# Existence of Conformal Metrics with Prescribed Q-Curvature

Authors: Mohammed Ali Al-Ghamdi; Hichem Chtioui; Afef Rigane
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider the problem of existence of conformal metrics with prescribed Q-curvature on standard sphereSn,n≥5. Under the assumption that the order of flatness at critical points of prescribed Q-curvature function K(x) is β∈]1,n−4], we give precise estimates on the losses of the compactness, and we prove new existence and multiplicity results through an Euler-Hopf type formula.


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## Body

## 1. Introduction and Main Result

The Paneitz operator onn-Riemannian manifold (M,g0) is a fourth-order differential operator which arises in conformal geometry and satisfies a certain covariance property (see [1]). For n≥5 it is defined by
(1)Pg0nu=Δg02u-divg0(anSg0g0+bnRicg0)du+n-42Qg0nu,
where
(2)an=(n-2)2+42(n-1)(n-2),bn=-4n-2,Qg0n=-12(n-1)Δg0Sg0+n3-4n2+16n-168(n-1)2(n-2)2Sg02-2(n-2)2|Ricg0|2.
Such a Qg0n is a fourth-order invariant called Q-curvature or Paneitz curvature. See [2, 3] for details about the properties of Pg0n.Ifg=u4/(n-4)g0 is a conformal metric to g0, where u is a smooth positive function, then the conformal covariance property of the Paneitz operator reads as follows:
(3)∀ψ∈C∞(M):Pg0n(uψ)=u(n+4)/(n-4)Pgn(ψ).
If one prescribes the Q-curvature for the metric g by a function K, this leads to the following equation:
(4)Pg0n(u)=u(n+4)/(n-4)Pgn(1)=n-42Ku(n+4)/(n-4).
The literature for the existence of solutions of the prescribed Q-curvature problem on compact manifolds is considerably bigger. In [4–7], existence results for the constant Q-curvature problem in 4-dimensional manifolds are given. On manifolds of dimension greater than 4, existence results were given for Einstein manifolds in [3]. On the sphere Sn, we refer to results of [8–14] and the references therein.In this paper we continue to study the problem of prescribing Q-curvature on the standard sphere(Sn,g0), n≥5. The Paneitz operator in this case is coercive on the Sobolev space H22(Sn) and has the following expression:
(5)𝒫(u)∶=Pg0n(u)=Δg02u-cnΔg0u+dnu,
where
(6)cn=12(n2-2n-4),dn=n-416n(n2-4).
According to problem (4), the problem can be formulated as follows. Given a smooth function K on Sn, we look for solutions of
(7)𝒫(u)=n-42Ku(n+4)/(n-4),u>0onSn.
The special case where the manifold is a sphere endowed with its standard metric deserves particular attention. Indeed due to Kazdan-Warner type obstructions, see [3], conditions have to be imposed on the function K.To state our result we set up the following conditions and notation.(
f
)
β Assume that K:Sn→ℝ, n≥5 be a C1 positive function such that for any critical point y of K, there exists some real number β=β(y) such that in some geodesic normal coordinate system centered at y (through the exponential map), we have
(8)K(x)=K(y)+∑k=1nbk|(x-y)k|β+R(x-y),
where bk=bk(y)≠0, forallk=1,…,n, ∑k=1n‍bk≠0 and ∑s=0[β]‍|∇sR(x-y)||x-y|-β+s=o(1) as x tends to y. Here ∇s denotes all possible derivatives of order s and [β] is the integer part of β. For each critical point y of K, we denote
(9)i~(y)=:♯{bk,k=1,…,n,suchthatbk<0}.
Let,
(10)𝒦={y∈Sn,∇g0K(y)=0}𝒦+={y∈𝒦,-∑k=1n‍bk>0},𝒦n-4={y∈𝒦,β=β(y)=n-4}.
For each p-tuple, p≥1 of distinct points τp:=(yl1,…,ylp) such that for all i=1,…,p, yli∈𝒦+∩𝒦n-4, we define a p×p symmetric matrix M(τp)=(mij) by
(11)mii=n-4nc~1-∑k=1nbk(yli)K(yli)n/4,mij=2(n-4)/2c1-G(yli,ylj)[K(yli)K(ylj)](n-4)/8,
where
(12)c1=∫ℝn‍dx(1+|x|2)(n+4)/2,c~1=c02n/(n-4)∫ℝn‍|x1|n-4(1+|x|2)ndx,
and G is the Green function for the operator 𝒫(u) on Sn. Here x1 is the first component of x in some geodesic normal coordinate system.Letρ(τp) be the least eigenvalue of M(τp). It was first pointed out by Bahri [15] that when the interaction between the different bubbles is of the same order as the self interaction, the least eigenvalue of some matrices like (11) plays a fundamental role in the existence of solutions to problems like (7). Regarding problem (7), such kind of phenomenon appears under (f)β condition in 𝒦n-4; see [11].(
A
1
) Assume that ρ(τp)≠0 for each distinct point yl1,…,ylp∈𝒦+∩𝒦n-4.Now, we introduce the following set:(13)𝒞∞∶={τp=(yl1,…,ylp)∈(𝒦+)p,p≥1,s.t.yi≠yj∀i≠j,andif{yl1,…,ylp}∩𝒦n-4≠∅,wedenotebyyi1,…,yiqallelementsof{yl1,…,ylp}withβ(yij)=n-4foreachj=1,…,qthenρ(yi1,…,yiq)>0}.
The main result of this paper is the following.Theorem 1.
Assume thatK satisfies (A1) and (f)β, with
(14)1<β≤n-4.
If
(15)d=∑(y1,…,yp)∈𝒞∞‍(-1)p-1+∑k=1p‍n-i~(yk)-1≠0.
Then (7) has at least one solution.
Moreover if(n-4)/2<β≤n-4, for generic K one has
(16)♯S≥|d|,
where S denotes the set of solutions to (7).Our argument uses a careful analysis of the lack of compactness of the Euler Lagrange functionalJ associated with problem (7). Namely, we study the noncompact orbits of the gradient flow of J the so-called critical points at infinity following the terminology of Bahri [15]. These critical points at infinity can be treated as usual critical points once a Morse lemma at infinity is performed from which we can derive just as in the classical Morse theory the difference of topology induced by these noncompact orbits and compute their Morse index. Such a Morse lemma at infinity is obtained through the construction of suitable pseudogradient for which the Palais-Smale condition is satisfied along the decreasing flow lines, as long as these flow lines do not enter the neighborhood of a finite number y1,…,yp of critical points of K such that (y1,…,yp)∈𝒞∞.Similar Morse lemma at infinity has been established for the problem (7) on the sphere Sn, n≥5, under the hypothesis that the order of flatness at critical points of K is β∈[n-4,n[; see [11].The rest of this paper is organized as follows. In Section2, we set up the variational problem and we recall the expansion of the gradient of the associated Euler-Lagrange functional near infinity. In Section 3, we characterize the critical points at infinity of the associated variational problem. Section 4 is devoted to the proof of the main result Theorem 1, while we give in Section 5 a more general statement than Theorem 1.


