Yamabe systems, optimal partitions, and nodal solutions to the Yamabe equation

We give conditions for the existence of regular optimal partitions, with an arbitrary number $\ell\geq 2$ of components, for the Yamabe equation on a closed Riemannian manifold $(M,g)$. To this aim, we study a weakly coupled competitive elliptic system of $\ell$ equations, related to the Yamabe equation. We show that this system has a least energy solution with nontrivial components if $\dim M\geq 10$, $(M,g)$ is not locally conformally flat and satisfies an additional geometric assumption whenever $\dim M=10$. Moreover, we show that the limit profiles of the components of the solution separate spatially as the competition parameter goes to $-\infty$, giving rise to an optimal partition. We show that this partition exhausts the whole manifold, and we prove the regularity of both the interfaces and the limit profiles, together with a free boundary condition. For $\ell=2$ the optimal partition obtained yields a least energy sign-changing solution to the Yamabe equation with precisely two nodal domains.


Introduction and statement of results
Consider the Yamabe equation where (M, g) is a closed Riemannian manifold of dimension m ≥ 3, S g is its scalar curvature, ∆ g := div g ∇ g is the Laplace-Beltrami operator, κ m := m−2 4(m−1) , and 2 * := 2m m−2 is the critical Sobolev exponent.We assume that the quadratic form induced by the conformal Laplacian L g is coercive.
If Ω is an open subset of M , we consider the Dirichlet problem Let H 1 g (M ) be the Sobolev space of square integrable functions on M having square integrable first weak derivatives, and let H 1 g,0 (Ω) be the closure of C ∞ c (Ω) in H 1 g (M ).The (weak) solutions of (1.2) are the critical points of the C 2 -functional J Ω : H 1 g,0 (Ω) → R given by The nontrivial ones belong to the Nehari manifold N Ω := {u ∈ H 1 g,0 (Ω) : u = 0 and J ′ Ω (u)u = 0}, which is a natural constraint for J Ω .So, a minimizer for J Ω over N Ω is a nontrivial solution of (1.2), called a least energy solution.Such a solution does not always exist.If Ω is the whole manifold M , it provides a solution to the celebrated Yamabe problem.In this case its existence was established thanks to the combined efforts of Yamabe [61], Trudinger [59], Aubin [3] and Schoen [44].A detailed account is given in [39].Set c Ω := inf u∈NΩ J Ω (u).
In this paper, given ℓ ≥ 2, we consider the optimal ℓ-partition problem inf where 3) is an ℓ-tuple {Ω 1 , . . ., Ω ℓ } ∈ P ℓ such that c Ωi is attained for every i = 1, . . ., ℓ, and We call it an optimal ℓ-partition for the Yamabe equation on (M, g).Optimal partitions do not always exist.In fact, there is no optimal ℓ-partition for the Yamabe equation on the standard sphere S m for any ℓ ≥ 2. This is because c Ω is not attained in any open subset Ω of S m whose complement has nonempty interior.Indeed, by means of the stereographic projection Σ : S m {q} → R m from a point q ∈ S m Ω, problem (1.2) translates into It is well known that this problem does not have a least energy solution; see, e.g., [52,Theorem III.1.2].
Our aim is to give conditions on (M, g) which guarantee the existence of an optimal ℓ-partition for every ℓ.To this end, we follow the approach introduced by Conti, Terracini and Verzini in [20,21] and Chang, Lin, Lin and Lin in [13] relating optimal partition problems with variational elliptic systems having large competitive interaction.
We consider the competitive elliptic system where λ ij = λ ji < 0, α ij , β ij > 1, α ij = β ji , and α ij + β ij = 2 * .Firstly, we provide sufficient conditions for (1.4) to have a least energy solution with nontrivial components; secondly, in the case α ij = β ij and λ ij ≡ λ, we study the asymptotic profiles of such solutions as λ → −∞.As a byproduct, we obtain the existence of a regular optimal ℓ-partition of (1.3), and the existence of a sign-changing solution of (1.1) with two nodal domains.Our results read as follows.
Note that, as α ij ∈ (1, 2 * − 1), it satisfies 8 m−2 < α ij < 2(m−4) m−2 when m > 9.By a least energy fully nontrivial solution we mean a minimizer of the variational functional for the system (1.4) on a suitable constraint that contains only solutions with nonzero components, see Section 2 below.Theorem 1.2.Assume that (A3) (M, g) is not locally conformally flat and dim M ≥ 10.If dim M = 10 then where W g (q) is the Weyl tensor of (M, g) at q.
(iv) Γ := M ℓ i=1 Ω i = R ∪ S , where R ∩ S = ∅, R is an (m − 1)-dimensional C 1,α -submanifold of M and S is a closed subset of M with Hausdorff measure ≤ m − 2. In particular, M = ∪ ℓ i=1 Ω i .Moreover, -given p 0 ∈ R there exist i = j such that where p → p ± 0 are the limits taken from opposite sides of R, -and for p 0 ∈ S we have lim p→p0 |∇ g u i (p)| 2 = 0 for every i = 1, . . ., ℓ.
From Theorems 1.1 and 1.2 we immediately obtain the following results.
Theorem 1.4.Assume (A3).Then there exists a least energy sign-changing solution to the Yamabe equation (1.1) having precisely two nodal domains.
