STATIC KLEIN-GORDON-MAXWELL-PROCA SYSTEMS IN 4-DIMENSIONAL CLOSED MANIFOLDS. II

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1 STATIC KLEIN-GORDON-MAXWELL-PROCA SYSTEMS IN 4-DIMENSIONAL CLOSED MANIFOLDS. II OLIVIER DRUET, EMMANUEL HEBEY, AND JÉRÔME VÉTOIS Abstract. We discuss static Klein-Gordon-Maxwell-Proca systems in the critical case of 4-dimensional closed manifolds in the continuation of Hebey- Truong [0]. We prove phase stability for all possible phases in the nonpositively curved case, and in the positively curved case when the phase lies in some set that we show to be maximal in the case of the round sphere. We investigate in this paper phase stability for the electrostatic Klein-Gordon- Maxwell-Proca system in 4-dimensional closed manifolds. The full Klein-Gordon- Maxwell-Proca system in short KGMP is a massive version of the more traditional Klein-Gordon-Maxwell system. It provides a dualistic model for the description of the interaction between a charged relativistic matter scalar field and the electromagnetic field that it generates. The external vector field ϕ, A in the system inherits a mass and is governed by the Proca action which generalizes that of Maxwell. Let M, g be a closed 4-dimensional Riemannian manifold. Writing the matter scalar field in polar form as ψx, t = ux, te isx,t, choosing the nonlinearity in the model to be pure and critical in terms of Sobolev embeddings, the full Klein-Gordon-Maxwell-Proca system is written as 2 u S t + qϕ 2 S qa 2 t + 2 g u + m 2 0u = u 3 + S t t + qϕ u 2. S qa u 2 = 0. A t + ϕ + m 2 ϕ + q S t + qϕ u 2 = 0 g A + A t t + ϕ + m 2 A = q S qa u 2, u 0. where g = div g is the Laplace-Beltrami operator, g = δd is half the Laplacian acting on forms, and δ is the codifferential. In its electrostatic form we assume A and ϕ do not depend on the time variable. Looking for standing waves solutions ψx, t = uxe iωt, letting ϕ = ωv, there necessarily holds that A 0 and the system reduces to the two following critical equations { g u + m 2 0u = u 3 + ω 2 qv 2 u g v + m 2 + q 2 u 2 v = qu In the above, m 0, m > 0 are masses m 0 is the mass of the particle, m is the Proca mass, and q > 0 is the electric charge of the particle. The Proca formalism comes with the assumption m > 0. The system 0.2, in Proca form in closed manifolds, has been investigated in Druet and Hebey [5] and Hebey and Wei [] in the case of 3-dimensional manifolds, and Hebey and Truong [0] in the critical dimension n = 4 4 is the dimension for which the second equation in 0.2 is also Date: October 5, 203. Published in Journal für die reine und angewandte Mathematik ,

2 2 OLIVIER DRUET, EMMANUEL HEBEY, AND JÉRÔME VÉTOIS critical by the u 2 v term. A sequence u α e iωαt of standing waves is said to have α finite energy if u α H = O. We keep here the origin of the writing in polar form and always assume in the sequel that the amplitude u of a standing wave ue iωt is nonnegative. A priori bounds were proved in Hebey and Truong [0]. We aim here in proving phase stability with the objective of getting more phases and more geometries by relaxing the a priori bound compactness property in Hebey and Truong [0]. Given a sequence ω α α of phases, we consider the family of equations { g u + m 2 0u = u 3 + ωα 2 qv 2 u g v + m 2 + q 2 u 2 v = qu We define phase stability as follows. Definition 0.. A phase ω m 0, +m 0 is said to be stable if for any sequence of phases ω α α converging to ω, any sequence u α e iωαt of finite energy standing α waves, u α 0, and any sequence v α α of gauge potentials, solutions of 0.3 for all α, there holds that, up to a subsequence, u α u and v α v in C 2 as α +, where u and v solve 0.2 with phase ω. When a phase ω is stable, 0.2 is automatically compact in the finite energy setting. But phase stability implies more and actually measures how much 0.2 is robust with respect to perturbations of ω. We let RgS g be the range interval [ RgS g = min M S g, max S g M ], 0.4 where S g is the scalar curvature of g. The first result of this paper,theorem 0., establishes that phases are stable as long as they do not enter in resonance with the ambient geometry through the condition m 2 0 ω 2 6 RgS g. The second result of the paper, Theorem 0.2, establishes that this condition is sharp. Theorem 0.. Let M, g be a closed 4-dimensional manifold, and m 0, m, q > 0 be positive real numbers. Any phase ω m 0, +m 0 such that m 2 0 ω 2 6 RgS g is a stable phase for 0.2, where RgS g is given by 0.4. In particular, if g has nonpositive scalar curvature, then any phase ω m 0, +m 0 is stable. A priori bounds were established in Hebey and Truong [0] under the condition that m 0 and ω satisfy that m 2 0 ω 2 < 6 min M S g, and thus that m 2 0 ω 2 sits on the left part of R + \ 6 RgS g. While a priori bounds are stronger than phase stability, the condition required to obtain such bounds is quite restrictive. It does not cover the full phases ω by missing the right part of R + \ 6 RgS g, and it implies that the background space has positive scalar curvature, which is not required by Theorem 0.. Basically, Theorem 0. establishes that by relaxing the a priori bound property to the stable phase property, we gain more phases and more geometries with respect to what was handled in Hebey and Truong [0]. Theorem 0. coupled with Theorem 0.3 in Hebey and Truong [0] provide a complete description of the various compactness properties one can associate to 0.2. Theorem 0.2 below complements the above picture by showing that both Theorem 0. and Theorem 0.3 in Hebey and Truong [0] are optimal. The second part of Theorem 0. obviously includes the model cases of flat torii T 4, g and compact hyperbolic spaces H 4, g. Any phase ω m 0, +m 0 is stable for these manifolds. The model case of the round sphere S 4, g is contained in the first part of Theorem 0.. In the case of the round sphere S 4, g, S g = 2 and

