In this paper we deal with the scalar curvature problem under minimal boundary mean curvature condition
on the standard 3-dimensional half-sphere. Using tools related to the theory of critical points at infinity, we give existence
results under perturbative and nonperturbative hypothesis, and with the help of some “Morse inequalities at infinity”, we provide multiplicity results for our problem.
1. New Results on Scalar Curvature Problem
In this paper, we revisit a problem having a geometric origin. Namely, let
(1)𝕊+3={x=(x1,x2,x3,x4)∈ℝ4,s.t.|x|=1,x4>0}
be the standard 3-dimensional half-sphere endowed with its standard Riemannian metric g. Given a function K:𝕊+3→ℝ, we consider the problem of finding a metric g~ in the conformal class of g such that ℛg~=K and hg~=0, where ℛg~ is the scalar curvature of 𝕊+3 and hg~is the mean curvature of ∂𝕊+3 with respect to g~. Let g~=u4g be such a metric “conformal” to g; then the above problem amounts to find a smooth positive solution to the following PDE:
(2)-Δgu+34u=18Ku5in𝕊+3,u>0in𝕊+3,∂u∂ν=0on∂𝕊+3,
where ν is the outward normal vector with respect to the metric g. Problems related to (2) were widely studied by various authors [1–12].
Note that, to solve the problem (2), the function K has to be positive somewhere. Moreover, there exist topological obstructions, as Kazdan-Warner obstructions for the scalar curvature problem on 𝕊n (see [13]). Therefore we are led to seek sufficient conditions to set on K, so that the problem (2) has solutions. In addition to the existence problem, we address, in this paper, the question of the number of such metrics g~in the conformal class of g, with prescribed scalar curvature K and zero boundary mean curvature.
In [3, 4, 6, 11], the authors have studied the problem (2). The methods of [6, 11] involve a fine blow-up analysis of some subcritical approximations and the use of topological degree tools. However, the methods of [3, 4] make use of algebraic topological and dynamical tools, coming from the theory of critical points at infinity (see [14]); we also have addressed this problem in [12], using similar tools.
The main contribution of the present work is to generalize certain previous existence results of [3, 12] and to give new existence results to which we add multiplicity results, using tools coming from the theory of critical points at infinity.
In the first part of this paper, we provide existence and multiplicity results under perturbative hypothesis.
In order to state our results, we introduce the following notations and assumptions.
Through the whole of this paper, we assume that K1=K/∂𝕊+3has a finite set of nondegenerate critical points,
(3)𝒦={y∈∂𝕊+3s.t.∇K1(y)=0}={y0,y1,…,yℓ},
ordered such that
(4)K1(y0)⩾K1(y1)⩾⋯⩾K1(yℓ).
We define the following sets:
(5)ℐ+={y∈𝒦s.t.∂K∂ν(y)>0},ℐ+0={y∈𝒦s.t.∂K∂ν(y)=0,-ΔgK(y)>0}.
Let Z be a pseudogradient of K1 of Morse-Smale type; that is, the intersections of unstable and stable manifolds of the critical points of K1, with respect to Z, are transverse. For y∈𝒦, we denote by Wu(y) and Ws(y), respectively, the unstable manifold of y and the stable manifold of y with respect to Z, and we denote by ind(K1,y) the Morse index of K1 at y, that is, the dimension of the submanifold Wu(y).
We introduce the following assumptions:
Wu(yj)∩Ws(yi)=∅for each yj∈𝒦∖(ℐ+∪ℐ+0) and yi∈ℐ+∪ℐ+0.
For each critical point y of K1 such that (∂K/∂ν)(y)=0 we have ΔgK(y)≠0; furthermore, there is a constant r->0 such that -ΔgK(y)(∂K/∂ν)(a)⩾0, for all a∈B(y,r-)∩∂𝕊+3.
Forally∈ℐ+∪ℐ+0, ind(K1,y)⩾1.
We then have the following perturbative result.
Theorem 1.
Under assumptions, (A0), (A1), and (A2), there exists a constant c0>0 independent of the function K such that if
(6)∥K1-1∥L∞(∂𝕊+3)<c0,
and if
(7)∑y∈ℐ+∪ℐ+0(-1)2-ind(K1,y)≠1,
then the problem (2) has at least one solution.