## 2. General Framework and Some Known Facts

### 2.1. Variational Structure and Lack of Compactness

Our problem (7) enjoys a variational structure. Indeed, solutions to (7) correspond to positive critical points of the functional
(17)J(u)=∫Sn‍𝒫uudvg0(∫Sn‍K|u|2n/(n-4)dvg0)(n-4)/2n,
defined on
(18)Σ={u∈H22(Sn),∥u∥2∶=∫Sn‍𝒫uudvg0=1}.However, it is delicate from a variational viewpoint because the functionalJ does not satisfy the Palais-Smale condition. This means that there exist sequences along which J is bounded, its gradient goes to zero and which not converge. The analysis of sequences failing Palais-Smale condition can be analyzed along the ideas introduced in [16, pages 325 and 334]. In order to describe such a characterization in our case, we need to introduce some notations. For a∈Sn and λ>0, let
(19)δ(a,λ)(x)=c012(n-4)/2×λ(n-4)/2(1+((λ2-1)/2)(1-cos(d(x,a))))(n-4)/2,
where d is the geodesic distance on (Sn,g) and c0 is chosen so that δ(a,λ) is the family of solutions of the problem
(20)𝒫(u)=u(n+4)/(n-4),u>0,onSn
(see [17]).We define now the set of potential critical points at infinity associated with the functionJ. For ε>0 and p∈ℕ*, let us define
(21)V(p,ε)={u∈Σ/∃a1,…,ap∈Sn,∃λ1,…,λp>ε-1,∃α1,…,αp>0with∥u-∑i=1p‍αiδ(ai,λi)∥<ε,|J(u)n/(n-4)αi8/(n-4)K(ai)-1|<ε∀i,εij<ε∀i≠j.
Here, εij=[(λi/λj)+(λj/λi)+(λiλj/2)(1-cosd(ai,aj))]-(n-4)/2.Forw a solution to (7) we also define V(p,ε,w) as
(22)V(p,ε,w)={u∈Σ/∃α0>0s.t.u-α0w∈V(p,ε),|α08/(n-4)J(u)n/(n-4)-1|<ε.
If u is a function in V(p,ε,w), one can find an optimal representation, following the ideas introduced in Proposition 5.2 of [15] and [16, pages 348–350]. Namely, we have the following.Proposition 2.
For anyp∈ℕ*, there is εp>0 such that if ε≤εp and u∈V(p,ε,w), then the following minimization problem
(23)minαi>0,λi>0,ai∈Sn,h∈Tw(Wu(w)).∥u-∑i=1p‍αiδ(ai,λi)-α0(w+h)∥
has a unique solution (α,λ,a,h), up to a permutation.In particular, we can writeu as follows:
(24)u=∑i=1p‍αiδ(ai,λi)+α0(w+h)+v,
where v belongs to H22(Sn)∩Tw(Ws(w)) and it satisfies (V0), Tw(Wu(w)) and Tw(Ws(w)) are the tangent spaces at w of the unstable and stable manifolds of w for a decreasing pseudogradient of J, and (V0) is the following:
(V0)〈v,ψ〉=0forψ∈{δi,∂δi∂λi,∂δi∂ai,i=1,…,p}〈v,w〉=0,〈v,h〉=0∀h∈TwWu(w).
Here, δi=δ(ai,λi) and 〈·,·〉 denotes the scalar product defined on H22(Sn) by
(25)〈u,v〉=∫Sn‍Δg0uΔg0vdvg0+cn∫Sn‍∇g0u∇g0vdvg0+dn∫Sn‍uvdvg0.
Notice that Proposition 2 is also true if we take w=0 and, therefore, h=0 and u in V(p,ε).We are ready now to state the characterization of the sequences failing the Palais-Smale condition. For technical reasons, we introduce the following subsets ofΣ. Let Σ+={u∈Σ,u≥0} and let for η positive very small,
(26)Vη(Σ+)={u∈Σ,J(u)(2n-4)/(n-4)e2J(u)∥u-∥L2n/(n-4)8/(n-4)<η},
where u-=max(0,-u). Using the idea introduced in [16, 18], see also [19], we have the following proposition.Proposition 3.
Let(uk) be a sequence in Vη(Σ+) such that J(uk) is bounded and ∂J(uk) goes to zero. Then there exists an integer p∈ℕ*, a sequence (εK)>0, εk tends to zero, and an extracted subsequence of uk's, again denoted (uk), such that uk∈V(p,εk,w) where w is zero or a solution to (7).Now arguing as in [16] (pages 326, 327, and 334), we have the following Morse lemma which completely gets rid of the v-contributions and shows that it can be neglected with respect to the concentration phenomenon.Proposition 4.
There is a𝒞1-map which to each (αi,ai,λi,h) such that ∑i=1p‍αiδ(ai,λi)+α0(w+h) belongs to V(p,ε,w) associates v¯=v¯(α,a,λ,h) such that v¯ is unique and satisfies
(27)J(∑i=1p‍αiδ(ai,λi)+α0(w+h)+v¯)=minv∈(V0){J(∑i=1p‍αiδ(ai,λi)+α0(w+h)+v)}.
Moreover, there exists a change of variables v-v¯→V such that
(28)J(∑i=1p‍αiδ(ai,λi)+α0(w+h)+v)=J(∑i=1p‍αiδ(ai,λi)+α0(w+h)+v¯)+∥V∥2.The following proposition gives precise estimates ofv¯.Proposition 5 (see [11, Lemma 3.1]).
Letu=∑i=1p‍αiδi+α0(w+h)∈V(p,ε,w) and let v¯ be defined in Proposition 4. One has the following estimates: there exists c>0 independent of u such that the following holds:
(29)∥v¯∥≤c∑i=1p‍[1λin/2+1λiβ+|∇K(ai)|λi+(logλi)(n+4)/2nλi(n+4)/2]+c{∑k≠r‍εkr(n+4)/2(n-4)(logεkr-1)(n+4)/2n,ifn≥12∑k≠r‍εkr(logεkr-1)(n-4)/n,ifn<12.At the end of this subsection, we give the following definition.Definition 6 (see [16, 18]).
A critical point at infinity ofJ on Vη(Σ+) is a limit of a flow line u(s) of the equation
(30)∂u∂s=-∂J(u(s)),u(0)=u0,
such that u(s) remains in V(p,ε(s),w) for s≥s0. Here w is either zero or a solution to (7) and ε(s) is some positive function tending to zero when s→+∞. Using Proposition 2, u(s) can be written as
(31)u(s)=∑i=1p‍αi(s)δ(ai(s),λi(s))+α0(s)(w+h(s))+v(s).
Denoting α~i∶=lims→+∞αi(s),y~i∶=lims→+∞ai(s), we denote by
(32)∑i=1p‍α~iδ(y~i,∞)+α~0wor(y~1,...,y~p,w)∞
such a critical point at infinity. If w≠0, it is called of w-type or mixed type.Notice thatVη(Σ+) remains invariant under the flow generated by (-∂J) (see Lemma 5.1 of [20]; see also Lemma 4.1 of [18]).