The main difficulty in proving Theorem 1.1 lies in the lack of compactness of the variational functional for the system (1.4).Least energy fully nontrivial solutions are given by minimization on a suitable constraint, but minimizing sequences may blow up, as it happens for instance when (M, g) is the standard sphere.To prove Theorem 1.1 we establish a compactness criterion (Proposition 2.8) that generalizes the condition given by Aubin for the Yamabe equation [4, Théorème 1].To verify this criterion we introduce a test function and we make use of fine estimates established in [26] to show that, under assumptions (A1) and (A2), a minimizer exists.
The components of least energy fully nontrivial solutions to the system (1.4) may also blow up as the parameters λ ij go to −∞.The standard sphere is again an example of this behavior.So, to prove Theorem 1.2, we establish a condition that prevents blow-up (see Lemma 4.2).To verify this condition we need to estimate the energy of suitable test functions.Rather delicate estimates are required, particularly in dimension 10 -where not only the exponents but also the coefficients of the energy expansion play a roleleading to the geometric inequality stated in assumption (A3).These estimates are derived in Appendix A.
But the occurrence of blow-up is not the only delicate issue in proving Theorem 1.2.To obtain an optimal ℓ-partition we need that the limit profiles of the components of the solutions to (1.5) are continuous.To this end, we show that the components (u n,i ) are uniformly bounded in the α-Hölder norm.This requires subtle regularity arguments which are well known in the flat case, see e.g.[9,41,47,52].We adapt some of these arguments (for instance, a priori bounds, blow-up arguments and monotonicity formulas) to obtain uniform Hölder bounds for general systems involving an anisotropic differential operator.This result (Theorem B.2) is interesting in itself.
In order to prove the optimal regularity of the limiting profiles u i , the regularity of the free boundaries M ℓ i=1 Ω i and the free boundary condition, we use local coordinates.This reduces the problem to the study of segregated profiles satisfying a system involving divergence type operators with variable coefficients.Using information arising from the variational system (1.5), we deduce limiting compatibility conditions between the u i 's which allow to prove an Almgren-type monotonicity formula and to perform a blow-up analysis, combining what is known in case of the pure Laplacian [9,47,55] with some ideas from papers dealing with variable coefficient operators [30,31,37,50].This result (which we collect in a more general setting in Theorem C.1) is also interesting in its own right.
As we mentioned before, optimal ℓ-partitions on the standard sphere S m do not exist.However, if one considers partitions with the additional property that every set Ω i is invariant under the action of a suitable group of isometries, then optimal ℓ-partitions of this kind do exist and they give rise to sign-changing solutions to the Yamabe equation (1.1) with precisely ℓ-nodal domains for every ℓ ≥ 2; as shown in [18].
Already in 1986 W.Y. Ding [25] established the existence of infinitely many sign-changing solutions to (1.1) on S m , and quite recently Fernández and Petean [28] showed that there is a solution with precisely ℓ nodal domains for each ℓ ≥ 2. These results, as those in [18], make use of the fact that there are groups of isometries of S m that do not have finite orbits.Looking for solutions which are invariant under such isometries allows avoiding blow-up.On the other hand, sign-changing solutions to (1.1) which blow-up along some special minimal submanifolds of the sphere S m have been found by Del Pino, Musso, Pacard and Pistoia in [22,23].The existence of a prescribed number of nodal solutions on some manifolds (M, g) with symmetries having finite orbits is established in [17].
However, the existence of nodal solutions to the Yamabe equation (1.1) on an arbitrary manifold (M, g) is largely an open problem.In [2] Ammann and Humbert established the existence of a least energy signchanging solution when (M, g) is not locally conformally flat and dim M ≥ 11.Theorem 1.4 recovers and extends this result (see Remark 4.11).We also note that an optimal ℓ-partition {Ω 1 , . . ., Ω ℓ } gives rise to what in [2] is called a generalized metric ḡ := ū2 * −2 g conformal to g by taking ū := u 1 + • • • + u ℓ with u i a positive solution to (1.1) in Ω i .So Theorem 1.3 may be seen as an extension of the main result in [2].
We close this introduction with references to related problems.The study of elliptic systems like (1.4) with critical exponents in euclidean spaces has been the subject of intensive research in the past two decades, starting from [14][15][16]; without being exhaustive, we refer to the recent contributions [24,56] for a state of the art and further references.On the other hand, optimal partition problems is another active field of research: see for instance the book [7] for an overview on a general theory using quasi-open sets and other relaxed formulations.Particular interest has been shown when the cost involves Dirichlet eigenvalues (leading to spectral optimal partitions) both in Euclidean spaces (see for instance the survey [6,33] or the recent [1,42,57] and references therein), and in the context of metric graphs (see e.g.[34,36] and references).