3 STABLE PHASES FOR THE KGMP SYSTEM 3 Theorem 0. asserts that any phase ω m 0, +m 0 is stable except, possibly, if m 0 2 and ω is one of the two solutions of m 2 0 ω 2 2. Theorem 0.2 asserts that these two solutions are indeed unstable phases, and thus that the set in Theorem 0. is maximal in the case of the round sphere. Theorem 0.2. Let S 4, g be the round 4-sphere, and m 0, m, q > 0 be positive real numbers with m Let ω R be such that m 2 0 ω 2 = 2. Then ω is an unstable phase for 0.2 which gives rise to resonant states in the sense that for such ω s there exist a sequence u α e iωαt α, u α 0, of standing waves, and a sequence v α α of gauge potentials, solving 0.3 for all α, such that ω α ω and u α L + as α +. Theorems 0. and 0.2 complement each other. They are distinct in nature. Theorem 0. has to do with a priori estimates. Theorem 0.2 has to do with constructive approaches. They are proved by using two different methods. Theorem 0. is proved by passing through the C 0 -theory for blow-up, the analysis of the range of influence, and then by getting a contradiction through a Pohozaev type identity. Theorem 0.2 is proved by going into the finite dimensional Lyapounov- Schmidt reduction method. We prove Theorem 0. in Sections to 3. We prove Theorem 0.2 in Section 4. We refer to Hebey and Truong [0] for the physics origin of the problem and the building of the equations.. A priori L -estimates We let M, g be a smooth compact Riemannian 4-manifold, and m 0, m, q > 0 be positive real numbers. Following a very nice idea going back to Benci and Fortunato [], we introduce the map Φ : H H solution of g Φu + m 2 + q 2 u 2 Φu = qu 2. for all u H. We refer to Hebey and Truong [0], see also Section 4, for the existence and regularity of the map Φ in the critical dimension n = 4, in the Riemannian context. We let ω m 0, +m 0, u α e iωαt be an arbitrary sequence α of finite energy standing waves, u α 0, and v α α be an arbitrary sequence of gauge potentials, satisfying that { g u α + m 2 0u α = u 3 α + ωα 2 qv α 2 u α g v α + m 2 + q 2 uα 2 vα = qu 2.2 α. for all α, and that ω α ω in R as α +. Obviously, v α = Φu α, and by the maximum principle, there holds that 0 v α q in M for all α. We aim in proving that, passing to a subsequence, the u α s and v α s converge in C 2 as α. By standard elliptic regularity theory, it suffices to prove that, passing to a subsequence, u α L = O. We proceed by contradiction and assume that u α L +..3 The second equation in.2 is critical of maximal homogeneity 3 in 4-space dimension and it acts as an auto-inductive perturbation the perturbation depends on the solution itself of the first nonlinear Schrödinger equation through its potential term h α = m 2 0 ωα 2 qv α 2.4 = m 2 0 ωα 2 qφu α 2.

