem 5.1].
4 A further comparison and uniqueness result
In this section, we prove a comparison result and a corresponding uniqueness result based on a spectral property of the operator L=Δ+a(x). As we have seen, the assumption λ1L(M)<0 facilitates the search of solutions of equation (1.3). Somehow the opposite assumption seems to limit the existence of solutions.
We recall that L has finite index if and only if there exists a positive solution u of the differential inequality
(4.1)Lu≤0,
outside a compact set K. In what follows we shall denote with (M,〈⋅,⋅〉,G) a triple with the following properties: (M,〈⋅,⋅〉) is a complete manifold with a preferred origin o and G∈C2(M∖{o}), G:M∖{o}→ℝ+, is such that
(4.2){(i)ΔG≤0on M∖{o},(ii)G(x)→+∞as x→o,(iii)G(x)→0as x→+∞.
Clearly a good candidate for G is the (positive) Green kernel at o on a non-parabolic complete manifold, which, however, might not satisfy (4.2) (iii). Observe that, for instance by the work of Li and Yau [14], (iii) is satisfied by the Green kernel if Ric≥0.
Other examples always concerning the Green kernel, are given by non-parabolic complete manifolds supporting a Sobolev inequality of the type
(4.3)S(α)-1(∫Mv21-α)1-α≤∫M|∇v|2 for all v∈Cc∞(M)
for some α∈(0,1) with S(α)>0.
For further examples see [19] and the references therein. Note that, in these results, the authors also describe the behavior of G(x) at infinity from above and below. The behavior of |∇G(x)| from above can often be obtained by classical gradient estimates. This is helpful, for instance, in Theorem 4.5 below.
However, since we only require superharmonicity of G, under a curvature assumption, we can use transplantation from a non-parabolic model. The argument is as follows. Assume (M,〈⋅,⋅〉) is a m-dimensional manifold with a pole o and with radial sectional curvature (with respect to o) Krad satisfying
Krad≤-F(r(x)) on M
with r(x)=distM(x,o), F∈C0(ℝ0+). Let g be a C2-solution of the problem
(4.4){g′′-F(r)g≤0,g(0)=0,g′(0)=1,
and suppose that g>0 on ℝ+. Note that this request is easily achieved by bounding appropriately F from above. See, for instance, [5]. Then, by the Laplacian comparison theorem,
(4.5)Δr≥(m-1)g′(r)g(r) on M∖{o}
and weakly on M.
Consider the C2-model Mg defined by g with the metric
〈⋅,⋅〉g=dr2+g2(r)dθ2
on M∖{o}=ℝ+×Sm-1, where Sm-1 is the unit sphere and dθ2 is its canonical metric. Then, M∖{o} is non-parabolic if and only if 1gm-1∈L1(+∞).
Now we transplant the positive Green function on M∖{o} evaluated at (y,o) to M, that is, we let
G(x)=∫r(x)+∞dsg(s)m-1>0 on M∖{o}.
An immediate computation yields
ΔG(x)=-1g(r(x))m-1{Δr(x)-(m-1)g′(r(x))g(r(x))} on M∖{o}.
Hence, (4.5) implies (4.2) (i). The remaining of (4.2) is trivially satisfied.
Thus, we solve the problem by looking for a solution of (4.4) satisfying 1gm-1∈L1(+∞). For a detailed analysis we refer also to [6].
On M∖{o}, we define
(4.6)t(x)=-12logG(x),
and, for s∈ℝ, we set
Λs={x∈M∖{o}:t(x)<s}∪{o},
so that ∂Λs={x∈M∖{o}:t(x)=s}.
Note that, because of (4.2) (ii), Λs is open and {Λs}s∈ℝ is an exhausting family of open sets. Property (4.2) (iii) and the completeness of (M,〈⋅,⋅〉) implies that Λs¯ is compact for each s∈ℝ.
We are now ready to prove the following theorem.
Theorem 4.1Let (M,〈⋅,⋅〉,G) be as above and suppose that a(x)∈C0(M) satisfies
(4.7)a(x)≤{1+1log2G(x)[1+1log2(-logG(x))]}|∇logG(x)|24,
outside a compact set K. Then, the operator L=Δ+a(x) has finite index.
Remark 4.2Observe that condition (4.7) is meaningful outside a sufficiently large compact set K because of (4.2) (iii).
Proof.