Further, if one assumes that all the solutions of (2) are nondegenerate, then one has
(8)#𝒞⩾|1-∑y∈ℐ+∪ℐ+0(-1)2-ind(K1,y)|,
where #𝒞 is the cardinality of the set of solutions of (2).
Note 1.
We recall here that for a generic function, K, it follows from the Sard-Smale theorem that all the solutions of (2) are nondegenerate. See, for example [15], for a related discussion on this. Thereby, in the assumptions of multiplicity results, one may replace expression “Further, if we assume that all the solutions of (2) are nondegenerate, then we have…” by expression “Further, for a generic function K, we have…”.
We recall also that w∈𝒞 is said to be a nondegenerate solution, if zero is not eigenvalue of the associated linearized operator -Δg+3/4-(1/8)Kw4.
Remark 2.
The existence result of Theorem 1 is slightly different from the one of Theorem 1.3 of [3], since the required assumptions by the two theorems are not all the same. Further, Theorem 1 also gives us a multiplicity result.
We will provide a more general result than Theorem 1. For this, we define, for each index i, 0⩽i⩽ℓ, the set
(9)Xi=⋃0⩽j⩽iyj∈ℐ+∪ℐ+0Ws¯(yj).
Under the assumption (A2), we see that Xi⫋∂𝕊+3=𝕊2; hence Xi is contractible in 𝕊2. Let us denote by θ:[0,1]×Xi→𝕊2 this contraction and let
(10)ci=minK∘θ.
We now introduce the following assumptions:
|K(y0)/ci-1|<c0, where c0 is given in Theorem 1.
Forally∈{yi+1,…,yℓ}∩ℐ+∪ℐ+0, K(y)<ci.
We then have the following existence and multiplicity result.
Theorem 3.
Under the conditions, (A0), (A1), (A2), (A3), and (A4), if
(11)∑yj∈ℐ+∪ℐ+00⩽j⩽i(-1)2-ind(K1,yj)≠1,
then the problem (2) has at least one solution.
Further, if one assumes that all the solutions of (2) are nondegenerate, then one has
(12)#𝒞⩾|1-∑yj∈ℐ+∪ℐ+00⩽j⩽i(-1)2-ind(K1,yj)|.
Remark 4.
Theorem 3 is more general than Theorem 1 for two reasons. First, because, in Theorem 3, K is close to a constant only in a prescribed region of ∂𝕊+3 and not in all ∂𝕊+3. Secondly, the count-index formula of Theorem 3 is more general than the one of Theorem 1, since the formula of Theorem 1 is obtained for i=ℓ from the one of Theorem 3.
In the second part of this work, we will establish nonperturbative results. For this, denoting by d the geodesic distance on 𝕊+3, let G1 be the function defined, for x≠y, by
(13)G1(x,y)=11-cosd(x,y).
And now, denoting by #ℐ+ the cardinality of ℐ+, let us introduce, for any integer p, 1⩽p⩽#ℐ+, the set
(14)ℱp={(yi1,…,yip)∈(ℐ+)p,yij≠yikforj≠k}.
For all p-tuple (yi1,…,yip)∈ℱp we define the matrix M(yi1,…,yip)=(mjk)1⩽p⩽j1⩽k⩽p by
(15)mjj=1K(yij)3/2∂K∂ν(yij),mjk=-42G1(yij,yik)(K(yij)K(yik))1/4nnnnnnnnnnnforj≠k.
Now we formulate the following assumptions:
∀(yij,yik)∈ℱ2; we have mjjmkk<mjk2.
∀ critical point y of K1, we have (∂K/∂ν)(y)⩽0.
We then have the following.
Theorem 5.
Under the conditions, (A0), (A1), and (A5) or (A6), if there exists an index i, 1⩽i⩽ℓ, such that
and if
(17)∑yj∈ℐ+∪ℐ+00⩽j⩽i(-1)2-ind(K1,yj)≠1,
then the problem (2) has at least one solution.
If in addition one assumes that all the solutions of (2) are nondegenerate, then
(18)#𝒞⩾|1-∑yj∈ℐ+∪ℐ+00⩽j⩽i(-1)2-ind(K1,yj)|.