### 2.2. Expansion of the Gradient of the Functional

In this subsection, we recall the expansion of the gradient of the functionalJ in V(p,ε), p≥1.Proposition 7 (see [11]).
For anyu=∑j=1p‍αjδj in V(p,ε), the following expansion holds
(i)(33)〈∂J(u),λi∂δi∂λi〉=-2c2J(u)∑i≠j‍αjλi∂εij∂λi+o(∑i≠j‍εij)+o(1λi),
where c2=c02n/(n-4)∫ℝn‍dy/(1+|y|2)(n+4)/2.
(ii) Ifai∈B(yji,ρ), yji∈𝒦 and ρ is a positive constant small enough, one has
(34)〈∂J(u),λi∂δi∂λi〉=2J(u)[-c2∑j≠i‍αjλi∂εij∂λi+n-42nc02n/(n-4)βαiK(ai)×∑k=1nbkλiβ∫ℝn‍signe(xk+λi(ai-yji)k)×|xk+λi(ai-yji)k|β-1×xk(1+|x|2)ndx+o(∑j≠i‍εij+∑j=1p‍1λjβ)].
(iii) Furthermore ifλi|ai-yji|<δ, for δ very small, one then has
(35)〈∂J(u),λi∂δi∂λi〉=2J(u)×[(∑j≠i‍εij+∑j=1p‍1λjβ)n-42nβc3αiK(ai)∑k=1nbkλiβ-c2∑j≠i‍αjλi∂εij∂λi+o(∑j≠i‍εij+∑j=1p‍1λjβ)],
where c3=c02n/(n-4)∫Sn‍|x1|β/(1+|x|2)ndx.Proposition 8 (see [11]).
Lettingu=∑j=1p‍αjδj∈V(p,ε), one has
(i)(36)〈∂J(u),1λi∂δi∂ai〉=-c5(J(u))2(n-2)/(n-4)αi(n+4)/(n-4)∇K(ai)λi+O(∑i≠j‍1λi|∂εij∂ai|)+o(∑i≠j‍εij+1λi),
where c5=∫ℝn‍dy/(1+|y|2)n.
(ii) Ifai∈B(yji,ρ), yji∈𝒦, one has
(37)〈∂J(u),1λi∂δi∂(ai)k〉=-2(n-4)c02n/(n-4)αi(n+4)/(n-4)(J(u))2(n-2)/(n-4)1λiβ×∫ℝn‍bk|xk+λi(ai-yji)k|βxk(1+|x|2)n+1dy+o(∑i≠j‍εij)+o(∑i=1p‍1λiβ)+O(∑i≠j‍1λi|∂εij∂ai|),
where k=1,…,n and (ai)k is the kth component of ai in some geodesic normal coordinates system.