4 4 OLIVIER DRUET, EMMANUEL HEBEY, AND JÉRÔME VÉTOIS Since 0 v α q for all α, the h α s are bounded in L. Noting that the v α s are bounded in H L, they converge, up to a subsequence, in L p for all p, and so do the h α s. We let h be the limit of the h α s, e.g. in L 2, so that h α L = O and h α h in L 2.5 as α + and, as we easily infer, the convergence in.4 holds in L p for all < p < +. No further control is a priori available on the h α s, and this is going to be a serious issue in what follows. There holds that h L and h m 2 0 ω 2.6 in M. In particular, g + h is coercive, where h is as in.5, and by Robert [4], the C 0 and C -estimates on the Green s function of g + h hold true. Following standard terminology, we define a bubble B α α as a sequence of functions in M given in dimension 4 by B α x =.7 µ 2 α + dgxα,x2 8 for all x M and all α, where x α α is a converging sequence of points in M, and µ α α is a sequence of positive real numbers converging to 0 as α +. We refer to the x α s in.7 as the centers of B α α, and to the µ α s as the weights of B α α. Then, see Druet-Hebey-Robert [6] and Hebey [9], the H -theory for blowup, as developed in Struwe [5], can be applied. And, even more, the upper control of the C 0 -theory developed in Druet-Hebey-Robert [6] see also Druet-Hebey [4] and Hebey [9] for more recent expositions also holds true. By the H -theory we get that there exist a solution u, Φu of 0.2 with phase ω, k N, and k-bubbles Bα i α, i =,..., k, such that, up to a subsequence, u α = u + µ α k Bα i + R α.8 i= in M for all α, where R α H 0 as α +, and µ j,α + µ j,α + d gx, x j,α 2 µ j,α +.9 as α +, for all i j, where the x s are the centers of Bα i α, and the s are the weights of Bα i α. To state the C 0 -theory, in the form we are going to use, we need to introduce some more notions. Given i, j {,..., k }, i j, we let s i,j,α be given by s 2 i,j,α = d g x, x j,α 2 + µ j,α.0 µ j,α 8 for all α, and then we define the range of influence r of the blow-up point x by min s i,j,α if u 0 j A i r = { min min s i,j,α ; }. if u 0, j A i where A i = { } j {,..., k}, j i s.t. = O µ j,α..2

5 STABLE PHASES FOR THE KGMP SYSTEM 5 If A i = and u 0, we adopt the convention that r = 2 i g, where i g is the injectivity radius of M, g, and if A i = and u 0, we adopt the convention that r =. From now on we order the blow-up points such that µ,α µ k,α.3 for all α. It follows from the structure equation.9 that r +.4 as α +. If j A i and i A j, we let λ i,j > 0 be given by λ i,j = lim α + µ j,α..5 Given i {,..., k}, we define { } B i = j {,..., k}, j i s.t. d g x, x j,α = O r if r 0, { } B i = j {,..., k}, j i s.t. x j B xi g 2 i if r 0 and, for j B i, we let z i,j be given by z i,j =.6 lim α + r exp x x j,α..7 All the limits involved in these definitions are assumed to exist, which is always possible passing to a subsequence. We let δ i > 0 be such that for any i and any j B i, z i,j 0 z i,j 0δ i..8 We also define C i to be the subset of B i given by { } C i = j B i s.t. z i,j = 0 A c i,.9 where A c i is the complementary set of A i, and thus the set consisting of the j s which are such that µ j,α = o. Then, passing to a subsequence, and for any i, there exist a subset D i of C i and a family R i,j j Di of positive real numbers such that for any j, j 2 D i, j j 2, d g x j,α, x j2,α s j, +.20 as α +, and such that for any j C i there exists a unique j D i such that lim sup α + d g x j,α, x j,α s j, R i,j 20 and lim sup α + where C i is as in.9. We let µ α = µ,α be given by s j, R i,j s j, 20,.2 µ α = max i=,...,k..22 As shown in Druet and Hebey [4], see also Hebey [9] for an exposition in book form, the C 0 -theory then gives that for any i {,..., k}, there exists C > 0 and a sequence ε α α of positive real numbers converging to zero such that, passing to a subsequence, uα Bα i C H i α + ε α B i α + C CB i α r 2 + j D i B j α.23

6 6 OLIVIER DRUET, EMMANUEL HEBEY, AND JÉRÔME VÉTOIS in B x 4δ i r \ Ri,j j D i B xj,α 0 s j,, where x i is the limit of the x s as α +, Hα i is given by H i αx = + ln d g x, x 2.24 d g x, x in M\{x }, and Hαx i = 0 for all α and all i, δ i is as in.8, the D i s and R i,j s are as in.20.2, and r is the range of influence of x as in.. When r 0 as α +, it follows from.23 that there exist C > 0 and a sequence ε α α of positive real numbers converging to zero such that uα x B i αx εα B i αx + C r 2 + j D i B j αx.25 for all x N α and all α. From.23 and standard elliptic theory we then get that for any i {,..., k}, there exists C > 0 such that, up to a subsequence, u α C r 2 and u α C r 3.26 in B x 2δ i r \B x 2 δ ir, where δi is as in.8, and r is the range of influence of x as in.. We also get that there exists C > 0 such that, up to a subsequence, for any j D i, u α Cµ j,α s 2 j, and u α Cµ j,α s 3 j,.27 in B xj,α 5R i,j s j, \B xj,α 5 R i,js j,, where the Di s and R i,j s are given by Still by the C 0 -theory, we can prove that given i {,..., k}, if µ r = o µ α, then, up to a subsequence, r 2 µ u α exp x r z 8 in Cloc 2 B 0 2δ i \ {0} as α +, where H i z = lim α + r2 µ j A i B i,i A j z 2 + H iz.28 λ i,j z z i,j 2 + X i.29 is a smooth function in B 0 2δ i satisfying that H i 0 0, δ i is as in.8, the λ i,j s are as in.5, the z i,j s are as in.7, and the X i s are nonnegative real numbers given by 2 X i = u x i +,.30 j A i\b i Θ i lim α + r s i,j,α where we adopt the convention that the first term in the right hand side of.30 is zero if u 0, that the second term is zero if A i \B i Θ i =, and where Θ i = {j A i s.t. i A j }. 2. Sharp estimates on the range of influence We let M, g be a smooth compact Riemannian 4-manifold, and m 0, m, q > 0 be positive real numbers. We let ω m 0, +m 0, u α e iωαt be an arbitrary sequence of finite energy standing waves, u α 0, and v α α α be an arbitrary sequence of gauge potentials, satisfying that they solve.2 for all α, and that ω α ω in R as α +. We assume that.3 holds true, and we want to get a contradiction.