As a corollary of Theorem 5, one has the following.
Corollary 6.
Under the conditions, (A0), (A1), and (A5) or (A6), if
(19)∑y∈ℐ+∪ℐ+0(-1)2-ind(K1,y)≠1,
then the problem (2) has at least one solution.
Furthermore, if we assume that all the solutions of (2) are nondegenerate, then
(20)#𝒞⩾|1-∑y∈ℐ+∪ℐ+0(-1)2-ind(K1,y)|.
Remark 7.
The result of Corollary 6 can be recovered by Theorem 1.1 and Corollary 1.2 in [3] and by Theorem 1.2 and Corollary 1 in [4], and it completes the existence result of Corollary 1.1 in [12] by a multiplicity result.
Commentary. We point out that the main new contribution of Theorem 5 (as well as that of Theorem 3) is that we address here the case where the total sum
(21)∑y∈ℐ+∪ℐ+0(-1)2-ind(K1,y)
is equal to 1, but a partial sum
(22)∑yj∈ℐ+∪ℐ+00⩽j⩽i(-1)2-ind(K1,yj)
is not equal to 1. The main issue is to take advantage of such information to prove the existence of solutions to the problem (2). Notice that an interpretation of the fact that the total sum is different from one is that the topological contribution of the critical points at infinity to the topology of the level sets of the associated Euler-Lagrange functional is not trivial.
In view of such an interpretation, we raise the following question: what happens if the total contribution is trivial, but a subset of critical points at infinity induce a nontrivial difference of topology; can we still use such a topological information to prove existence of solutions?
With respect to the above question, Theorem 5 (and also Theorem 3) gives a sufficient condition to be able to derive from such local information the existence of solutions for the problem (2).
As pointed out above, our result does not only give existence results but also, under generic conditions, gives a lower bound on the number of solutions of (2). Such a result is reminiscent to the celebrated Morse Theorem, which states that the number of critical points of a Morse function defined on a compact manifold is lower-bounded in terms of the topology of the underlying manifold. Our result can be seen as a kind of Morse Inequality at Infinity. Indeed, it gives a lower bound on the number of metrics with prescribed scalar curvature and zero boundary mean curvature, in terms of the topology at infinity.
In what follows we show a situation where Corollary 6 does not work, while Theorem 5 allows having solutions to problem (2).
Example.
Let K1:𝕊2→ℝ be such that
ℐ+0={y0,y1,y2,y3},
ℐ+={y4},
ind(K1,y0)=ind(K1,y3)=ind(K1,y4)=2,
and
ind(K1,y1)=ind(K1,y2)=1.
It is easy to compute that
(23)∑y∈ℐ+∪ℐ+0(-1)2-ind(K1,y)=1,
and then Corollary 6 does not work. But if we have
(24)1K(y3)1/2>1K(y0)1/2+1K(y2)1/2,
using Theorem 5 with i=2, then we have
(25)|1-∑yj∈ℐ+∪ℐ+00⩽j⩽2(-1)2-ind(K1,yj)|=2≠1,
and thus (2) has at least a solution. If solutions of (2) are assumed to be nondegenerate, we derive the existence of at least two solutions.
The rest of our paper is organized as follows. In Section 2, we set the general framework and recall some basic known facts. Section 3 is devoted to the proofs of Theorem 1 and Theorem 3. Finally, we prove Theorem 5 and Corollary 6 in Section 4.
2. Known Facts about Scalar Curvature Problem
In this Section we recall the variational formulation of the problem (2), as well as some previous useful results. We introduce on H1(𝕊+3,ℝ) the norm
(26)∥u∥2=∫𝕊+3(-Δgu+34u)udvg
associated with the Yamabe operator -Δg+3/4, where dvg is the volume element of g on 𝕊+3. Now we define ∑={u∈H1,∥u∥=1} to be the unit sphere of H1(𝕊+3,ℝ) and ∑+={u∈∑,u≥0}. The Euler functional on H1(𝕊+3,ℝ) associated with the problem (2) is
(27)J(u)=∥u∥2(∫𝕊+3Ku6dvg)1/3.