Compactness for the Yamabe system
We write • , • and | • | for the Riemannian metric and the norm in (M, g) and for v, w ∈ H 1 g (M ) we define where ∇ g denotes the weak gradient.Since we are assuming that the conformal Laplacian L g is coercive, • g is a norm in H 1 g (M ), equivalent to the standard one, and the Yamabe invariant of (M, g) is positive.We write |u| g,r := M |u| r dµ g 1/r for the norm in L r g (M ), r ∈ [1, ∞).Set H := (H 1 g (M )) ℓ and let J : H → R be given by This functional is of class C 1 and its partial derivatives are Hence, the critical points of J are the solutions to the system (1.4).Note that every solution u to the Yamabe equation (1.1) gives rise to a solution of the system (1.4) whose i-th component is u and all other components are 0. We are interested in fully nontrivial solutions, i.e., solutions (u 1 , . . ., u ℓ ) such that every u i is nontrivial.They belong to the Nehari-type set A fully nontrivial solution u to (1.4) is called a least energy fully nontrivial solution if J (u) = c.
where Y g is the Yamabe invariant of (M, g).Hence, N is a closed subset of H.
then there exists s u ∈ (0, ∞) ℓ such that s u u ∈ N .
(ii) If there exists s u ∈ (0, ∞) ℓ such that s u u ∈ N , then s u is unique and Moreover, s u depends only on the values . ., ℓ, and it depends continuously on them.
Proof.Define J u : (0, ∞) ℓ → R by If u i = 0 for all i = 1, . . ., ℓ, then, as .Hence, for every 0 Set T := {u ∈ H : u i g = 1, ∀i = 1, . . ., ℓ}, and let (iii) Let u ∈ U.Then, u is a critical point of Ψ if and only if s u u is a fully nontrivial critical point of J .
Proof.The proof is identical to that of [19,Theorem 3.3].
Corollary 2.6.If u ∈ N and J (u) = c, then u is a fully nontrivial solution to the system (1.4).
Proof.Since Ψ u u g = inf U Ψ and U is an open subset of the smooth Hilbert manifold T , we have that u u g is a critical point of Ψ.By Proposition 2.5, u is a critical point of J .
Recall that the operator L g is conformally invariant, i.e., if g = ϕ 2 * −2 g, ϕ > 0, is a metric conformal to g, then So, changing the metric within the conformal class of g does not affect our problem.Let S m be the standard m-sphere and p ∈ S m .Since the stereographic projection S m {p} → R m is a conformal diffeomorphism, the Yamabe invariant of S m is the best constant for the Sobolev embedding where ω m denotes the volume of S m [3,53].
It is shown in [19,Proposition 4.6] that c = inf N J is not attained if M = R m .Therefore, from the previous paragraph, c is also not attained if M is the standard sphere S m .Our aim is to investigate whether this infimum is attained in some other cases, at least for some values of α ij and β ij .
To this end, we establish a compactness criterion for the system (1.4) that extends a similar well known criterion for the Yamabe equation [39, Theorem A].The key ingredient is the following result of T. Aubin.
For each Z ⊂ {1, . . ., ℓ}, let (S Z ) be the system of ℓ − |Z| equations where |Z| denotes the cardinality of Z.The fully nontrivial solutions of (S Z ) are the solutions (u 1 , . . ., u ℓ ) of (1.4) which satisfy u i = 0 iff i ∈ Z.We write J Z and N Z for the functional and the Nehari set associated to (S Z ), and define The following compactness criterion is inspired by [19,Lemma 4.10].
Proposition 2.8.Assume that Then c is attained by J on N .
Proof.By Ekeland's variational principle and Proposition 2.5 there is a sequence (u n ) in N such that J (u n ) → c and J ′ (u n ) → 0.Then, (u n ) is bounded in H and, after passing to a subsequence, u n,i ⇀ ūi weakly in H 1 g (M ), u n,i → ūi strongly in L 2 g (M ) and u n,i → ūi a.e. on M , where u n = (u n,1 , . . ., u n,ℓ ).A standard argument shows that ū = (ū 1 , . . ., ūℓ ) is a solution of the system (1.4).
Proof.Let s > 0 and assume that where the last inequality is given by Theorem 2.7 and C 1 , C 2 are positive constants independent of L and K. Since . Now, starting with s such that 2(1 + s) = 2 * and iterating this argument, we conclude that u i ∈ L r g (M ) for every r ≥ 1.Since u i is a weak solution of the equation from elliptic regularity [39, Theorem 2.5] and the Sobolev embedding theorem [39, Theorem 2.2] we get that u i ∈ C 0,γ (M ) for any γ ∈ (0, 1).Then, f i ∈ C 0,γ (M ) for any γ ∈ (0, 1) such that γ < β ij − 1 for all j = i, and, by elliptic regularity again, we conclude that u i ∈ C 2,γ (M ) for any such γ.

The choice of the test function
To prove the strict inequality in Proposition 2.8 we need a suitable test function.We follow the approach of Lee and Parker [39].