7 STABLE PHASES FOR THE KGMP SYSTEM 7 We follow the analysis in Druet and Hebey [4] but we need to face the lost of C - control on the h α s in.4 which is going to lead to quite serious difficulties. As in the preceding section, we order the blow-up points such that.3 holds true. We aim in proving that the following proposition holds true. µ Proposition 2.. Let i {,..., k}. If r = o µ α, where r is the range of influence of x as in., and µ α is as in.22, then, up to a subsequence, m 2 0 ω 2 6 S gx i + o r 2 ln = 2H i 0 + o, 2. where H i is as in.29, ω is the limit of the ω α s, and x i is the limit of the x s. Moreover, there holds that H i 0 0 if we assume in addition that m 2 0 ω 2 µ 6 RgS g. In case r o µ α, there holds that m 2 0 ω 2 6 S gx i + o where µ α is as in.22. µ 2 ln r = O µ α + o µ 2 ln, 2.2 We prove Proposition 2. by finite inverse induction on i. We let i {,..., k }, and in case i < k, we assume that for any j = i +,..., k, 2. holds true for j as soon as µ α r j,α = o µj,α. H i If i = k we do not assume anything. Assuming H i we aim to prove that 2. holds true for i. Let i {,..., k }, i < k, be arbitrary. Assuming that H i holds true we get that for any j D i, s 2 j, = O ln 2.3 µ j,α where D i is as in Indeed, if j D i, then j > i. Moreover, for any j D i, we have that i A j so that s j, r j,α, and we clearly have that s 2 j, = o µ j,α µ = o µj,α µ α. The first equality is obvious. For the second one we remark that if i >, then for j D i, d g x, x j,α 2 = o r 2 = o µ µ α since A i. In particular, µ α r j,α = o µj,α, and 2.3 is a direct consequence of H i since H i 0 0. In what follows we introduce the subsets Ω of M given by Ω = B x δ i r \ Ω i,j,α, 2.4 j D i where Ω i,j,α = B xj,α R i,j s j, 2.5 for all j D i, δ i is as in.8, and the D i s and R i,j s are given by It follows from.20 that the Ω i,j,α s are disjoint for α sufficiently large, and it is easily checked that s j, = or for all j D i. We also have that D i = if i = k so that no Ω i,j,α s have to be considered in that case. We define X α to be the smooth -form given by X α x = 6n Rc g x f α x, f α x f α x, 2.6

8 8 OLIVIER DRUET, EMMANUEL HEBEY, AND JÉRÔME VÉTOIS where f α x = 2 d g x, x 2, and Rc g is the 0, 2-tensor field we get from the 2, 0-Ricci tensor Rc g thanks to the musical isomorphism. We let X α u α be given by X α u α = X α, u α, where, is the pointwise scalar product for -forms. Applying the Pohozaev identity in Druet and Hebey [4] to the u α s in Ω with the -forms X α, we get that h α u α X α u α dv g + g div g X α u 2 Ω 8 αdv g Ω + div g X α h α u 2 4 αdv g Ω = Q α Q j α + R,α + R 2,α R j 2,α, j D i j D i 2.7 where, if ν = ν α stands for the unit outer normal to Ω, the Q α s are given by Q α = div g X α ν u α u α dσ g 4 B x δ ir X α ν u α 2 X α u α ν u α dσ g, B x δ ir the Q j α s are given by Q j α = div g X α ν u α u α dσ g 4 Ω i,j,α Ω i,j,α 2 X αν u α 2 X α u α ν u α dσ g, 2.9 where Ω i,j,α is as in 2.5, the R,α s are given by R,α = X α 4 div gx α g u α, u α dv g, 2.0 the R 2,α s are given by Ω R 2,α = 4 8 B x δ ir B x δ ir X α ν u 2 α dσ g ν div g X α u 2 αdσ g, 2. and the R j 2,α s are given by R j 2,α = X α ν u 2 α dσ g ν div g X α u 2 4 Ω i,j,α 8 αdσ g. 2.2 Ω i,j,α We split the proof of Proposition 2. in several lemmas. All what follows is up to a subsequence. The first lemma is contained in Druet and Hebey [4]. We note that for X α as in 2.6, X α x = O d g x, x, div g X α x 4 = O d g x, x 2, div g X α x = O d g x, x, and g div g X α x = 4 3 S g x + O d g x, x. 2.3