## 2.1. Variational Structure and Lack of Compactness

Our problem (7) enjoys a variational structure. Indeed, solutions to (7) correspond to positive critical points of the functional
(17)J(u)=∫Sn‍𝒫uudvg0(∫Sn‍K|u|2n/(n-4)dvg0)(n-4)/2n,
defined on
(18)Σ={u∈H22(Sn),∥u∥2∶=∫Sn‍𝒫uudvg0=1}.However, it is delicate from a variational viewpoint because the functionalJ does not satisfy the Palais-Smale condition. This means that there exist sequences along which J is bounded, its gradient goes to zero and which not converge. The analysis of sequences failing Palais-Smale condition can be analyzed along the ideas introduced in [16, pages 325 and 334]. In order to describe such a characterization in our case, we need to introduce some notations. For a∈Sn and λ>0, let
(19)δ(a,λ)(x)=c012(n-4)/2×λ(n-4)/2(1+((λ2-1)/2)(1-cos(d(x,a))))(n-4)/2,
where d is the geodesic distance on (Sn,g) and c0 is chosen so that δ(a,λ) is the family of solutions of the problem
(20)𝒫(u)=u(n+4)/(n-4),u>0,onSn
(see [17]).We define now the set of potential critical points at infinity associated with the functionJ. For ε>0 and p∈ℕ*, let us define
(21)V(p,ε)={u∈Σ/∃a1,…,ap∈Sn,∃λ1,…,λp>ε-1,∃α1,…,αp>0with∥u-∑i=1p‍αiδ(ai,λi)∥<ε,|J(u)n/(n-4)αi8/(n-4)K(ai)-1|<ε∀i,εij<ε∀i≠j.
Here, εij=[(λi/λj)+(λj/λi)+(λiλj/2)(1-cosd(ai,aj))]-(n-4)/2.Forw a solution to (7) we also define V(p,ε,w) as
(22)V(p,ε,w)={u∈Σ/∃α0>0s.t.u-α0w∈V(p,ε),|α08/(n-4)J(u)n/(n-4)-1|<ε.
If u is a function in V(p,ε,w), one can find an optimal representation, following the ideas introduced in Proposition 5.2 of [15] and [16, pages 348–350]. Namely, we have the following.Proposition 2.
For anyp∈ℕ*, there is εp>0 such that if ε≤εp and u∈V(p,ε,w), then the following minimization problem
(23)minαi>0,λi>0,ai∈Sn,h∈Tw(Wu(w)).∥u-∑i=1p‍αiδ(ai,λi)-α0(w+h)∥
has a unique solution (α,λ,a,h), up to a permutation.In particular, we can writeu as follows:
(24)u=∑i=1p‍αiδ(ai,λi)+α0(w+h)+v,
where v belongs to H22(Sn)∩Tw(Ws(w)) and it satisfies (V0), Tw(Wu(w)) and Tw(Ws(w)) are the tangent spaces at w of the unstable and stable manifolds of w for a decreasing pseudogradient of J, and (V0) is the following:
(V0)〈v,ψ〉=0forψ∈{δi,∂δi∂λi,∂δi∂ai,i=1,…,p}〈v,w〉=0,〈v,h〉=0∀h∈TwWu(w).
Here, δi=δ(ai,λi) and 〈·,·〉 denotes the scalar product defined on H22(Sn) by
(25)〈u,v〉=∫Sn‍Δg0uΔg0vdvg0+cn∫Sn‍∇g0u∇g0vdvg0+dn∫Sn‍uvdvg0.
Notice that Proposition 2 is also true if we take w=0 and, therefore, h=0 and u in V(p,ε).We are ready now to state the characterization of the sequences failing the Palais-Smale condition. For technical reasons, we introduce the following subsets ofΣ. Let Σ+={u∈Σ,u≥0} and let for η positive very small,
(26)Vη(Σ+)={u∈Σ,J(u)(2n-4)/(n-4)e2J(u)∥u-∥L2n/(n-4)8/(n-4)<η},
where u-=max(0,-u). Using the idea introduced in [16, 18], see also [19], we have the following proposition.Proposition 3.
Let(uk) be a sequence in Vη(Σ+) such that J(uk) is bounded and ∂J(uk) goes to zero. Then there exists an integer p∈ℕ*, a sequence (εK)>0, εk tends to zero, and an extracted subsequence of uk's, again denoted (uk), such that uk∈V(p,εk,w) where w is zero or a solution to (7).Now arguing as in [16] (pages 326, 327, and 334), we have the following Morse lemma which completely gets rid of the v-contributions and shows that it can be neglected with respect to the concentration phenomenon.Proposition 4.
There is a𝒞1-map which to each (αi,ai,λi,h) such that ∑i=1p‍αiδ(ai,λi)+α0(w+h) belongs to V(p,ε,w) associates v¯=v¯(α,a,λ,h) such that v¯ is unique and satisfies
(27)J(∑i=1p‍αiδ(ai,λi)+α0(w+h)+v¯)=minv∈(V0){J(∑i=1p‍αiδ(ai,λi)+α0(w+h)+v)}.
Moreover, there exists a change of variables v-v¯→V such that
(28)J(∑i=1p‍αiδ(ai,λi)+α0(w+h)+v)=J(∑i=1p‍αiδ(ai,λi)+α0(w+h)+v¯)+∥V∥2.The following proposition gives precise estimates ofv¯.Proposition 5 (see [11, Lemma 3.1]).
Letu=∑i=1p‍αiδi+α0(w+h)∈V(p,ε,w) and let v¯ be defined in Proposition 4. One has the following estimates: there exists c>0 independent of u such that the following holds:
(29)∥v¯∥≤c∑i=1p‍[1λin/2+1λiβ+|∇K(ai)|λi+(logλi)(n+4)/2nλi(n+4)/2]+c{∑k≠r‍εkr(n+4)/2(n-4)(logεkr-1)(n+4)/2n,ifn≥12∑k≠r‍εkr(logεkr-1)(n-4)/n,ifn<12.At the end of this subsection, we give the following definition.Definition 6 (see [16, 18]).
A critical point at infinity ofJ on Vη(Σ+) is a limit of a flow line u(s) of the equation
(30)∂u∂s=-∂J(u(s)),u(0)=u0,
such that u(s) remains in V(p,ε(s),w) for s≥s0. Here w is either zero or a solution to (7) and ε(s) is some positive function tending to zero when s→+∞. Using Proposition 2, u(s) can be written as
(31)u(s)=∑i=1p‍αi(s)δ(ai(s),λi(s))+α0(s)(w+h(s))+v(s).
Denoting α~i∶=lims→+∞αi(s),y~i∶=lims→+∞ai(s), we denote by
(32)∑i=1p‍α~iδ(y~i,∞)+α~0wor(y~1,...,y~p,w)∞
such a critical point at infinity. If w≠0, it is called of w-type or mixed type.Notice thatVη(Σ+) remains invariant under the flow generated by (-∂J) (see Lemma 5.1 of [20]; see also Lemma 4.1 of [18]).


## 2.2. Expansion of the Gradient of the Functional

In this subsection, we recall the expansion of the gradient of the functionalJ in V(p,ε), p≥1.Proposition 7 (see [11]).
For anyu=∑j=1p‍αjδj in V(p,ε), the following expansion holds
(i)(33)〈∂J(u),λi∂δi∂λi〉=-2c2J(u)∑i≠j‍αjλi∂εij∂λi+o(∑i≠j‍εij)+o(1λi),
where c2=c02n/(n-4)∫ℝn‍dy/(1+|y|2)(n+4)/2.
(ii) Ifai∈B(yji,ρ), yji∈𝒦 and ρ is a positive constant small enough, one has
(34)〈∂J(u),λi∂δi∂λi〉=2J(u)[-c2∑j≠i‍αjλi∂εij∂λi+n-42nc02n/(n-4)βαiK(ai)×∑k=1nbkλiβ∫ℝn‍signe(xk+λi(ai-yji)k)×|xk+λi(ai-yji)k|β-1×xk(1+|x|2)ndx+o(∑j≠i‍εij+∑j=1p‍1λjβ)].
(iii) Furthermore ifλi|ai-yji|<δ, for δ very small, one then has
(35)〈∂J(u),λi∂δi∂λi〉=2J(u)×[(∑j≠i‍εij+∑j=1p‍1λjβ)n-42nβc3αiK(ai)∑k=1nbkλiβ-c2∑j≠i‍αjλi∂εij∂λi+o(∑j≠i‍εij+∑j=1p‍1λjβ)],
where c3=c02n/(n-4)∫Sn‍|x1|β/(1+|x|2)ndx.Proposition 8 (see [11]).
Lettingu=∑j=1p‍αjδj∈V(p,ε), one has
(i)(36)〈∂J(u),1λi∂δi∂ai〉=-c5(J(u))2(n-2)/(n-4)αi(n+4)/(n-4)∇K(ai)λi+O(∑i≠j‍1λi|∂εij∂ai|)+o(∑i≠j‍εij+1λi),
where c5=∫ℝn‍dy/(1+|y|2)n.
(ii) Ifai∈B(yji,ρ), yji∈𝒦, one has
(37)〈∂J(u),1λi∂δi∂(ai)k〉=-2(n-4)c02n/(n-4)αi(n+4)/(n-4)(J(u))2(n-2)/(n-4)1λiβ×∫ℝn‍bk|xk+λi(ai-yji)k|βxk(1+|x|2)n+1dy+o(∑i≠j‍εij)+o(∑i=1p‍1λiβ)+O(∑i≠j‍1λi|∂εij∂ai|),
where k=1,…,n and (ai)k is the kth component of ai in some geodesic normal coordinates system.