On ℝ+, we define the function
(4.8)κ(s)=1+14s2[1+1log2s],
so that inequality (4.7) can be rewritten as
(4.9)a(x)≤κ(t(x))|∇logG(x)|24 on M∖K.
To prove the theorem we need to provide a positive solution u of (4.1) on M∖K^ for some compact K^. To this end, we look for u of the form
(4.10)u(x)=G(x)β(t(x))=e-t(x)β(t(x))
on M∖ΛT for some T>0 sufficiently large and with β:[T,+∞)→ℝ+. Now a simple computation shows that u satisfies
(4.11)Δu+[1-β¨β(t(x))]|∇logG(x)|24u=12ΔGG(x)[β(t(x))-β˙(t(x))]
on M∖ΛT, where the dot denotes the derivative with respect to t. Thus, using (4.9) and (4.11), we obtain
Δu+a(x)u≤[κ(t(x))-1+β¨β(t(x))]|∇logG(x)|24u+ΔG2G(x)[β(t(x))-β˙(t(x))].
Hence, using (4.2) (i), we have that (4.1) is satisfied on M∖ΛT for u as in (4.10) if we show the existence of a positive solution β of
(4.12)β¨+[κ(t)-1]β=0,
satisfying the further requirement
(4.13)β-β˙≥0 on [T,+∞) for some T>0.
In other words, we have to show that (4.12) is non-oscillatory and that (4.13) holds at least in a neighborhood of +∞. As for non oscillation, applying [5, Theorem 6.44], we see that this is the case if
κ(t)-1≤14t2[1+1log2t]
on [T,+∞) for some T>0 sufficiently large. This is guaranteed by the definition of κ (4.8).
To show the validity of (4.13), we use the following trick. Fix n≥3 and define ρ∈ℝ+ via the prescription
t=t(ρ)=log(n-2ρn-22).
Note that
t(0+)=-∞,t(+∞)=+∞,t′(ρ)=n-221ρ on ℝ+.
We then define
z(ρ)=e-t(ρ)β(t(ρ)).
If β is a solution of (4.12) on [T,+∞), having set R=ρ(T)>0 with ρ(t) the inverse function of t(ρ), we have that z satisfies
(ρn-1z′)′+κ(t(ρ))(n-2)24ρ2ρn-1z=0 on [R,+∞).
We can also fix the initial conditions
z(R)=1,z′(R)=0.
Hence, since κ≥0 on [R,+∞), a first integration of the solution z of the above Cauchy problem yields
z′(ρ)≤0 on [R,+∞).
But
z′(ρ)=n-221ρn2{β˙(t(ρ))-β(t(ρ))},
and therefore (4.13) is satisfied.
∎
Remark 4.3We have just proved that the equation
(4.14)β¨+14t2[1+1log2t]β=0 on [T,+∞)
(say T≥e) is non-oscillatory. This is not a consequence of the usual Hille–Nehari criterion (see [27]). Indeed, setting h(t) to denote the coefficient of the linear term in (4.14), the condition of the classical criterion to guarantee the non-oscillatory character of the equation is that h(t)≥0 for t≫1 and
(4.15)lim supt→+∞t∫t+∞h(s)𝑑s<14.
However, in this case we have
14<t∫t+∞ds4s2<t∫t+∞h(s)𝑑s<14+14∫t+∞dsslog2s=14+14logt,
so that (4.15) is not satisfied.
We shall now see how to get non-oscillation of (4.14) following the idea in the proof of the aforementioned [5, Theorem 6.44]. This will enable us to determine the asymptotic behavior of a solution β of (4.14) at +∞, and therefore of u defined in (4.10) and
satisfying (4.1). This will be later used in Theorem 4.5.
To this end, we consider the function
w(t)=tlogt,
which is a solution of Euler’s equation
w¨+14t2w=0
on [T,+∞), T>0, and positive on [T,+∞) for T>1. Then, the function
z=βw
satisfies
(4.16)(w2z˙)⋅+(κ(t)-1-14t2)w2z=0 on [T,+∞)
for T≫1.
Since 1w2∈L1(+∞) we can define the critical curve χw2 relative to w2 as in [5, equation (4.21)]. A computation yields
χw2(t)=141t2log2t for t≫1,
so that
κ(t)-1-14t2=χw2(t).