The problem (2) then amounts to find a critical point of J under the constraint u∈∑+. The difficulty in this problem comes from the fact that the functional J fails to satisfy the Palais-Smale condition on ∑+. This failure was studied by various authors; see for example [16–18]. To characterize the sequences violating the Palais-Smale condition on ∑+, we need to fix some notations. For (a,λ)∈𝕊+3¯×(0,+∞) let
(28)δ(a,λ)(x)=σλλ2+1-(λ2-1)cosd(a,x),
where d is the geodesic distance on (𝕊+3,g) (δ(a,λ)is known to be the solution of the Yamabe problem on the Sphere 𝕊3) and σ is chosen so that
(29)-Δgδ(a,λ)+34δ(a,λ)=δ(a,λ)5
is satisfied on 𝕊+3. Observe that, if a∈∂𝕊+3, we have ∂δ(a,λ)/∂ν=0 on ∂𝕊+3. However, ∂δ(a,λ)/∂ν≠0 if a∉∂𝕊+3; then in this case we need to introduce another function φ(a,λ) which satisfies
(30)-Δgφ(a,λ)+34φ(a,λ)=δ(a,λ)5in𝕊+3,∂φ(a,λ)∂ν=0on∂𝕊+3.
We will write in the sequel δi for δ(ai,λi) and φi for φ(ai,λi).
Let w be a nondegenerate solution of (2) or zero. Then, for ϵ>0 and integers p,q such that 1⩽q+p, we define
(31)V(p,q,ϵ,w)={∥u-α0w-∑i=1pαiδi-∑i=p+1p+qαiφi∥u∈∑+/∃a1,…,ap+q∈𝕊+3¯,nnnnnnn∃λ1,…,λp+q>1ϵ,nnnnnnn∃α0,…,αp+q>0,nnnnnnns.t.|α04J(u)3-1|<ϵ,nnnnnnn∥u-α0w-∑i=1pαiδi-∑i=p+1p+qαiφi∥<ϵ,nnnnnnnεij<ϵ,|αi4K(ai)αj4K(aj)-1|<ϵnnnnnnnfor1⩽i≠j⩽p+q,nnnnnnnλidi<ϵfor1⩽i⩽p,nnnnnnnλidi>1ϵforp+1⩽i⩽p+q∑i=1pαiδi∑+∑i=p+1p+q∥},
where di=d(ai,∂𝕊+3)and εij=(λi/λj+λj/λi+λiλj(1-cosd(ai,aj)))-1/2. Note that, when w=0, to write the definition of V(p,q,ϵ,0), we replace w by 0 and we remove α0 and the condition |α04J(u)3-1|<ϵ in the definition of V(p,q,ϵ,w) above.
The failure of the Palais-Smale condition can be described (see Proposition 1 in [4]), as follows.
Proposition 8.
Let (uk) be a sequence in ∑+ such that J(uk) is bounded and ∂J(uk)→0. Then, there exist integers p,q such that 1⩽q+p, a sequence ϵk>0(ϵk→0), and an extracted subsequence of (uk), again denoted by (uk), such that ∀k∈ℕ,uk∈V(p,q,ϵk,w), where w is either solution of (2) or zero.
Here ∂J is the gradient of J with respect of the H1-inner product:
(32)(u,v)=∫𝕊+3(-Δgu+34u)vdvg.
We consider the following minimization problem for a function u belonging to V(p,q,ϵ,w), with ϵ small:
(33)min∥u-α0(w+h)-∑i=1pαiδi-∑i=p+1p+qαiφi∥
for
{α0,αi,λi>0,1⩽i⩽p+qai∈∂𝕊+3,1⩽i⩽pai∈𝕊+3,p+1⩽i⩽p+qh∈Tw(Wu(w)),
where Tw(Wu(w)) is the tangent space at w to Wu(w) the unstable manifold of w for a pseudogradient of J. We then have the next proposition which defines a parameterization of the set V(p,q,ϵ,w). It follows from the corresponding statement in [19] (see also Proposition 4 in [4]).
Proposition 9.
For any integers p,q, such that 1⩽q+p, there exists ϵ(p,q)>0 such that if ϵ<ϵ(p,q) and u∈V(p,q,ϵ,w), the minimization problem (33) has a unique solution (up to permutation).