## 3. Characterization of the Critical Points at Infinity

This section is devoted to the characterization of the critical points at infinity inV(p,ε),p≥1, under β-flatness condition with 1<β≤n-4. This characterization is obtained through the construction of a suitable pseudogradient at infinity for which the Palais-Smale condition is satisfied along the decreasing flow-lines as long as these flow lines do not enter in the neighborhood of finite number of critical points yi,i=1,…,p of K such that (y1,…,yp)∈𝒞∞.Now we introduce the following main result.Theorem 9.
Assume thatK satisfies (A1) and (f)β, 1<β≤n-4.
Letβ∶=max{β(y)/y∈𝒦}. For p≥1, there exists a pseudogradient W in V(p,ε) so that the following holds.
There exists a constantc>0 independent of u=∑i=1p‍αiδ(aiλi)∈V(p,ε) such that
(38)(i)〈∂J(u),W(u)〉≤-c(∑i=1p‍1λiβ+∑i=1p‍|∇K(ai)|λi+∑j≠i‍εij).(ii)〈∂J(u+v¯),W(u)+∂v¯∂(αi,ai,λi)(W(u))〉≤-c(∑i=1p‍1λiβ+∑i=1p‍|∇K(ai)|λi+∑j≠i‍εij).
Furthermore |W| is bounded and the only case where the maximum of the λi's is not bounded is when ai∈B(yli,ρ) with yli∈𝒦, foralli=1,…,p, (yl1,…,ylp)∈𝒞∞.We will prove Theorem9 at the end of the section. Now we state two results which deal with two specific cases of Theorem 9. Let
(39)V1(p,ε)={u=∑i=1p‍αiδi∈V(p,ε)s.t.,ai∈B(yli,ρ),yli∈𝒦∖𝒦n-4∀i=1,…,p∑i=1p‍∑i=1p‍},V2(p,ε)={u=∑i=1p‍αiδi∈V(p,ε)s.t.,ai∈B(yli,ρ),yli∈𝒦n-4∀i=1,…,p∑i=1p‍}.
We then have the following.Proposition 10 (see [11], Proposition 3.7).
Forp≥1 there exists a pseudogradient W2 in V2(p,ε) such that for all u=∑i=1p‍αiδi∈V2(p,ε), one has
(40)〈∂J(u),W2(u)〉≤-c(∑i=1p‍1λin-4+∑i≠j‍εij+∑i=1p‍|∇K(ai)|λi),
where c is a positive constant independent of u. Furthermore, one has |W2| which is bounded and the only case where the maximum of λi’s is not bounded is when ai∈B(yli,ρ),yli∈𝒦+, for all i=1,…,p, with ρ(yl1,…,ylp)>0.Proposition 11.
Forp≥1, there exists a pseudogradient W1 in V1(p,ε) so that the following holds.
There existsc>0 independent of u=∑i=1p‍αiδi∈V1(p,ε) such that
(41)〈∂J(u),W1(u)〉≤-c(∑i=1p‍1λiβ+∑i≠j‍εij+∑i=1p‍|∇K(ai)|λi).
Furthermore |W1| is bounded and the only case where the maximum of the λi's is not bounded is when ai∈B(yli,ρ) with yli∈𝒦+, for all i=1,…,p, and yi≠yj for all i≠j.Observe that inV1(p,ε) the interaction of two bubbles is negligible with respect to the self-interaction. Similar phenomena occur for the scalar curvature problem, see [21], so the proof of Proposition 11 is similar to the corresponding statement in [21].Before giving the proof of Theorem9, we state the following notations extracted from [11].LetM1 be a fixed positive constant large enough and let u=∑i=1p‍αiδ(ai,λi)∈V(p,ε) such that ai∈B(ai,ρ), yli∈𝒦, for all i=1,…,p. For any index i, 1≤i≤p, we define the following vector fields:
(42)Zi(u)=αiλi∂δ(ai,λi)∂λi,Xi=αi∑k=1n‍1λi∂δ(ai,λi)∂(ai)k∫ℝn‍bk|xk+λi(ai-yli)k|βi(1+λi|(ai-yli)k|)βi-1×xk(1+|x|2)n+1dx.
We claim that Xi is bounded. Indeed, the claim is trivial if λi|ai-yli|≤M1. If λi|ai-yli|>M1, for any k, 1≤k≤n, such that λi|(ai-yli)k|>M1/n, we have the following estimate:
(43)∫ℝn|xk+λi(ai-yli)k|βixk(1+|x|2)n+1dx=c(signeλi(ai-yli)k)(λi|(ai-yli)k|)βi-1×(1+o(1)).
Hence our claim follows. Next, we will say that
(44)i∈L1ifλi|ai-yli|≤M1,i∈L2ifλi|ai-yli|>M1,
and we will denote by ki the index satisfying
(45)|(ai-yli)ki|=max1≤k≤n|(ai-yli)k|.
It easy to see that if i∈L2, then λi|(ai-yli)ki|>M1/n.Now, we introduce the following two lemmas.Lemma 12.
Letu=∑i=1p‍αiδ(ai,λi)∈V(p,ε), such that ai∈B(yli,ρ), yli∈𝒦, for all i=1,…,p. One then has
(46)〈∂J(u),Zi(u)〉=-2c2J(u)∑j≠i‍αiαj∂εij∂λi+O(1λiβi)+[O(|(ai-yli)ki|βi-2λi2),ifi∈L2]+o(∑j≠i‍εij)+o(∑j=1p‍1λjβj),