9 STABLE PHASES FOR THE KGMP SYSTEM 9 Then we compute the right hand side in 2.7. Lemma 2.. Let i {,..., k} be arbitrary, and in case i < k, assume that H i holds true. Let Iα i = h α u α X α u α dv g + g div g X α u 2 Ω 8 αdv g Ω div g X α h α u 2 4 αdv g, Ω where Ω is as in 2.4 and X α is as in 2.6. Then Iα i = 28ω 3 H i 0 + o µ 2 r 2 + o µ 2 ln µ if r = o µ 2 ln 2.5 µ α, and Iα i = O µ α + o otherwise, where r is the range of influence of x as in., H i is as in.29, and µ α is as in.22. The next lemma in the proof of Proposition 2. is as follows. Lemma 2.2. Let i {,..., k} be arbitrary, and in case i < k, assume that H i holds true. Let Iα i be as in 2.4. There holds that Iα i = 64ω 3 m 2 0 ω 2 6 S gx i + o µ 2 ln r + o µ 2 r 2 + o µ 2 ln µ O v α u 2 αdv g + O v α u α X α u α dv g, Ω Ω where r is the range of influence of x as in., µ α is as in.22, and x i is the limit of the x s. Proof of Lemma 2.2. Let h α be as in.4. Then, and we get by 2.3 and 2.7 that h α = m 2 0 ω 2 α + O v α 2.7 Iα i = m 2 0 ω 2 + o u α X α u α dv g Ω + g div g X α u 2 8 αdv g Ω + m ω 2 + o div g X α u 2 αdv g Ω + O Ω v α u 2 αdv g + O Using.23 it is easily checked that Ω d g x, x u 2 αdv g = o Ω v α u α X α u α dv g. 2.8 µ 2 ln. 2.9

10 0 OLIVIER DRUET, EMMANUEL HEBEY, AND JÉRÔME VÉTOIS Integrating by parts, thanks to and 2.3, there also holds that u α X α u α dv g = 2 u 2 αdv g + o µ 2 r 2 + o µ 2 ln Ω Ω Then, by , we get that Iα i = m 2 0 ω 2 6 S gx i + o u 2 αdv g Ω + o µ 2 r 2 + o µ 2 ln + O Ω v α u 2 αdv g + O Ω v α u α X α u α dv g. 2.2 We have that Bα i 2 dv g = o µ 2 ln 2.22 Ω i,j,α for all j D i. Thanks to 2.2 and 2.22, and thanks to.23, we then get that Iα i = m 2 0 ω 2 6 S g x i + o Bα i 2 dv g B x δ ir + o µ 2 r 2 + o µ 2 ln O v α u 2 αdv g + O v α u α X α u α dv g. Ω Ω We have that B x δ ir Bα i 2 dv g = 64ω 3 µ 2 ln r + o µ 2 ln Combining 2.23 and 2.24, this ends the proof of Lemma 2.2. At this point it remains to handle the two terms involving the gauge potentials v α in 2.6. We let A α = v α u 2 αdv g and B α = v α u α X α u α dv g 2.25 Ω Ω for all α. The following lemma reduces the problem of controlling these two terms to the problem of controlling the first term. Lemma 2.3. Let i {,..., k} be arbitrary, and in case i < k, assume that H i holds true. There exists C > 0 such that Bα 2 C µ 2 ln + o µ 2 µ r 2 A α 2.26 for all α, where A α and B α are as in 2.25, and r is the range of influence of x as in.. Proof of Lemma 2.3. By the Cauchy-Schwarz inequality, since the v α s are bounded in L, and by 2.3, there exists C > 0 such that B 2 α CA α Ω d g x, 2 u α 2 dv g 2.27