Fix N > m.Given p ∈ M , there is a metric g on M conformal to g such that in g-normal coordinates at p; see [39,Theorem 5.1].These coordinates are called conformal normal coordinates at p. Since the Yamabe invariant Y g is positive, the Green function G p for the conformal Laplacian L g exists at every p ∈ M and is strictly positive.Fix p ∈ M and define the metric This metric is asymptotically flat of some order τ > 0 which depends on M .If m = 3, 4, 5, or (M, g) is locally conformally flat, the Green function has the asymptotic expansion in conformal normal coordinates (x i ) at p, where b m = (m − 2)ω m−1 and ω m−1 is the volume of S m−1 .The constant A(p) is related to the mass of the manifold ( M , g).It follows from the positive mass theorems of Schoen and Yau [45,46] that A(p) > 0 if the manifold (M, g) is not conformal to the standard sphere S m and, either m < 6, or (M, g) is locally conformally flat.In the other cases the expansion of the Green function G p involves the Weyl tensor W g (p) of (M, g) at p; see [39, Section 6] for details.For δ > 0, let , written in conformal normal coordinates (x i ) at p and, for suitably small r > 0, define The following estimates were proved by Esposito, Pistoia and Vétois in [26, Proof of Lemma 1].If m = 3, 4, 5, or (M, g) is locally conformally flat, then If (M, g) is not locally conformally flat and m = 6, If (M, g) is not locally conformally flat and m ≥ 7, ) c m is a positive constant depending only on m.In particular, From these estimates we derive the following result.
Lemma 3.1.Assume that (M, g) is not conformal to the standard sphere S m .Then, there exist p ∈ M and for all δ > 0 sufficiently small, where where .
If m = 3, 4, 5, or (M, g) is locally conformally flat, the positive mass theorem ensures that A(p) > 0 for any p ∈ M , and from (3.2) we get that where C is a positive constant.If (M, g) is not locally conformally flat and m ≥ 6, we choose and, if m > 6, from (3.4) we derive for some positive constant C. Since s δ → 1 as δ → 0, our claim is proved.
if and only if • or (M, g) is not locally conformally flat, m ≥ 9, and 2 α.From (3.1) we deduce Therefore, Proof.We prove this statement by induction on ℓ.
If ℓ = 1 the system reduces to the Yamabe equation (1.1), and (3.6) is equivalent to Y g < σ m .This inequality follows from Lemma 3.1 taking δ small enough.

Phase separation and optimal partitions
In this section we restrict to the case λ ij = λ and α ij = β ji = 2 * 2 =: β.Our aim is to study the behavior of least energy fully nontrivial solutions to the system (1.5) as λ → −∞ and to derive the existence and regularity of an optimal partition.
Let Ω be an open subset of M .As mentioned in the introduction, the nontrivial solutions of (1.2) are the critical points of the restriction of the functional to the Nehari manifold Lemma 4.1.Assume there exists (u 1 , . . ., u ℓ ) ∈ M ℓ such that u i ∈ C 0 (M ) and Set Ω i := {p ∈ M : u i (p) = 0}.Then {Ω 1 , . . ., Ω ℓ } is an optimal ℓ-partition for the Yamabe problem on (M, g), each Ω i is connected and J Ωi (u i ) = c Ωi for all i = 1, . . ., ℓ.
Proof.As u i is continuous and nontrivial, we have that Ω i is an open nonempty subset of M and To prove the last two statements of the lemma we argue by contradiction.If, say, Ω 1 were the disjoint union of two nonempty open sets Θ 1 and Θ 2 , then, setting ū1 (p which is again a contradiction.Hence, J Ωi (u i ) = c Ωi for all i = 1, . . ., ℓ and inf This shows that {Ω 1 , . . ., Ω ℓ } is an optimal ℓ-partition and concludes the proof.
Lemma 4.2.Let λ n < 0 and u n = (u n,1 , . . ., u n,ℓ ) be a least energy fully nontrivial solution to the system (1.5).Assume that λ n → −∞ as n → ∞ and that u n,i ≥ 0 for all n ∈ N. Assume further that Then there exists (u ∞,1 , . . ., u ∞,ℓ ) ∈ M ℓ such that, after passing to a subsequence, u n,i → u ∞,i strongly in Proof.To highlight the role of λ n , we write J n and N n for the functional and the Nehari set associated to the system (1.5) and we define So, after passing to a subsequence, we get that u n,i ⇀ u ∞,i weakly in g (M ) and u n,i → u ∞,i a.e. on M , for each i = 1, . . ., ℓ.Hence, u ∞,i ≥ 0.Moreover, as Hence, u ∞,j u ∞,i = 0 a.e. on M whenever i = j.On the other hand, as So, passing to the limit as n → ∞ we obtain We claim that To prove this claim, let Z := {i ∈ {1, . . ., ℓ} : u ∞,i = 0}.After reordering, we may assume that Z is either empty or Z = {k + 1, . . ., ℓ} for some 0 ≤ k < ℓ.Then, arguing as we did to prove (2.2), we get that On the other hand, if i ∈ Z, there exists But then assumption (4.1) implies that k = ℓ, i.e., Z = ∅ and claim (4.4) is proved.Moreover, (4.5) becomes Hence, t i = 1 and, so, (u ∞,1 , . . ., u ∞,ℓ ) ∈ M ℓ , and and u n,i → u ∞,i strongly in H 1 g (M ) and L 2 * g (M ), we obtain (4.2).
Let s ≥ 0 and set w n,i := u 1+s n,i .Since Now, for any K > 0, we have that where . Fix ε > 0, and choose K s , n s such that 1+ǫ σm (1 + s) 2 η(K s , n) < 1 2 for every n ≥ n s .From Theorem 2.7 and inequalities (4.7) and (4.8) we obtain where C s , C s and C ′ s are positive constants depending on s but not on n, Iterating this inequality, starting with s = 0, we conclude that, for any r ∈ [2, ∞), where Cr is a positive constant independent of n.Now, we fix 2R > 0 smaller than the injectivity radius of M .Since M is covered by a finite number of geodesic balls of radius R and u n,i satisfies we derive from [32, Theorem 8.17] that (u n,i ) is uniformly bounded in L ∞ (M ), as claimed.