where ki is defined in (45).Proof.
Using Proposition7, we have
(47)〈∂J(u),Zi(u)〉=-2c2J(u)∑j≠i‍αiαjλi∂εij∂λi+n-42nc02n/(n-4)βαi2K(ai)×∫ℝn‍signe(xk+λi(ai-yji)k)|xk+λi(ai-yji)k|β-1×xk(1+|x|2)ndx+o(∑j≠i‍εij)+o(∑j=1p‍1λjβj).
Observe that for k∈{1,…,n}, if λi|(ai-yli)k|>M1/n, we have
(48)∫ℝn‍signe(xk+λi(ai-yji)k)×|xk+λi(ai-yli)k|βi-1xk(1+|x|2)ndx=csigne(λi(ai-yji)k)×(λi|(ai-yli)k|)βi-2(1+o(1)),
taking M1 large enough. If not, we have
(49)∫ℝn‍|xk+λi(ai-yli)k|βi-1|xk|(1+|x|2)ndx=O(1).
Using the fact that ki defined in (45) satisfies λi|(ai-yli)ki|>M1/n, if i∈L2, Lemma 12 follows.Lemma 13.
Foru=∑i=1p‍αiδ(ai,λi)∈V(p,ε), such that ai∈B(yli,ρ), yli∈𝒦, for all i=1,…,p, one has
(50)〈∂J(u),Xi(u)〉≤O(∑j≠i‍1λi|∂εij∂ai|)+O[(1λiβi),ifi∈L1]+[-c(1λiβi+|(ai-yli)ki|βi-1λi),ifi∈L2]+o(∑j=1p‍1λjβj),
where ki is defined in (45).Proof.
Using Proposition8, we have
(51)〈∂J(u),Xi(u)〉=-2(n-4)c02n/(n-4)αi2n/(n-4)1λiβi×∑k=1n‍(∫ℝn‍bk|xk+λi(ai-yli)k|βi(1+λi|(ai-yli)k|)(βi-1)/2×xk(1+|x|2)n+1dx|xk+λi(ai-yli)k|βi(1+λi|(ai-yli)k|)(βi-1)/2)2+O(∑j≠i‍1λi|∂εij∂ai|)+o(∑j=1p‍1λjβj)≤-c1λiβi(∫ℝn‍bki|xk+λi(ai-yli)ki|βi(1+λi|(ai-yli)ki|)(βi-1)/2×xki(1+|x|2)n+1dx|xk+λi(ai-yli)ki|βi(1+λi|(ai-yli)ki|)(βi-1)/2)2+O(∑j≠i‍1λi|∂εij∂ai|)+o(∑j=1p‍1λjβj).
Using (43) and the fact that λi|(ai-yli)ki|>M1/n, if i∈L2, Lemma 13 follows.Proof of Theorem9.
In order to complete the construction of the pseudogradientW suggested in Theorem 9, it only remains (using Propositions 10 and 11) to focus attention on the two following subsets of V(p,ε).
Subset 1.We consider here the case of u=∑i=1p‍αiδ(aiλi)=∑i∈I1‍αiδ(aiλi)+∑i∈I2‍αiδ(aiλi) such that
(52)I1≠∅,I2≠∅,∑i∈I1‍αiδ(aiλi)∈V1(♯I1,ε),∑i∈I2‍αiδ(aiλi)∈V2(♯I2,ε).
Without loss of generality, we can assume that
(53)λ1≤⋯≤λp.
We distinguish three cases.
Case  1. u1∶=∑i∈I1‍αiδ(aiλi)∉V11(♯I1,ε) = {u=∑j=1♯I1‍αjδ(aj,λj), aj∈B(ylj,ρ), ylj∈𝒦+ for all j=1,…,♯I1 and ylj≠ylk for all j≠k}.
LetW1~ be the pseudogradient on V(p,ε) defined by W1~(u)=W1(u1), where W1 is the vector filed defined by Proposition 11 in V1(♯I1,ε). Note that if u1∉V11(♯I1,ε), then the pseudo-gradient W1(u1) does not increase the maximum of the λi's, i∈I1. Using Proposition 11, we have
(54)〈∂J(u),W1~(u)〉≤-c(∑i∈I1‍1λiβi+∑j≠i,i,j∈I1‍εij+∑i∈I1‍|∇K(ai)|λi)+O(∑i∈I1,j∈I2‍εij).
An easy calculation yields
(55)εij=O(1(λiλj)(n-4)/2)=o(1λiβi)+o(1λjβj),
Fixi0∈I1, we denote by
(56)J1={i∈I2,s.t,λin-4≥12λi0βi0},J2=I2∖J1.
Using (54) and (55), we find that
(57)〈∂J(u),W1~(u)〉≤-c(∑i∈I1∪J1‍1λiβi+∑i∈I1‍|∇K(ai)|λi+∑j≠i∈I1‍εij)+o(∑i=1p‍1λiβi).
From another part, by Lemma 12 we have
(58)〈∂J(u),∑i∈J1‍-2iZi(u)〉≤c∑j≠i,i∈J1‍2iλi∂εij∂λi+O(∑i∈J1‍1λiβi)+O(∑i∈J1∩L2‍|(ai-yli)ki|βi-2λi2).
Observe that using a direct calculation, we have
(59)λi∂εij∂λi≤-cεij,ifλi≥λjorλi~λjor|ai-aj|≥δ0>0.
Since for i<j, we have
(60)2iλi∂εij∂λi+2jλj∂εij∂λj≤-cεij,
and for i∈J1 and j∈J2 we have λj≤λi, we obtain λi(∂εij/∂λi)≤-cεij. These estimates yield
(61)〈∂J(u),∑i∈J1‍-2iZi(u)〉≤-c∑j≠i,i∈J1,j∈J1∪J2‍εij+O(∑i∈J1‍1λiβi)+O(∑i∈J1∩L2‍|(ai-yli)ki|βi-2λi2)+O(∑i∈J1,j∈I1‍εij).
It is easy to see that for any index i∈L2, we have
(62)|(ai-yli)ki|βi-2λi2≤nM1|(ai-yli)ki|βi-1λi,
where ki is defined in (45) and M1 is large enough. Thus, we derive that
(63)|(ai-yli)ki|βi-2λi2=o(|(ai-yli)ki|βi-1λi),foranyi∈L2.
Letm1>0 be small enough; using Lemma 13, (63), and (55), we get
(64)〈∂J(u),∑i∈J1‍-2iZi(u)+m1∑i∈J1∩L2‍Xi(u)〉≤-c(∑j≠i,i∈J1,j∈J1∪J2‍εij+∑i∈J1‍|∇K(ai)|λi)+O(∑i∈J1‍1λiβi)+o(∑i=1p‍1λiβi),
and by (57) we obtain
(65)〈∂J(u),W1~(u)+m1(∑i∈J1‍-2iZi(u)+m1∑i∈J1∩L2‍Xi(u))〉≤-c(∑i∈I1∪J1‍|∇K(ai)|λi∑i∈I1∪J1‍1λiβi+∑i≠j∈I1‍εij+∑j≠i,i∈J1,j∈J1∪J2‍εij+∑i∈I1∪J1‍|∇K(ai)|λi∑i∈I1∪J1‍1λiβi+)+o(∑i=1p‍1λiβi).
