Existence and Multiplicity Results for the Scalar Curvature Problem on the Half-Sphere
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 be the standard -dimensional half-sphere endowed with its standard Riemannian metric . Given a function , we consider the problem of finding a metric in the conformal class of such that and , where is the scalar curvature of and is the mean curvature of with respect to . Let be such a metric “conformal” to ; then the above problem amounts to find a smooth positive solution to the following PDE: where is the outward normal vector with respect to the metric . Problems related to (2) were widely studied by various authors [1–12].
Note that, to solve the problem (2), the function has to be positive somewhere. Moreover, there exist topological obstructions, as Kazdan-Warner obstructions for the scalar curvature problem on (see ). Therefore we are led to seek sufficient conditions to set on , 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 in the conformal class of , with prescribed scalar curvature 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 ); we also have addressed this problem in , 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 has a finite set of nondegenerate critical points, ordered such that We define the following sets:
Let be a pseudogradient of of Morse-Smale type; that is, the intersections of unstable and stable manifolds of the critical points of , with respect to , are transverse. For , we denote by and , respectively, the unstable manifold of and the stable manifold of with respect to , and we denote by the Morse index of at , that is, the dimension of the submanifold .
We introduce the following assumptions:for each and . For each critical point of such that we have ; furthermore, there is a constant such that , for all ., .
We then have the following perturbative result.
Theorem 1. Under assumptions, , , and , there exists a constant independent of the function such that if
then the problem (2) has at least one solution.
Further, if one assumes that all the solutions of (2) are nondegenerate, then one has where is the cardinality of the set of solutions of (2).
Note 1. We recall here that for a generic function, , it follows from the Sard-Smale theorem that all the solutions of (2) are nondegenerate. See, for example , 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 , we have…”.
We recall also that is said to be a nondegenerate solution, if zero is not eigenvalue of the associated linearized operator .
Remark 2. The existence result of Theorem 1 is slightly different from the one of Theorem 1.3 of , 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 , , the set Under the assumption , we see that ; hence is contractible in . Let us denote by this contraction and let We now introduce the following assumptions: , where is given in Theorem 1. , .
We then have the following existence and multiplicity result.
Remark 4. Theorem 3 is more general than Theorem 1 for two reasons. First, because, in Theorem 3, is close to a constant only in a prescribed region of and not in all . 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 from the one of Theorem 3.
In the second part of this work, we will establish nonperturbative results. For this, denoting by the geodesic distance on , let be the function defined, for , by And now, denoting by the cardinality of , let us introduce, for any integer , , the set For all -tuple we define the matrix by Now we formulate the following assumptions: ; we have . critical point of , we have . We then have the following.
Theorem 5. Under the conditions, , , and or , if there exists an index , , such that ()
then the problem (2) has at least one solution.
If in addition one assumes that all the solutions of (2) are nondegenerate, then
As a corollary of Theorem 5, one has the following.
Remark 7. The result of Corollary 6 can be recovered by Theorem 1.1 and Corollary 1.2 in  and by Theorem 1.2 and Corollary 1 in , and it completes the existence result of Corollary 1.1 in  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 is equal to 1, but a partial sum 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?
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.
Example. Let be such that and It is easy to compute that
and then Corollary 6 does not work. But if we have
using Theorem 5 with , then we have
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 the norm associated with the Yamabe operator , where is the volume element of on . Now we define to be the unit sphere of and . The Euler functional on associated with the problem (2) is The problem (2) then amounts to find a critical point of under the constraint . The difficulty in this problem comes from the fact that the functional 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 let where is the geodesic distance on (is known to be the solution of the Yamabe problem on the Sphere ) and is chosen so that is satisfied on . Observe that, if , we have on . However, if ; then in this case we need to introduce another function which satisfies We will write in the sequel for and for .
Let be a nondegenerate solution of (2) or zero. Then, for and integers such that , we define where and . Note that, when , to write the definition of , we replace by and we remove and the condition in the definition of above.
The failure of the Palais-Smale condition can be described (see Proposition 1 in ), as follows.
Proposition 8. Let be a sequence in such that is bounded and . Then, there exist integers such that , a sequence , and an extracted subsequence of , again denoted by , such that , where is either solution of (2) or zero.
Here is the gradient of with respect of the -inner product: We consider the following minimization problem for a function belonging to , with small: where is the tangent space at to the unstable manifold of for a pseudogradient of . We then have the next proposition which defines a parameterization of the set . It follows from the corresponding statement in  (see also Proposition 4 in ).
Proposition 9. For any integers , such that , there exists such that if and , the minimization problem (33) has a unique solution (up to permutation).
Thus, we can write any uniquely as follows: where is the solution of the minimization problem (33) and is as follows: where , , and is the tangent space at to the stable manifold of for a pseudogradient of .