Lemma 4.5.Assume that (M, g) is not locally conformally flat, m ≥ 10, and there exists (u 1 , . . ., u ℓ−1 ) ∈ M ℓ−1 such that u i ∈ C 0 (M ), u i ≥ 0 and If m = 10, assume further that there exists p ∈ M such that Then Proof.It suffices to show that Set Ω i := {q ∈ M : u i (q) > 0}.Then, Ω i is open and Ω i ∩ Ω j = ∅ if i = j.Since (M, g) is not locally conformally flat and m ≥ 4, there exists p ∈ M such that the Weyl tensor W g (p) at p does not vanish.After reordering, we may assume that, either p ∈ Ω 1 , or p ∈ M ∪ ℓ−1 i=1 Ω i .First, we consider the case where p ∈ Ω 1 .If m = 10 we take p satisfying (4.9).Fix r > 0 suitably small so that the closed geodesic ball centered at p is contained in Ω 1 and let written in conformal normal coordinates around p, where V δ,p is the function in (3.1).If (M, g) is not locally conformally flat and m ≥ 7 the estimates (3.4) yield with cm as in (3.5).Now, set Note that v i = 0 and v 1 v ℓ = 0 on M , and For m ≥ 10 from Remark 2.4 and Lemma A.1 we derive → 1 as δ → 0. Similarly, using (4.11), we obtain + o(δ 4 ). Since → 1 as δ → 0, and m−4 and m−2 2 > 4 when m ≥ 11, we have that, for δ small enough, with C > 0. On the other hand, if m = 10, then m−2 2 = 4. Recalling that ω m is the volume of the standard m-sphere S m and using (3.5) we obtain by assumption (4.9).Hence, for δ small enough, with C > 0. From (4.12), (4.13), (4.14) and (4.15) we derive for δ small enough.This proves (4.10) when p ∈ Ω 1 .
If p ∈ M ∪ ℓ−1 i=1 Ω i , we fix r > 0 small enough so that the closed geodesic ball of radius r centered at p is contained in M ∪ ℓ−1 i=1 Ω i and define u ℓ := t ℓ V δ,p with V δ,p as above and t ℓ > 0 such that u ℓ as claimed.
Remark 4.6.The argument given above does not carry over to m < 10 or to the case where (M, g) is locally conformally flat.Indeed, as can be seen from identities (3.2), (3.3) and (3.4) and Lemma A.1, in these cases ), with Cu 1 (p) > 0, for δ small enough.
• For ℓ = 2, inequality (4.9) holds true if Indeed, choosing p to be a minimum of u 1 , since u 1 is a positive solution to the Yamabe equation (1.1) we have that κ m S g (p) Lemma 4.8.Assume that (M, g) satisfies the following conditions: (A4) (M, g) is not locally conformally flat and dim M ≥ 10.If dim M = 10, then there exist a positive least energy fully nontrivial solution ū to the Yamabe equation (1.1) and a point p ∈ M such that ū(p) < 5 567 |W g (p)| 2 g and, in addition, |W g (q)| g = 0 for every q ∈ M if ℓ ≥ 3.
Let λ n < 0 and u n = (u n,1 , . . ., u n,ℓ ) be a least energy fully nontrivial solution to the system (1.5).Assume that λ n → −∞ as n → ∞ and that u n,i ≥ 0 for all n ∈ N. Then there exists (u ∞,1 , . . ., u ∞,ℓ ) ∈ M ℓ with u ∞,i ∈ C 0,α (M ) for every α ∈ (0, 1) such that, after passing to a subsequence, u n,i → u ∞,i strongly in H 1 g (M ) ∩ C 0,α (M ), u ∞,i ≥ 0, and Proof.The proof is by induction on ℓ.Let ℓ = 2. Then u 1 := ū satisfies the hypotheses of Lemma 4.5.Therefore, the inequality (4.1) holds true and Lemma 4.2 yields the existence of (u ∞,1 , . . ., u ∞,ℓ ) ∈ M ℓ such that, after passing to a subsequence, u n,i → u ∞,i strongly in H 1 g (M ), u ∞,i ≥ 0, and From Lemmas 4.3 and 4.4 we get that (u n,i ) is uniformly bounded in C 0,α (M ).Therefore, the family {u n,i } is equicontinuous and, as u n,i → u ∞,i a.e. on M , the Arzelà-Ascoli theorem yields u n,i → u ∞,i in C 0 (M ).Now, let ℓ ≥ 3 and assume that the statement holds true for ℓ − 1.Then, by Remark 4.7, the hypotheses of Lemma 4.5 are satisfied and, consequently, (4.1) holds true for ℓ.The same argument we gave for ℓ = 2 yields the result for ℓ.Remark 4.9.Observe that to prove the previous lemma for ℓ, we need it to be true for ℓ − 1, because the inequality (4.1) must hold true in order to apply Lemma 4.2.Therefore, the inequality ū(p) < 5 567 |W g (p)| 2 g is required for every ℓ ≥ 2.