Hence, from [5, Theorem 5.1 and Proposition 5.7] we deduce that the solution z(t) of (4.16) satisfies
z(t)∼Clogtloglogt as t→+∞
for some constant C>0, and therefore
β(t)∼Ctlogtloglogt as t→+∞.
Using the above, we finally obtain the asymptotic behavior of u in (4.10), that is,
u(x)∼φ(x) as x→∞ in M
with
(4.17)φ(x)=CG(x)-logG(x)log(-logG(x))loglog(-logG(x))
as x→∞ on M.
In particular, the behavior of u at infinity is known once that of G(x) is known.
Next we prove a version of [5, Theorem 5.20] for equation (1.3).
Theorem 4.4Let (M,〈⋅,⋅〉) be a complete manifold, a(x), b(x), c(x)∈C0(M), σ>1, τ<1, and assume (1.4) and (3.31). Let Ω be a relatively compact open set and assume the existence of w∈C2(M∖Ω¯), a positive solution of
(4.18)Lw=Δw+a(x)w≤0 on M∖Ω¯.
Suppose that u and v are positive C2 solutions on M of
(4.19){Δu+a(x)u-b(x)uσ+c(x)uτ≤0,Δv+a(x)v-b(x)vσ+c(x)vτ≥0.
If
(4.20)u-v=o(w) as x→∞,
then v≤u on M.
Proof.
The idea of the proof is the same as that of [5, Theorem 5.20]. We report it here for the sake of completeness and for some minor differences. First we extend w to a positive function w~ on M. Towards this end, let Ω′ be a relatively compact open set such that Ω¯⊂Ω′. Fix a cut-off function ψ, 0≤ψ≤1, such that ψ≡1 on Ω and ψ≡0 on M∖Ω′. Define w~=ψ+(1-ψ)w. Note that w~>0 on M and w~=w on M∖Ω′¯, so that Lw~≤0 on M∖Ω′¯. For notational convenience we write again w and Ω in place of w~ and Ω′, but this time w>0 on M.
Let ε>0 and define uε=u+εw on M. Then, uε is a solution on M of
Δuε+a(x)uε≤b(x)uσ-c(x)uτ+εLw.
Therefore, interpreting the differential inequality in the weak sense, we have that for each φ∈Liploc(M), φ≥0,
-∫M〈∇uε,∇φ〉+∫Ma(x)uεφ≤∫Mb(x)uσφ-∫Mc(x)uτφ+ε∫MφLw.
Now, by the second Green formula,
∫MφLw=∫Ma(x)wφ+∫MwΔφ=∫MwLφ,
and therefore we can rewrite the above inequality as
(4.21)-∫M〈∇uε,∇φ〉+∫Ma(x)uεφ≤∫Mb(x)uσφ-∫Mc(x)uτφ+ε∫MwLφ.
Similarly, interpreting the second differential inequalitty of (4.19) in the weak sense, we have
(4.22)-∫M〈∇v,∇φ〉+∫Ma(x)vφ≥∫Mb(x)vσφ-∫Mc(x)vτφ
with φ as above.
Next, by contradiction suppose that
Γ={x∈M:v(x)>u(x)}≠∅.
Then, for ε>0 sufficiently small,
(4.23)Γε={x∈M:v(x)>uε(x)}≠∅.
We now consider the Lipschitz function γε=(v2-uε2)+. Condition (4.20) implies that γε has compact support in M and it is not identically zero because of (4.23). Thus, the functions φ1=γεuε and φ2=γεv are admissible, respectively, for (4.21) and (4.22). Substituting we have
-∫M〈∇uεuε,∇γε〉-|∇uε|2uε2γε-a(x)γε≤∫Mb(x)uσuεγε-c(x)uτuεγε+εwL(γεuε)
and -∫M〈∇vv,∇γε〉-|∇v|2v2γε-a(x)γε≥∫Mb(x)vσ-1γε-c(x)vτ-1γε.
Thus, subtracting the second from the first, we deduce
-∫Γε〈∇uεuε-∇vv,∇γε〉+∫Γε(|∇uε|2uε2-|∇v|2v2)γε
≤∫Γεb(x)(uσuε-vσ-1)γε-∫Γεc(x)(uτuε-vτ-1)γε+ε∫MwL(γεuε).