Proposition 10. There exists a map which, to each such that , with small, associates , satisfying Moreover, there exists such that
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 which remain in , where is a function tending to when (see ).
Proposition 11. Assume that, for any , , the matrix is nondegenerate. Under the assumption, , the critical points at infinity of are in . More precisely consider the following. (1)If , they correspond to single bubbles . The Morse index of at its critical point at infinity is where is the Morse index of at its critical point .(2)If , they correspond to combinations , where is such that , where is the smallest eigenvalue of the matrix .
Proof. Using Corollary 3 of , there are no critical points at infinity for in with . Hence they are in . Using Corollary 5 of , there are no critical points at infinity for in with ; that is, there are no bubbles (or blow-ups) with all concentration points interior to . Then, using Proposition 3.1 of , we rule out the existence of critical points at infinity in with (the so-called mixed bubbles), which signifies that bubbles concentrate uniquely in points of the boundary ; that is to say, critical points at infinity are in , . Finally, using Corollary 1.1 of , we derive the result.
We start the proofs by recalling the following results.
Lemma 12. For such that a small enough neighborhood of a critical point of , there is a change of variables such that if , and if , after another change of variable , where and are positive constants and is the Sobolev constant for . (We recall that and are of the same order of size).
Lemma 13 (see , Lemma 4.1). Let , and . For large enough positive parameter, let . Then
Lemma 14. For small enough and , one has the expansion
Proof of Theorem 1. Let
Using the expansion of provided by Lemma 14, there exists independent of such that if , then for all, and with , . Using Proposition 11, the only critical points at infinity of under the level correspond to single bubbles, where. We let in the sequel. Now define
where is the stable manifold of for a decreasing pseudogradient of the function . By hypothesis , we have . Thus, is contractible in . We denote by the associated contraction. Let
where is the unstable manifolds at infinity, for the pseudogradient provided by Proposition 3.1 in  (or the pseudogradient provided by Proposition 9 in ), of the critical point at infinity . It can be described, using the expansion of given by Lemma 12, as the product of , for the pseudogradient of , by which is the domain of the variable , for some large enough positive real .
The contraction gives rise to a contraction Using Lemma 12, , one has Since is close to 1, we derive that , for large enough. Denoting by the range of , that is, , then we derive that is below the level ; that is to say, the contraction is performed under the level and thus (see , Sections 7 and 8) where is the set of critical points of or, equivalently, of the solutions of (2). Now, denoting by the Euler-Poincaré characteristic, and using the fact that is a contractible set, it holds that where (resp., ) denotes the Morse index of at (resp., ). Since and since , we derive that . Hence where is the cardinality of q.e.d.
Proof of Theorem 3. Using the same arguments of the proof of Theorem 1, let
where is defined by (9). As we remarked it above, can be parameterized by . Let
where and is the contraction
that is, for all , and a fixed point in .
We now claim that is contractible in . Indeed, the contraction induces the following contraction: Thus, by Lemma 12, we derive, for large enough, Since and , we have ; then, for large enough, and thus is contractible in . Our claim follows.
On the other side, using Proposition 11, under the assumptions, and , the critical points at infinity under the level are , where and . Thus, denoting by the range of , that is, , we derive that where is the set of the critical points of . Our claim thus implies that Hence, arguing as above, we derive that , and that Our theorem is thereby proved.
Proof of Theorem 5. Let
Using Proposition 11, we observe that under the assumptions, and or , there are no critical points at infinity of multiple masses, that is, in for . Indeed, on one hand, under , we derive that, for any and for any , the matrix is definite negative, and then its smallest eigenvalue is . On the other hand, under , and therefore for all .
Thus the critical points at infinity of our variational problem lie in . Using the assumption of Theorem 5, it follows that the only critical points at infinity of under the level for small enough are , where , for .
The unstable manifolds at infinity, for the pseudogradient (provided by Proposition 3.1 in ), of such critical points at infinity, which we denote by , can be described as the product of , for the pseudogradient of , by , domain of the variable , for some large enough positive real . Let The set can be parameterized by , where is given by (9). We now claim that is contractible in , where Indeed, let , , and large enough. For , we have, by Lemma 13, the following estimate: Let The function is continuous, and it satisfies , . Now, since , for any , it follows from inequality (66) that for all . Thus, the contraction is performed under the level . We derive that is contractible in , and our claim follows. On the other hand, denoting by the range of , that is, , we have where is the set of the critical points of , and since is contractible, it yields Hence, we derive as above that and that where is the number of solutions of (2). Theorem 5 follows.
Proof of Corollary 6. Recall that has only nondegenerate critical points , such that . Observe that the assumption of Theorem 5 serves uniquely to exclude points of such that , from the count-index formula. So, for the index , the assumption vanishes since all the critical points of are here taken into consideration. Thus, our Corollary follows from Theorem 5, taking .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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