We need to add the remainder indices i∈J2. Note that u~:=∑j∈J2‍αjδj∈V2(♯J2,ε). Thus using Proposition 10, we apply the associated vector field which we will denote by W2~. We then have the following estimate:
(66)〈∂J(u),W2~(u)〉≤-c(∑j∈J2‍1λjβj+∑i≠j,i,j∈J2‍εij+∑j∈J2‍|∇K(aj)|λj)+O(∑j∈J2,i∈J1‍εij)+o(∑i=1p‍1λiβi),
since |ai-aj|≥ρ for j∈J2 and i∈I1.
Let in this caseW=W1~+m1(W2~+∑i∈J1‍-2iZi+m1∑i∈J1∩L2‍Xi).
From (65) and (66) we obtain
(67)〈∂J(u),W(u)〉≤-c(∑i=1p‍1λiβi+∑i=1p‍|∇K(ai)|λi+∑i≠j‍εij).
Case  2. u1∶=∑i∈I1‍αiδ(aiλi)∈V11(♯I1,ε) and u2:=∑i∉I2‍αiδ(aiλi)∉V21(♯I2,ε):={u=∑j=1♯I2‍αjδ(aj,λj), aj∈B(ylj,ρ), ylj∈𝒦+, for all j=1,…,♯I2 and ρ(yl1,…,y♯I2)>0}.
Sinceu2∈V2(♯I2,ε), by Proposition 10, we can apply the associated vector field which we will denote by V1. We get
(68)〈∂J(u),V1(u)〉≤-c(∑i∈I2‍1λiβi+∑i∈I2‍|∇K(ai)|λi+∑i≠j,i,j∈I2‍εij)+O(∑i∈I2,j∈I1‍εij).
Observe that V1(u) does not increase the maximum of the λi's, i∈I2, since u2∉V21(♯I2,ε). Fix i0∈I2 and let
(69)J1~={i∈I1,s.t,λiβi≥12λi0n-4},J2~=I1∖J1~.
Using (68) and (55), we get
(70)〈∂J(u),V1(u)〉≤-c(∑i∈I2∪J1~‍1λiβi+∑i∈I2‍|∇K(ai)|λi+∑i≠j,i,j∈I2‍εij)+o(∑i=1p‍1λiβi).
We need to add the indices i, i∈J2~. Letting u~:=∑j∈J~2‍αjδ(ajλj), since u~∈V1(♯J~2,ε), we can apply the associated vector field given by Proposition 10. Let V2 be this vector field. By Proposition 11 we have
(71)〈∂J(u),V2(u)〉≤-c(∑j∈J2~‍1λjβj+∑j∈J2~‍|∇K(aj)|λj+∑i≠j,i,j∈J2~‍εij)+O(∑j∈J2~,i∉J2~‍εij).
Observe that I1=J1~∪J2~ and we are in the case where for all i≠j∈I1, we have |ai-aj|≥ρ. Thus by (55), we get
(72)O(∑j∈J2~,i∉J2~‍εij)=o(∑i=1p‍1λiβi),
and hence
(73)〈∂J(u),V1(u)+V2(u)〉≤-c(∑i=1p‍1λiβi+∑i∈I2∪J2~‍|∇K(ai)|λi+∑i≠j‍εij).
Let in this case W=V1+V2+m1∑i∈J1~‍Xi(u), m1 small enough.
Using the previous estimate and Lemma13, we find that
(74)〈∂J(u),W(u)〉≤-c(∑i=1p‍1λiβi+∑i=1p‍|∇K(ai)|λi+∑i≠j‍εij).
Case  3. u1∈V11(♯I1,ε) and u2∈V21(♯I2,ε).
LetV1~(resp., V2~) be the pseudo-gradient in V(p,ε) defined by V1~(u)=W1(u1) (resp., V2~(u)=W2(u2)) where W1 (resp., W2) is the vector field defined by Proposition 11 (resp., Proposition 10) in V11(♯I1,ε) (resp., V21(♯I2,ε)) and let in this case
(75)W=V1~+V2~.
Using Proposition 10, Proposition 11, and (55) we get
(76)〈∂J(u),W(u)〉≤-c(∑i=1p‍1λiβi+∑i=1p‍|∇K(ai)|λi+∑i≠j‍εij).
Notice that in the first and second cases, the maximum of theλi's, 1≤i≤p, is a bounded function and hence the Palais-Smale condition is satisfied along the flow lines of W. However in the third case all the λi's, 1≤i≤p, will increase and go to +∞ along the flow lines generated by W.
Subset 2. We consider the case of u=∑i=1p‍αiδi∈V(p,ε), such that there exist ai satisfying ai∉∪y∈𝒦B(y,ρ).
In this region, the construction of the pseudo-gradientW proceeds exactly as the proof of Theorem 3.2, of subset 2, of [11].
Finally, observe that our pseudogradientW in V(p,ε) satisfies claim (i) of Theorem 9 and it is bounded, since ∥λi(∂δ(ai,λi)/∂λi)∥ and ∥(1/λi)(∂δ(ai,λi)/∂ai)∥ are bounded. From the definition of W, the λi's, 1≤i≤p, decrease along the flow lines of W as long as these flow lines do not enter in the neighborhood of finite number of critical points yli, i=1,…,p, of 𝒦 such that (yl1,…,ylp)∈𝒞∞.
Now, arguing as in Appendix 2 of [16], see also Appendix B of [18], claim (ii) of Theorem 9 follows from (i) and Proposition 5. This completes the proof of Theorem 9.Corollary 14.
Letp≥1. The critical points at infinity of J in V(p,ε) correspond to
(77)(yl1,…,ylp)∞∶=∑i=1p‍1K(yli)(n-4)/2δ(yli,∞),
where (yl1,…,ylp)∈𝒞∞. Moreover, such a critical point at infinity has an index equal to i(yl1,…,ylp)∞=p-1+∑i=1p‍n-i~(y).