Thus, we can write any u∈V(p,q,ϵ,w)uniquely as follows:
(34)u=∑i=1pα-iδ(a-i,λ-i)+∑i=p+1p+qα-iφ(a-i,λ-i)+v+α-0(w+h-),
where (α-0,α-1,…,α-p+q,a-1,…,a-p+q,λ-1,…,λ-p+q,h-) is the solution of the minimization problem (33) and v∈H1(𝕊+3)∩Tw(Ws(w)) is as follows:
(35)(v,ψ)=0forψ∈{δi,∂δi∂ai,∂δi∂λi}i=1i=p⋃{φi,∂φi∂ai,∂φi∂λi}i=p+1i=p+q(v,w)=0,(v,h)=0∀h∈Tw(Wu(w)),∥v∥<ϵ,
where δi=δ(ai,λi), φi=φ(ai,λi), and Tw(Ws(w)) is the tangent space at w to Ws(w) the stable manifold of w for a pseudogradient of J.
We also consider the case, where w=0; then the condition (35) becomes
(36)(v,ψ)=0forψ∈{δi,∂δi∂ai,∂δi∂λi}i=1i=p⋃{φi,∂φi∂ai,∂φi∂λi}i=p+1i=p+q,JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ∥v∥<ϵ.
From similar statements in [14, 20, 21] (see also Proposition 2.2 in [3] or Proposition 3 in [4]), we have then the following.
Proposition 10.
There exists a 𝒞1 map which, to each
(37)(α,a,λ)=(α1,…,αp+q,a1,…,ap+q,λ1,…,λp+q)
such that ∑i=1pαiδi+∑i=p+1p+qαiφi+v∈V(p,q,ϵ,0), with ϵ small, associates v-=v-(α,a,λ), satisfying
(38)J(∑i=1pαiδi+∑i=p+1p+qαiφi+v-)=minvverifying(36)J(∑i=1pαiδi+∑i=p+1p+qαiφi+v).
Moreover, there exists c>0 such that
(39)∥v-∥⩽c(∑i=1p|∇K(ai)|λi+1λi2+∑j=p+1p+q1λj2dj2nnnnnnnn+∑k≠lεkl(lnεkl-1)1/3∑i=1p).
The next proposition characterizes the critical points at infinity of the associated variational problem. We recall that critical points at infinity are the orbits of the gradient flow of J which remain in V(p,q,ϵ(s),w), where ϵ(s) is a function tending to 0 when s→+∞ (see [14]).
Proposition 11.
Assume that, for any (yi1,…,yip)∈ℱp, p⩾1, the matrix M(yi1,…,yip) is nondegenerate. Under the assumption, (A1), the critical points at infinity of J are in V(p,0,ϵ,0). More precisely consider the following.
If p=1, they correspond to single bubbles δ(y,+∞),y∈(ℐ+0∪ℐ+). The Morse index of J at its critical point at infinity δ(y,+∞) is
(40)ind(J,δ(y,+∞))=2-ind(K1,y),
where ind(K1,y) is the Morse index of K1 at its critical point y.
If p⩾2, they correspond to combinations ∑j=1p(1/K(yij)4)δ(yij,+∞), where (yi1,…,yip)∈ℱp is such that ρ(yi1,…yip)>0, where ρ(yi1,…yip) is the smallest eigenvalue of the matrix M(yi1,…,yip).
Proof.
Using Corollary 3 of [4], there are no critical points at infinity for J in V(p,q,ϵ,w) with w≠0. Hence they are in V(p,q,ϵ,0). Using Corollary 5 of [4], there are no critical points at infinity for J in V(0,q,ϵ,w) with q≠0; that is, there are no bubbles (or blow-ups) with all concentration points interior to 𝕊+3. Then, using Proposition 3.1 of [3], we rule out the existence of critical points at infinity in V(p,q,ϵ,w) with pq≠0 (the so-called mixed bubbles), which signifies that bubbles concentrate uniquely in points of the boundary ∂𝕊+3; that is to say, critical points at infinity are in V(p,0,ϵ,0), p⩾1. Finally, using Corollary 1.1 of [4], we derive the result.