11 STABLE PHASES FOR THE KGMP SYSTEM for all α. There also exists C > 0 such that d g x, 2 u α 2 dv g C d g x, 2 u α Bα i 2 dv g Ω Ω + C d g x, 2 Bα i 2 dv g Ω for all α, and we can conclude by noting that Ω d g x, x 2 B i α 2 dvg = O 2.28 µ 2 ln, 2.29 and that by.23, using Hölder s inequalities, d g x, x 2 uα Bα i 2 dvg = o µ 2 r 2 + o µ 2 ln Ω This ends the proof of Lemma Now we search for estimates on the A α s in We split v α into a quasiharmonic and a quasi-poisson part. We define 3 ˆΩ = B x 2 δ ir \ 2 3 R i,js j,, 2.3 j D i B xj,α where δ i is as in.8, and the D i s and R i,j s are given by The B xj,α 2 3 R i,js j, s are disjoint for α and we have that Ω ˆΩ for all i and all α. Then we let v α = w,α + w 2,α 2.32 for all α, where w,α is given by { g w,α + m 2 w,α = 0 in ˆΩ w,α = v α on ˆΩ 2.33 for all α, w 2,α is given by { g w 2,α + m 2 w 2,α = W α in ˆΩ w 2,α = 0 on ˆΩ 2.34 for all α, and W α = g v α + m 2 v α = q qv α u 2 α, 2.35 for all α. As shown in Hebey and Truong [0], working at the macroscopic level, we easily get that there exists β 0, ] and C > 0 such that v α Cu β α in M 2.36 for all α. Now the next lemma establishes a first set of L -estimates for the quasiharmonic part of the decomposition of the v α s. Lemma 2.4. Let i {,..., k} be arbitrary, and in case i < k, assume that H i µ holds true. Suppose r o µ α, where r is the range of influence of x as in.. Then w,α L ˆΩ 0 as α +, where ˆΩ is as in 2.3, and w,α is as in

12 2 OLIVIER DRUET, EMMANUEL HEBEY, AND JÉRÔME VÉTOIS µ Proof of Lemma 2.4. Assuming that r o µ α, we get that By and 2.37 we then get that = O r 2 µ α = o r max u α = o B x 3 2 δir Similarly, by 2.3, which follows from H i, and by.26.27, there also holds that max u α = o B xj,α 2 3 Ri,jsj, By the maximum principle, 0 w,α max ˆΩ w,α 2.40 in ˆΩ, and since w 2,α 0, we get from 2.36 and 2.40 that 0 w,α max ˆΩ v α max ˆΩ u β α 2.4 in ˆΩ. Combining 2.38, 2.39, and 2.4, this ends the proof of Lemma 2.4. The next lemma provides a pointwise estimate for the w 2,α s. Lemma 2.5. There exists C > 0 such that µ 2 ln 2 + dgx,x2 µ w 2,α x C 2 µ 2 + d gx, x for all α and all x ˆΩ, where ˆΩ is as in 2.3, and w 2,α is as in Proof of Lemma 2.5. Let G α be the Green s function of g + m 2 in ˆΩ with zero Dirichlet boundary condition on ˆΩ, and G be the Green s function of the same operator in B xi 2 3 i g with zero Dirichlet boundary condition on B xi 2 3 i g, where x i is the limit of the x s and i g is the injectivity radius of M, g. Let x ˆΩ, and G α,x, G x be the functions defined for y x by G α,x y = G α x, y and G x y = Gx, y. Since G α,x G x on ˆΩ we get that G α,x G x in ˆΩ \{x}. In particular, see Robert [4], we get that there exists C > 0 such that G α x, y C d g x, y for all x, y ˆΩ, x y. There holds that w 2,α x = G α,x g w 2,α + m 2 w 2,α dvg ˆΩ = G α,x W α dv g, ˆΩ 2.44 where W α is as in 2.35, and since W α Cu 2 α, it follows from 2.43 and 2.44 that there exists C > 0 such that u 2 w 2,α x C αdv g d g x, ˆΩ

13 STABLE PHASES FOR THE KGMP SYSTEM 3 for all α and all x ˆΩ. By.23 we then get that from 2.45 that B w 2,α x C αy i 2 dv g y ˆΩ d g x, y 2 B C αy i 2 dv g y B x 3 2 δir d g x, y and 2.42 easily follows from 2.46 by noting that there exists C > 0 such that Bαy i 2 dv g y µ 2 ln 2 + dgx,x2 µ B x 3 2 δir d g x, y 2 C 2 µ 2 + d gx, x 2 for all x and all α. This ends the proof of Lemma 2.5. Thanks to Lemmas 2.4 and 2.5 we can prove the following. Lemma 2.6. Let i {,..., k} be arbitrary, and in case i < k, assume that H i µ holds true. Suppose r o µ α, where r is the range of influence of x as in.. Then v α u 2 αdv g = o µ 2 ln, 2.47 ˆΩ where ˆΩ is as in 2.3. Proof of Lemma 2.6. By Lemma 2.5, for any R > 0 there exists ε R > 0 such that w 2,α L ˆΩ \B x R ε R 2.48 for all α, and such that ε R 0 as R +. By.23, Lemma 2.4, and 2.48, we then get that v α u 2 αdv g v α u 2 αdv g + ε R u ˆΩ ˆΩ B x R ˆΩ 2 αdv g v α u 2 αdv g + ε R µ 2 ln 2.49 ˆΩ B x R for α. By Hölder s inequality, v α u 2 αdv g ˆΩ B x R 3/ vαdv g/4 4 u B x R ˆΩ B 8/3 x Rµ α dv g and by.23, there exists C > 0 such that u 8/3 α dv g Cµ 4/3 2.5 ˆΩ B x R for all α. Also there holds that vαdv 4 g CVol g Bx R Cµ B x R