Inserting the expression for γε and rearranging, we finally have
∫Γε|∇uε-uεv∇v|2-|∇v-vuε∇uε|2
(4.24)≤∫Γεb(x)(uσuε-vσ-1)γε-∫Γεc(x)(uτuε-vτ-1)γε+ε∫MwL(γεuε).
Let V be a relatively compact open set with smooth boundary such that Ω¯⊂V and let ψ, 0≤ψ≤1, be a cut-off function such that ψ≡1 on Ω and ψ≡0 on M∖V¯. Then, using again the second Green formula and (4.18), we have
∫MwL(γεuε)=∫MwL(ψγεuε)+∫MwL((1-ψ)γεuε)
=∫MwL(ψγεuε)+∫M(1-ψ)γεuεLw
≤∫MwL(ψγεuε).
Now since uε is bounded from below by a positive constant on V¯, by applying the dominated convergence theorem, we deduce that
limε→0ε|∫MwL(ψγεuε)|≤limε→0ε[∫V|∇w||∇(ψγεuε)|+|a(x)wψγεuε|]=0.
Therefore, letting ε→0 in (4.24), using Fatou’s lemma and the last two inequalities, we get
0≤∫Γ|∇u-uv∇v|2+∫Γ|∇v-vu∇u|2
(4.25)≤∫Γb(x)(uσ-1-vσ-1)(v2-u2)-∫Γc(x)(uτ-1-vτ-1)(v2-u2)≤0.
Therefore vu is constant on any connected component of Γ. Clearly Γ must have no boundary because otherwise letting x→∂Γ we would deduce u=v on Γ which is a contradiction. By connectedness, v=Au on M for some A>1 and inserting into (4.25) we deduce
∫Γb(x)(1-Aσ-1)(1-A2)uσ+1+∫Γc(x)(Aτ-1-1)(1-A2)uτ+1≡0.
Since u>0, this contradicts assumptions (1.4) and (3.31). Hence, Γ=∅, that is, v≤u on M.
∎
Thus, considering φ defined in (4.17), as a consequence of Theorem 4.4, Theorem 4.1, and the subsequent discussion we have the following theorem.
Theorem 4.5Let (M,〈⋅,⋅〉,G) as in (4.2), a(x), b(x), c(x)∈C0(M), σ>1, τ<1, and assume (1.4), (3.31), and
a(x)≤{1+1log2G(x)[1+1log2(-logG(x))]}|∇logG(x)|24
outside a compact set. If u and v are positive C2 solutions of
Δu+a(x)u-b(x)uσ+c(x)uτ=0,
such that
u(x)-v(x)=o(φ(x)) as x→∞
with φ(x) as in (4.17), then u≡v on M.
It is reasonable that if we strengthen the upper bound (4.9) on a(x) the growth of u defined in (4.10) should improve in (4.17).
For the sake of simplicity let us suppose
a(x)≤λ|∇logG(x)|24
on M∖K for some constant λ∈(-∞,1]. We proceed as in the proof of Theorem 4.1 to arrive to (4.12) that now reads
β¨+[λ-1]β=0 on [T,+∞) for some T>0.
Positive solutions of the above are immediately obtained. Indeed, for λ=1 we let β(t)=Ct for some constant C>0 while for λ∈(-∞,1) we let β(t)=Ce1-λt, C>0. Thus, the positive solution u(x) of Lu≤0 given in (4.10) satisfies
u(x)∼{CG(x)log1G(x)if λ=1,CG(x)1-1-λ2if λ∈(-∞,1),
as x→∞ for some constant C>0.
Thus, going back to Theorem 4.5 we obtain the following version.
Theorem 4.6Let (M,〈⋅,⋅〉,G) as in (4.2), a(x), b(x), c(x)∈C0(M), σ>1, τ<1, and assume (1.4), (3.31), and
a(x)≤λ|∇logG(x)|24
outside a compact set, for some constant λ∈(-∞,1].
If u and v are positive C2 solutions of
Δu+a(x)u-b(x)uσ+c(x)uτ=0,
such that
u(x)-v(x)={o(G(x)log1G(x))if λ=1,o(G(x)1-1-λ2)if λ∈(-∞,1),
as x→∞, then u≡v on M.
As a final remark we observe that finiteness of the index of L=Δ+a(x) can be also deduced by the validity of a Sobolev-type inequality on M. Indeed, according to [22, Lemma 7.33], the validity of (4.3) and the assumption
a+(x)∈L1α(M)
imply that L has finite index.