3. Proofs of Theorems 1 and 3
We start the proofs by recalling the following results.
Lemma 12.
For u=αδ(a,λ)∈V(1,0,ϵ,0) such that a∈B(y) a small enough neighborhood of y∈ℐ+0∪ℐ+ a critical point of K1, there is a change of variables (a~,λ~)such that if y∈ℐ+,
(41)J(u)=ψ1(a~,λ~)=(S32)2/31K(a~)1/3(1+cλ~∂K∂ν(y)+o(1λ~)),
and if y∈ℐ+0, after another change of variable λ~↦λ~~,
(42)J(u)=ψ2(a~,λ~~)=(S32)2/31K(a~)1/3(1-c~ΔgK(y)λ~~2+o(1λ~~2)),
where c and c~ are positive constants and S3 is the Sobolev constant for 𝕊3. (We recall that λ,λ~ and λ~~ are of the same order of size).
Lemma 12 is easily deduced from Proposition 2.8 in [3].
Lemma 13 (see [3], Lemma 4.1).
Let a1,a2∈∂𝕊+3, and α1,α2>0. For λ large enough positive parameter, let u=α1δ(a1,λ)+α2δ(a2,λ). Then
(43)J(u)⩽(S32)2/3(1K(a1)1/2+1K(a2)1/2)2/3(1+o(1))=c∞(a1,a2)(1+o(1)).
Lemma 14.
For ϵ>0 small enough and u=(∑i=1pαiδ(ai,λi)+v)∈V(p,0,ϵ,0), one has the expansion
(44)J(u)=(S32)2/3∑i=1pαi2(∑i=1pαi6K(ai))1/3(1+o(1)).
Lemma 14 is a particular case (for n=1) of Proposition 2.4 of [3].
Proof of Theorem 1.
Let
(45)c1=32(S32)2/3.
Using the expansion of J provided by Lemma 14, there exists c0>0 independent of K1 such that if ∥K1-1∥L∞(∂𝕊+3)<c0, then for allu∈V(1,0,ϵ,0), J(u)<c1 and ∀u∈V(p,0,ϵ,0) with p⩾2, J(u)>c1. Using Proposition 11, the only critical points at infinity of J under the level c1 correspond to single bubblesδ(y,+∞), wherey∈(ℐ+0∪ℐ+). We let in the sequelδ(y,+∞)=y∞. Now define
(46)X=⋃y∈(ℐ+0∪ℐ+)Ws¯(y),
where Ws(y) is the stable manifold of y for a decreasing pseudogradient Z of the function K1. By hypothesis (A2), we have X⫋∂𝕊+3=𝕊2. Thus, X is contractible in 𝕊2. We denote by h:[0,1]×X→𝕊2 the associated contraction. Let
(47)X∞=⋃y∈(ℐ+0∪ℐ+)Wu¯(y∞),
where Wu(y∞) is the unstable manifolds at infinity, for the pseudogradient V provided by Proposition 3.1 in [3] (or the pseudogradient W provided by Proposition 9 in [4]), of the critical point at infinity y∞. It can be described, using the expansion of J(u) given by Lemma 12, as the product of Ws(y), for the pseudogradient Z of K1, by [A,+∞) which is the domain of the variable λ, for some large enough positive real A.
The contraction h gives rise to a contraction
(48)h~:[0,1]×X×[A,+∞)⟶Σ+(t,x,λ)⟼δ(h(t,x),λ)∥δ(h(t,x),λ)∥.
Using Lemma 12, ∀(t,x,λ)∈[0,1]×X×[A,+∞), one has
(49)J(h~(t,x,λ))=(S32)2/31K(h(t,x))1/3(1+O(1A)).