## 4. Proof of Theorem1

We prove the existence result by contradiction. Assume thatJ has no critical point in Vη(Σ+). It follows from Corollary 14 that the critical points at infinity of the associated variational problem are in one to one correspondence with the elements of 𝒞∞ defined in (13).Notice that, just like for usual critical points, it is associated with each critical point at infinityw∞ of J stable and unstable manifolds Ws∞(w∞) and Wu∞(w∞) (see [16, pages 356-357]). These manifolds can be easily described once a finite-dimensional reduction like the one we performed in Section 3 is established.For anyw∞=(yi1,…,yip)∈𝒞∞, let c(w)∞=Sn(∑j=1p‍1/K(yij)(n-4)/2)4/n denote the associated critical value. Here we choose to consider a simplified situation, where for any w∞≠w∞′, c(w)∞≠c(w′)∞, and thus order the c(w)∞'s, w∞∈𝒞∞ as
(78)c(w1)∞<⋯<c(wk0)∞.
By using a deformation lemma (see [22, proposition 7.24 and Theorem 8.2]), we know that if c(wk-1)∞<a<c(wk)∞<b<c(wk+1)∞, then
(79)Jb≃Ja∪Wu∞(wk)∞,
where Jb={u∈Vη(Σ+),J(u)≤b} and ≃ denotes retracts by deformation.We apply the Euler-Poincaré characteristic of both sides of (79); we find that
(80)χ(Jb)=χ(Ja)+(-1)i(wk)∞,
where i(wk)∞ denotes the index of the critical point at infinity (wk)∞. Let
(81)b1<c(w1)∞=minu∈Vη(Σ+)J(u)<b2<c(w2)∞<⋯<bk0<c(wk0)∞<bk0+1.
Since we have assumed that (4) has no solution, Jbk0+1 is a retard by deformation of Vη(Σ+). Therefore χ(Jbk0+1)=1, since Vη(Σ+) is a contractible set. Now using (80), we derive after recalling that χ(Jb1)=χ(∅)=0(82)1=∑j=1k0‍(-1)i(wj)∞.
Hence if (82) is violated, J has a critical point in Vη(Σ+).Now, arguing as in the proof of theorem of [18, pages 659-660], we prove that such a critical point is positive.To prove the multiplicity part of the statement, we observe that it follows from Sard-Smale theorem that for genericK's, the solutions to (7) are all nondegenerate, in the sense that the associated linearized operator does not admit zero as eigenvalue. We need to introduce the following lemma extracted from [11].Lemma 15 (see [11, Section 3.2]).
Letw be a solution to (7). Assume that the function K satisfies condition (f)β, with (n-4)/2<β≤n-4; then for each p∈ℕ*, there are neither critical points nor critical points at infinity in V(p,ε,w).Once the existence of mixed critical points at infinity is ruled out, it follows from the previous arguments that(83)Vη(Σ+)≃⋃j=1k0Wu∞(wj)∞∪⋃w,∂J(w)=0Wu(w).
Now using the Euler-Poincaré theorem, we derive that
(84)1=∑j=1k0‍(-1)i(wj)∞+∑w,∂J(w)=0‍(-1)morse(w).
Hence our theorem follows.


## 5. A General Existence Result

In this last section of this paper, we give a generalization of Theorem1. Namely, in view of the result of Theorem 1, one may think about the situation where the degree d given in Theorem 1 is equal to zero; that is, the total sum in d is equal to 1, but a partial one is not equal to 1. A natural question arises: is it possible in this case to use such information to derive an existence result? In the following theorem we give a partial answer to this question.Theorem 16.
LetK be a function satisfying (A1) and (f)β, with 1<β≤n-4. If there exists k∈ℕ such that
(85)(1)∀w∞∈𝒞∞,i(w)∞≠k+1(2)∑w∞∈𝒞∞,i(w∞)≤k‍(-1)i(w∞)≠1,
then the problem (7) has at least one solution.
Moreover for genericK's, if  (n-4)/2<β≤n-4, then the number of solutions is lower bounded by
(86)|1-∑w∞∈𝒞∞,i(w∞)≤k‍(-1)i(w∞)|.Letl♯ be the maximal index over all elements of 𝒞∞. Please observe that the integer k=l♯ satisfies (1) of Theorem 16; it follows that Theorem 1 is a corollary of Theorem 16.Proof of Theorem16.
We set(87)Xk∞=⋃x∞∈𝒞∞,i(x)∞≤k‍Wu∞(x∞)¯,
where Wu∞(x∞)¯ is the closure of the unstable manifolds of x∞. Observe that Xk∞ is a stratified set of top dimension k, which is contractible set in Vη(Σ+), since Vη(Σ+) is a contractible set. Let U denote the image of such a contraction. To prove the first part of Theorem 16, arguing by contradiction, we assume that (7) has no solution. Using the pseudo-gradient constructed in Theorem 9, we can deform U. By transversality arguments, we can assume that such a deformation avoids all critical points at infinity of index greater or equal to k+1. Note that, from assumption (1) of Theorem 16, there is no critical point at infinity with index k+1. It follows then from a theorem of Bahri and Rabinowitz [22] that
(88)U≃Xk∞.
Hence from the Euler-Poincaré theorem, we get
(89)1=χ(Xk∞)=∑w∞∈C∞,i(w∞)≤k‍(-1)i(w∞),
which is a contradiction with assumption (2) of Theorem 16.
Regarding the multiplicity result, we observe that for genericK's the functional J admits only nondegenerate critical points. Hence by Lemma 15, the set U will be deformed into
(90)U≃Xk∞∪⋃w∈CkWu(w),
where Ck denotes the set of the critical points of J of Morse index less than or equal to k, which are dominated by U. Finally by using the Euler-Poincaré theorem, the proof of Theorem 16 follows.



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