Proof of (v): If u ∈ H 1 g (M ) is a sign-changing solution of the Yamabe equation (1.1), then u + := max{u, 0} = 0, u − := min{u, 0} = 0 and J ′ M (u)[u ± ] = 0. Hence, u belongs to the set Moreover, as shown in [12, Lemma 2.6], any minimizer of J M on E M is a sign-changing solution of (1.1).For every u ∈ E M , we have that (u Hence, it is a sign-changing solution of (1.1), as claimed.
Remark 4.10.As can be seen from its proof, Theorem 1.2 is true under assumption (A4) and, consequently, so are Corollaries 1.3 and 1.4.As noted in Remark 4.7, (A4) is weaker that (A3), but it requires some knowledge on the least energy solution to the Yamabe equation (1.1) having precisely two nodal domains.
Remark 4.11.In [2], Ammann and Humbert defined the second Yamabe invariant of (M, g) as where λ 2 ( g) is the second eigenvalue of the operator κ −1 m L g and [g] is the conformal class of g.Using the variational characterization in [2, Proposition 2.1] one can easily verify that inf The invariant µ 2 (M, g) is not attained at a metric, but it is shown in [2] that, if (M, g) is not locally conformally flat and m ≥ 11, this invariant is attained at the generalized metric conformal to g which is given by a minimizer of J M in E M .So Corollary 1.4 recovers and extends this result.
Remark 4.12.It is interesting to compare our result with that proved by Robert and Vétois in [43] under assumptions which are complementary to ours.In fact, they establish the existence of a sign-changing solution to the subcritical perturbation of Yamabe equation (A.5) By (A.1), (A.2) and (A.3), there are positive constants c 1 , c 2 , c 3 such that (A.6) Lemma A.1.We have the following estimates: ), where if m ≥ 7 and (M, g) is not l.c.f., Proof.(i) : From (A.4) and (A.6) we obtain , and, using (A.6) again, where ∂ ν is the exterior normal derivative, if m = 6 and M is not l.c.f., if m ≥ 7 and M is not l.c.f.Indeed, arguing as in [26] we obtain s 8 ds if m = 6 and M is not l.c.f., if m ≥ 7 and M is not l.c.f.This concludes the proof of statement (i).
(ii) : Using the inequalities , where This concludes the proof of statement (ii).
(iii) : Using (A.5) and (A.6) we obtain )u dµ g see (A.13) where and, arguing as in [26], if m ≥ 7 and M is not l.c.f.This concludes the proof of statement (iii).
This concludes the proof of statement (iv).

B Uniform bounds in Hölder spaces
In this appendix we prove Lemma 4.4.Since it does not require additional effort, we consider the more general system where (M, g) is a closed Riemannian manifold of dimension m ≥ 1, λ < 0, γ > 0, and Lemma 4.4 is a particular case of the following result.
In local coordinates, the system (B.1)becomes where Ω is an open subset of R m , a(x) := |g(x)|, A(x) := |g(x)|(g kl (x)), (g kl ) is the metric written in local coordinates, (g kl ) is its inverse and |g| is the determinant of (g kl ).Observe that the second order differential operator is uniformly elliptic and, since M is compact, a is bounded away from 0. Therefore, we end up with a system of the form be the space of real symmetric m × m matrices.For the system (B.2) we prove the following result.
We now show that Theorem 1.4 follows from Theorem B.2.
Proof of Theorem B.1.Arguing by contradiction, assume that {u λ,i : λ < 0} is unbounded in C 0,α (M ) for some α ∈ (0, 1) and some i = 1, . . ., ℓ. Since, by assumption, this set is uniformly bounded in L ∞ (M ), there exist λ n → −∞ and p n = q n in M such that u n,i := u λn,i satisfies where d g is the geodesic distance in (M, g).As (u n,i ) is uniformly bounded in L ∞ (M ), this implies that d g (p n , q n ) → 0.Moreover, since M is compact, a subsequence satisfies p n → p in M .Hence, q n → p. Now, in local coordinates around p the system (B.1)becomes (B.2) with This is a contradiction to Theorem B.2.
Therefore, for the remainder of the appendix, our goal is to prove Theorem B.2.We follow very closely the proof of [47,Theorem 1.2], where the case A(x) = I was treated, mainly highlighting the differences that arise from having a divergence-type operator instead of the Laplacian.
We use the following notation for the seminorm in Hölder spaces:

B.1 Auxiliary lemmas
We present the following generalization of [10, Lemma 5.2], [49, Lemma 4.1] and [51, Lemma 2.2] to our setting.The first part of the lemma is required to treat the case γ ≤ 1, while the second part is needed for γ > 1 (see the upcoming proof of Lemma B.6 for more details).
Then there exists a constant c > 0, depending only on m, Θ and a 0 , such that Observe that, for x ∈ B 2R (0), where in the last inequality we used C ≥ 1 and the definition of L.Moreover, observe that there exist c 1 , c 2 > 0, depending on L (that is, on m, a 0 and Θ), such that for every x ∈ R m .
Therefore, given x 0 ∈ B R (0), by the comparison principle (which we can apply because C > 0) we have Evaluating the previous inequality at x = x 0 yields and the conclusion follows.