Since K is close to 1, we derive that J(h~(t,x,λ))<c1, for A large enough. Denoting by R(h~) the range of h~, that is, R(h~):=h~([0,1]×X∞), then we derive that R(h~) is below the level c1; that is to say, the contraction h~ is performed under the level c1 and thus (see [22], Sections 7 and 8)
(50)R(h~)≃⋃ℐ+0∪ℐ+Wu(y∞)⋃w∈𝒞R(h~)∩Ws(w)≠∅Wu(w),
where 𝒞 is the set of critical points of J or, equivalently, of the solutions of (2). Now, denoting by χ the Euler-Poincaré characteristic, and using the fact that R(h~) is a contractible set, it holds that
(51)1=χ(R(h~))=∑y∈ℐ+0∪ℐ+(-1)ind(J,y∞)lllllllllllllll+∑w∈𝒞R(h~)∩Ws(w)≠∅(-1)ind(J,w),
where ind(J,w) (resp., ind(J,y∞)) denotes the Morse index of J at w (resp., y∞). Since ind(J,y∞)=2-ind(K1,y) and since ∑y∈ℐ+0∪ℐ+(-1)2-ind(K1,y)≠1, we derive that 𝒞≠∅. Hence
(52)|1-∑y∈ℐ+0∪ℐ+(-1)2-ind(K1,y)|⩽|∑w∈𝒞R(h~)∩Ws(w)≠∅(-1)ind(J,w)|⩽#𝒞,
where #𝒞 is the cardinality of 𝒞 q.e.d.
Proof of Theorem 3.
Using the same arguments of the proof of Theorem 1, let
(53)Xi∞=⋃0⩽j⩽iyj∈ℐ+∪ℐ+0Ws¯(yj∞),
where Xi is defined by (9). As we remarked it above, Xi∞ can be parameterized by Xi×[A,+∞). Let
(54)ci∞=(S32)2/31ci1/3,
where ci=minK∘θ and θ is the contraction
(55)θ:[0,1]×Xi⟶𝕊2,(t,x)⟼θ(t,x)
that is, for all x∈Xi, θ(0,x)=x and θ(1,x)=a0 a fixed point in Xi.
We now claim that Xi∞ is contractible in Jci∞+ϵ. Indeed, the contraction θ induces the following contraction:
(56)θ~:[0,1]×Xi×[A,+∞)⟶Σ+(t,x,λ)⟼δ(θ(t,x),λ)∥δ(θ(t,x),λ)∥.
Thus, by Lemma 12, we derive, for λ large enough,
(57)J(δ(θ(t,x),λ))=(S32)2/31K(θ(t,x))1/3(1+O(1λ)).
Since ∀t∈[0,1] and ∀x∈Xi, we have K(θ(t,x))⩾ci; then, for λ⩾A large enough,
(58)J(δ(θ(t,x),λ))⩽(S32)2/31ci1/3(1+O(1A))⩽ci∞+ϵ,
and thus Xi∞ is contractible in Jci∞+ϵ. Our claim follows.
On the other side, using Proposition 11, under the assumptions, (A3) and (A4), the critical points at infinity under the level ci∞+ϵ are yj∞, where yj∈(ℐ+0∪ℐ+) and 0⩽j⩽i. Thus, denoting by R(θ~) the range of θ~, that is, R(θ~):=θ~([0,1]×Xi∞), we derive that
(59)R(θ~)≃⋃0⩽j⩽iyj∈ℐ+0∪ℐ+Wu(yj∞)⋃w∈𝒞R(θ~)∩Ws(w)≠∅Wu(w),
where 𝒞 is the set of the critical points of J. Our claim thus implies that
(60)1=χ(R(θ~))=∑0⩽j⩽iyj∈ℐ+0∪ℐ+(-1)ind(J,yj∞)jjjjjjjjjjjjjjjjjjkk+∑w∈𝒞R(θ~)∩Ws(w)≠∅(-1)ind(J,w).
Hence, arguing as above, we derive that 𝒞≠∅, and that
(61)|1-∑0⩽j⩽iyj∈ℐ+0∪ℐ+(-1)2-ind(K1,yj)|⩽|∑w∈𝒞R(θ~)∩Ws(w)≠∅(-1)ind(J,w)|⩽#𝒞.
Our theorem is thereby proved.
4. Proofs of Theorem 5 and Corollary 6 Proof of Theorem 5.
Let
(62)c∞(y0,yi)=(S32)2/3(1K(y0)1/2+1K(yi)1/2)2/3.