14 4 OLIVIER DRUET, EMMANUEL HEBEY, AND JÉRÔME VÉTOIS for all α. By we then get that v α u 2 αdv g = O µ 2 ˆΩ B x R 2.53 and since ε R 0 as R +, we get 2.47 by combining 2.49 and This ends the proof of Lemma 2.6. Now we handle the cases where r µ = o µ α. The first lemma in this direction is as follows. Lemma 2.7. Let i {,..., k} be arbitrary, and in case i < k, assume that H i µ holds true. Suppose r = o µ α, where r is the range of influence of x as in.. Then for all α, where ε 0 > 0 is independent of α. r 2 ln r ε Proof of Lemma 2.7. We proceed by contradiction and assume that r 2 r ln 0 µ as α +. Since there also holds that r 0 when r = o µ α, we get that r 2 ln as α +. By.23, since the v α s are bounded, there exists C > 0 such that ˆΩ v α u 2 αdv g Cµ 2 ln 2.56 for all α. By Lemmas 2. to 2.3, and 2.56, there holds that 28ω 3 H i 0 + o µ 2 r 2 = O µ 2 ln Combining 2.55 and 2.57 it follows that H i 0 = 0, a contradiction with the fact that H i 0 0. This ends the proof of Lemma 2.7. From Lemma 2.7 we get that the following key estimate holds true. Lemma 2.8. Let i {,..., k} be arbitrary, and in case i < k, assume that H i µ holds true. Suppose r = o µ α, where r is the range of influence of x as in.. Then v α u 2 αdv g = o µ 2 ln, 2.58 ˆΩ where ˆΩ is as in 2.3. Proof of Lemma 2.8. By Lemma 2.7, r 2 C ln, and by and 2.36 we then get that there exists C > 0 such that v α C ln β 2.59

15 STABLE PHASES FOR THE KGMP SYSTEM 5 on B 3 x 2 δ ir, for all α. Still by and 2.36, it follows from 2.3, and thus from H i, that there exists C > 0 such that v α C µ j,α s 2 β j, C µ j,α ln β 2.60 µ j,α on B xj,α 2 3 R i,js j, for all j Di and all α. By the maximum principle, the quasi-harmonic part w,α of the decomposition of the v α s satisfies that 0 w,α max ˆΩ w,α and since w,α v α for all α, it follows from 2.59 and 2.60 that w,α L ˆΩ as α +. Now we handle the quasi-poisson part w 2,α of the decomposition of the v α s. By.23, since the v α s are bounded, there exists C 0 > 0 such that W α C 0 B i α 2 in ˆΩ for all α, where W α is as in Define w α by { g w α + m 2 w α = C 0 B i α 2 in B x 3 2 δ ir w α = 0 on B x 3 2 δ ir By the maximum principle, 0 w 2,α w α in ˆΩ for all α, and by.23 and Hölder s inequality we then get that /4 3/4 w 2,α u 2 αdv g w2,αdv 4 g u 8/3 α dv g ˆΩ ˆΩ ˆΩ 3/4 C wαdv g/4 4 Bα i 8/3 dv g B iα B iα 2.63 for all α, where B i α = B 3 x 2 δ ir. There holds that Bα i 8/3 dv g = O, 2.64 B iα µ 4/3 and we can write using 2.64 that there exists C > 0 such that w α 2 H B iα C C B iα B iα g w α + m 2 w α wα dv g B i α 2 w α dv g 3/4 C wαdv g/4 4 Bα i 8/3 dv g B iα B iα /4 C wαdv 4 g B iα 2.65 for all α. By the Euclidean Sobolev inequality, w α L4 B iα K w α L2 B iα for all α and some K > 0 independent of α, and by 2.65 it follows that /4 wαdv 4 g = O B iα

16 6 OLIVIER DRUET, EMMANUEL HEBEY, AND JÉRÔME VÉTOIS Combining 2.63, 2.64, and 2.66, we then get that w 2,α u 2 αdv g = O µ 2 = o µ 2 ln ˆΩ We easily get 2.58 from 2.6 and 2.67 since v α = w,α + w 2,α and since, by.23, u α 2 = O µ 2 L 2 ˆΩ ln. This ends the proof of Lemma 2.8. At this point we can prove Proposition 2.. This is the subject of what follows. Proof of Proposition 2.. First we let i {,..., k} be arbitrary, and in case i < k, we assume that H i holds true. By Lemma 2.3, Lemma 2.6, and Lemma 2.8, we always have that A α = o µ 2 ln B α = o µ 2 ln + o µ 2 r 2, and + o 2.68 µ 2 r 2, where A α and B α are as in As is easily checked, 2. and 2.2 follow from Lemma 2., Lemma 2.2, and It remains to prove that H i 0 0 if we assume that r = o µ µ α and that m 2 0 ω 2 6 RgS g. Let Y be an arbitrary -form in R n. We apply again the Pohozaev identity in Druet and Hebey [4] to u α in Ω, but we choose here X = X α to be given in the exponential chart at x by X α κ = Y κ 2 3 R κjklx x j x k Y l, where Y l = Y l for all l and the R κjkl are the components of the Riemann tensor Rm g at x in the exponential chart. As is easily checked, still in geodesic normal coordinates at x, X α κj = R κjkl x x k Y l + O x 2 so that we obtain div g X α = O x 2. Then, thanks to the symmetries of the Riemann tensor, we get with the Pohozaev identity that Ω 2 X α ν u α 2 X α u α ν u α dσ g + h α u α X α u α dv g Ω = O u 2 αdv g + O d g x, x 2 u α 2 dv g 2.69 Ω Ω + O u 2 α dσ g + O u 2 αdσ g + O ν u α u α dσ g, Ω Ω Ω where ν = ν α stands for the unit outer normal to Ω. Estimating the right-hand side of 2.69 thanks to 2.3, and thanks to.23 and.26, we get that Ω 2 X α ν u α 2 X α u α ν u α dσ g + h α u α X α u α dv g = O µ 2 ln + O 2.70 µ 2 Ω µ r 2.