2. We follow the proof of [51, Lemma 2.2] (which deals with the the Laplace operator).Our main addition is the use of the mean value theorem for divergence operators, which reads as follows: Given Ω ⊂ R m there exist k, K > 0, only depending on θ, Θ > 0, such that for y ∈ Ω there exists an increasing family of sets D r (y) ⊂ Ω such that B kr (y) ⊂ D r (y) ⊂ B Kr (y) and, for every solution w of − div(A(x)∇w) ≤ 0 in Ω, we have w is increasing, and w(y w. (See [5,Theorem 6.3] for the proof of this result, which was previously stated in [8,11]).Now take a nonnegative solution v ∈ H 1 (B 2R (0)) of Using the uniform ellipticity and since C > 0, we can apply the maximum principle to deduce that 0), and take η R (x) := η(x/R).Then, Now let y ∈ B R (0).Since γ ≥ 1, by the mean value theorem presented above and Jensen's inequality, By the maximum principle we have that u ≤ v + (δ/C) 1/γ , from which the conclusion follows.
with a > 0 and γ > 0. Under this change of variables, we reduce the proof of this lemma to the case of the Laplace operator.Therefore, parts 1 and 3 follow from [41, Proposition 2.2 and Corollary 2.3], while part 2 follows from [47,Lemma A.3] (see also [48,] for the case γ ≥ 1).

B.2 A contradiction argument and a blow-up analysis
Fix α ∈ (0, 1).Without loss of generality, we assume that B 3 (0) ⊂ Ω.Under the assumptions of Theorem B.2, we aim at proving the uniform Hölder bound in B 1 (0).Fix Λ > 0 such that Our goal is to prove that there exists C > 0 such that We may assume that the maximum is attained at i = 1.Then, for each n, we fix a pair of points x n , y n ∈ B 2 (0) with x n = y n such that As (u n ) is uniformly bounded in L ∞ (B 2 (0)), this implies that |x n − y n | → 0. So (x n ) and (y n ) converge to the same point.We denote The contradiction argument is based on two blow-up sequences both defined in the scaled domain Ω n := (B 3 (0) − x n )/r n ; see [47,51,58,60].Here r n ∈ (0, 1), r n → 0, will be conveniently chosen later.Observe that B 1/rn (0) ⊂ Ω n , therefore Ω n approaches R m as n → ∞.Since η is positive in B 2 (0), the functions v n,i and vn,i are nonnegative and nontrivial in Ω ′ n := (B 2 (0) − x n )/r n .Note that, as x n ∈ B 2 (0), Ω ′ n approaches a limit domain Ω ∞ which is either a half-space or the whole R m , as n → ∞.
6.For any compact set K ⊂ R m there exists C > 0 such that In particular, (v n,i ) has uniformly bounded oscillation in any compact set.Then the sequence (v n (0)) is bounded in R ℓ , where v n := (v n,1 , . . ., v n,ℓ ).
Then, setting w n (x) := v n,1 (x)−v n,1 (0), by the Ascoli-Arzelà theorem we have that ). Observing that R may be taken arbitrarily large, we conclude that div Arguing as in as in [47, p. 401-402] and using (B.4), we see that In in B R (0), which gives again div(A n (x)∇v n,1 (x)) → 0 uniformly in B R (0), a contradiction.Finally, if γ > 1, one may argue exactly as in Case 1 of the proof of [47, Lemma 2.3], using this time the decay estimate Lemma B.3-2.In both cases, ī = 1 and ī > 1, we end up with div(A n (x)∇v n,1 (x)) → 0 locally uniformly in R m , leading as before to a contradiction.Lemma B.7.Up to a subsequence, we have that Proof.We follow [47,Lemma 2.5].Arguing by contradiction, assume that the sequence considered in the statement is bounded and take With this choice we have that (B.8) is satisfied and from Lemma B.6 we deduce that (v n (0)) is bounded in R ℓ .Combining this fact with Lemma B.5-1, we deduce the existence of (v 1 , . . ., v ℓ ) ∈ C 0,α (R m , R ℓ ) such that vn,i → v i in the α-Hölder norm as n → ∞.Under the previous choice of r n one has Λ n = −1.Hence, by elliptic regularity, the convergence of vn,i to v i is actually in C 1,α , and where Ω ∞ is the limit domain of Ω ′ n and A ∞ , a ∞ are defined in (B.7).In particular, for any i = j, the pair In particular, v i v j ≡ 0 in R m for every i = j.
Proof.Using Lemmas B.5, B.6 and B.7, in particular the smoothness, boundedness and uniform ellipticity of A n , the proof is obtained from a straightforward adaptation of that of [47, Lemma 2.6] (which, in turn, is based on [41, Lemma 3.6])).Observe that v i v j ≡ 0 is a direct consequence consequence of the strong convergence of v n , the convergence in (ii) and the fact that Lemma B.9.Let (v 1 , . . ., v ℓ ) be as in Lemma B.8 and A ∞ be as in (B.7).Then, The set Ω 1 is open and connected, and Proof.Using the previous lemma together with Lemma B.4, the proof follows exactly as the one of [ Proof.We test the i-th equation in Lemma B.5-2 against ∇v n,i , Y , integrate by parts and take the sum for all i = 1, . . ., ℓ to obtain Finally, following [31, Lemma A.5], we have (by item 5.), we have that 7. follows from 3., 4. and 6.