Using Proposition 11, we observe that under the assumptions, (A1) and (A5) or (A6), there are no critical points at infinity of multiple masses, that is, in V(p,0,ϵ,0) for p⩾2. Indeed, on one hand, under (A5), we derive that, for any p⩾2and for any (yi1,…,yip)∈ℱp, the matrix M(yi1,…,yip) is definite negative, and then its smallest eigenvalue ρ(yi1,…,yip) is <0. On the other hand, under (A6), ℐ+=∅ and therefore ℱp=∅ for all p.
Thus the critical points at infinity of our variational problem lie in V(1,0,ϵ,0). Using the assumption (Hi) of Theorem 5, it follows that the only critical points at infinity of J under the level
(63)ci=c∞(y0,yi)+ϵ
for ϵ small enough are δ(yj,+∞)=yj∞, where yj∈(ℐ+0∪ℐ+), for 0⩽j⩽i.
The unstable manifolds at infinity, for the pseudogradient V (provided by Proposition 3.1 in [3]), of such critical points at infinity, which we denote by Wu(yj∞), can be described as the product of Ws(yj), for the pseudogradient Z of K1, by [A,+∞), domain of the variable λ, for some large enough positive real A. Let
(64)Xi∞=⋃0⩽j⩽iyj∈(ℐ+0∪ℐ+)Wu¯(yj∞).
The set Xi∞ can be parameterized by Xi×[A,+∞), where Xi is given by (9). We now claim that Xi∞ is contractible in Jci, where
(65)Jci={u∈Σ+s.t.J(u)⩽ci}.
Indeed, let a1,a2∈∂𝕊+3, α1,α2>0, and λ large enough. For u=α1δ(a1,λ)+α2δ(a2,λ), we have, by Lemma 13, the following estimate:
(66)J(u)⩽(S32)2/3(1K(a1)1/2+1K(a2)1/2)2/3(1+o(1))=c∞(a1,a2)+o(1).
Let
(67)ψ:[0,1]×Xi×[A,+∞)⟶Σ+(t,x,λ)⟼tδ(y0,λ)+(1-t)δ(x,λ)∥tδ(y0,λ)+(1-t)δ(x,λ)∥.
The function ψ is continuous, and it satisfies ψ(0,x,λ)=δ(x,λ)/∥δ(x,λ)∥, ψ(1,x,λ)=δ(y0,λ)/∥δ(y0,λ)∥. Now, since K1(x)⩾K1(yi), for any x∈Xi, it follows from inequality (66) that J(ψ(t,x,λ))<ci for all (t,x,λ)∈[0,1]×Xi×[A,+∞). Thus, the contraction ψ is performed under the level ci. We derive that Xi∞ is contractible in Jci, and our claim follows. On the other hand, denoting by R(ψ) the range of ψ, that is, R(ψ):=ψ([0,1]×Xi∞), we have
(68)R(ψ)≃⋃0⩽j⩽iyj∈ℐ+0∪ℐ+Wu(yj∞)⋃w∈𝒞R(ψ)∩Ws(w)≠∅Wu(w),
where 𝒞 is the set of the critical points of J, and since R(ψ) is contractible, it yields
(69)1=χ(R(ψ))=∑0⩽j⩽iyj∈ℐ+0∪ℐ+(-1)ind(J,yj∞)jjjjjjjjjjjjjjjjjkk+∑w∈𝒞R(ψ)∩Ws(w)≠∅(-1)ind(J,w).
Hence, we derive as above that 𝒞≠∅ and that
(70)|1-∑0⩽j⩽iyj∈ℐ+0∪ℐ+(-1)2-ind(K1,yj)|⩽|∑w∈𝒞R(ψ)∩Ws(w)≠∅(-1)ind(J,w)|⩽#𝒞,
where #𝒞 is the number of solutions of (2). Theorem 5 follows.
Proof of Corollary 6.
Recall that K1 has only nondegenerate critical points y0,y1,…,yℓ, such that K1(y0)⩾⋯⩾K1(yℓ). Observe that the assumption (Hi) of Theorem 5 serves uniquely to exclude points yj of ℐ+0∪ℐ+ such that i+1⩽j⩽ℓ, from the count-index formula. So, for the index i=ℓ, the assumption (Hℓ) vanishes since all the critical points of ℐ+0∪ℐ+ are here taken into consideration. Thus, our Corollary follows from Theorem 5, taking i=ℓ.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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