Set B r := B r (0), u = (u 1 , . . ., u ℓ ), |u| 2 := Proof.For λ < 0, we have where the last identity is a consequence of testing the i-th equation in (B.2) by u λ,i , integrating by parts, and taking the sum over i. Passing to the limit as λ → −∞ and using assumption (H4), yields the claim.
Lemma C.5.Let ω ⋐ Ω be such that 0 ∈ ω.There exist constants C, r > 0, depending only on the dimension m, on ω and on an upper bound for DA ∞ , such that for every r ∈ (0, r).
Proof.We combine the proof of [ The conclusion now follows from the estimate where the constant C > 0 arises from items 3. and 6. in Lemma C.2.

Z(x)
From now on we use the summation convention for repeated indices, unless stated otherwise.
Lemma C.6 (Local Pohozaev-type identities).For every r > 0 such that B r ⊂ Ω, we have the following identity (where A = (a ij )) Proof.From system (B.2) we derive an identity for the u λ,i 's, and then pass to the limit as λ → −∞.For each i, from the divergence theorem and the definition of µ(x) and Z(x), we derive Following now [50, Lemma A.1] (see also [31,Lemma A.9]), we obtain Passing to limit as λ → −∞, the conclusion will follow once we prove the following claim In particular, given ω ⋐ Ω with 0 ∈ ω, there exist constants C, r > 0 (depending only on m, θ, ω and on an upper bound for DA ∞ ) such that, for every r ∈ (0, r), , div(Z − x) L ∞ (Br) ≤ Cr for some constant C > 0 depending only on the dimension m, on ω and on an upper bound for DA ∞ .Then, using also (H2), we obtain A∇u i , ∇u i = C E(r) (see equations (A.3)-(A.12) in [50] for more details).This completes the proof.
Remark C.8. Observe that identities (C.2) and (C.7), which can be seen as local Pohozaev-type identities, are equivalent.They correspond to the condition (G3) for the Laplacian stated in [47] and [55] respectively.
Proof of Theorem C.3.This result now follows from standard arguments.Here, as before, we mainly verify the dependence of the constants.Within this proof, O(1) will represent a bounded function of r depending only on m, θ, ω and on an upper bound for DA ∞ (but which is independent of u ∞ ).We have, by Lemma C.7,

C.2 Almgren's monotonicity formula: the general case
We have proved a monotonicity formula under the assumption that A(0) = Id.The general case can be reduced to this case in the following way: let A(x 0 ) 1 2 be the square root of the (positive definite) matrix A(x 0 ), that is, the unique positive definite matrix whose square is A(x 0 ).We recall that A(x 0 ) 1 2 is also symmetric, it commutes with A(x 0 ), it has real entries and that the map x 0 → A(x 0 ) 1 2 is continuous (see for instance [35]).Following [30,50], we set T x0 x := x 0 + A(x 0 ) Observe that A x0 (0) = Id.Let now N (x 0 , u, r) := E(x 0 , u, r) H(x 0 , u, r) , • we have an elliptic divergence type operator instead of the pure Laplacian operator, therefore the estimates will depend on the ellipticity constant θ; • the identity (C.2) plays the role of the identity in the last assumption of [47,Proposition 3.4], while the monotonicity formula (Theorem C.9) plays the role of the monotonicity formula [47,Theorem 3.3].
For related proofs of Lipschitz continuity in similar contexts, see also [41,  where x → x ± 0 are the limits taken from opposite sides of Γ; see [55, Section 2] for the details.For related proofs of regularity in similar contexts, see also [47,Theorem 1.7] or [54,Chapter 3].
Remark C.10.We remark that Theorem C.1 can be seen as a direct consequence of Theorem 7.1 in [55].However, since the latter result is presented without proof, we have decided to write this appendix and give all the necessary details.

Following [ 19 ,
Proposition 3.1], it is easy to see that U is a nonempty open subset of T .Define Ψ : U → R by Ψ(u) := J (s u u), with s u as in Proposition 2.3.This function has the following properties.Proposition 2.5.(i) Ψ ∈ C 1 (U, R). (ii) Let u n ∈ U.If (u n ) is a Palais-Smale sequence for Ψ, then (s un u n ) is a Palais-Smale sequence for J .Conversely, if (u n ) is a Palais-Smale sequence for J and u n ∈ N for all n ∈ N, then un un g is a Palais-Smale sequence for Ψ.
which looks like the difference between a positive solution u 0 to the Yamabe equation and a bubble.Their result holds true either in the locally conformally flat case, or in low dimensions 3 ≤ m ≤ 9, or if m = 10 provided u 0 (p) > 5 567 |W g (p)| 2 g for any p ∈ M.An interesting open problem would be to show that under these assumptions a least energy sign-changing solution to the Yamabe problem (1.1) does not exist, as suggested by Remark 4.6.and

Proof. 1 .
For the first statement we follow closely the proof of [49, Lemma 4.1], which considers the case of a constant matrix.Define z(x) with L := max{1, (a 0 m + Θ) 2 }.