17 STABLE PHASES FOR THE KGMP SYSTEM 7 By 2.7, Ω h α u α X α u α dv g = m 2 0 ω 2 α Ω u α X α u α dv g + O B α, 2.7 and, integrating by parts, it is easily checked that u α X α u α dv g = O µ 2 ln + O µ 2 Ω µ r 2. Coming back to 2.7, thanks to 2.68, it follows that Ω 2 X α ν u α 2 X α u α ν u α dσ g = O µ 2 ln + O 2.72 µ 2 µ r 2. By.26.29, thanks also to 2.3, we can write that 2 X α ν u α 2 X α u α ν u α dσ g Ω = 28ω 3 Y H i 0 + o µ 2 r 3 + ˆΘ α, where Y H i 0 = Y κ κ H i 0, and ˆΘ α = o µ 2 ln 3/ If we assume that m 2 0 ω 2 6 RgS g, then we get as a consequence of 2. that r ln /2 = O. Coming back to 2.72 and 2.73, it follows that Y H i 0 = 0, and since Y is arbitrary, we get that H i 0 0. This ends the proof of Proposition Proof of Theorem 0. We prove Theorem 0. in this section. We let M, g be a smooth compact Riemannian 4-manifold, and m 0, m, q > 0 be positive real numbers. We let ω m 0, +m 0, u α e iωαt be an arbitrary sequence of finite energy standing waves, α u α 0, and v α α be an arbitrary sequence of gauge potentials, satisfying that they solve.2 for all α, and that ω α ω in R as α +. We assume that m 2 0 ω 2 6 RgS g 3. and that.3 holds true, and we want to get a contradiction. As in the preceding section, we order the blow-up points such that.3 holds true. First we claim that u in.8 has to be zero. To prove this, we proceed by contradiction and assume that u 0. Then, by the definition of the range of influence, r,α = O µ α, where µ α = µ,α is as in.22. In particular r,α = o and we can apply Proposition 2. with i =. As we know, H 0 0 and, therefore, it follows from 2. in Proposition 2. that r 2,α ln µ α C, 3.2

18 8 OLIVIER DRUET, EMMANUEL HEBEY, AND JÉRÔME VÉTOIS where C > 0 is independent of α. Noting that 3.2 leads to a contradiction when combined with the estimate r,α = O µ α, this proves that u 0. Now we remark that by 3. and 2.2 in Proposition 2. there necessarily holds that r,α 0 as α +. In particular, A, where A is as in.2. Moreover, by 2. in Proposition 2. and 3. we get that for any i A {}, there exists C i > 0 such that r 2 ln C i 3.3 as α +. By 3.3, for any i A {}, = o r It follows from 3.4 that for any i A {}, A i B i, where the B i s are as in.6. By the explicit formula.30, H i z = j A i B i λ i,j z z i,j 2, 3.5 where H i is as in.29. Let E = A B {}. For any i A B, we have that A i B i = E \ {i}. Let i E be such that d g x,α, x d g x,α, x j,α for all j E and all α. Then the z i,j s all lie in a ball whose boundary contains zero. In particular they all lie in a half space and we get that there exists ν i R n, ν i =, such that ν i, z i,j > 0 for all j A i B i. By Proposition 2., H i 0.ν i = 0, and by 3.5 we have that H i 0.ν i = 2 j A i B i λ i,j z i,j 4 ν i, z i,j, a contradiction since λ ij > 0. This ends the proof of Theorem Proof of Theorem 0.2 We prove Theorem 0.2 in this section. The proof is based on a finite-dimensional reduction method, the so-called Lyapounov-Schmidt method introduced originally by Floer and Weinstein [8] in the one-dimensional case. An early reference on the Lyapounov-Schmidt method for a Sobolev critical equation is Rey [2]. Different techniques based on the finite dimensional reduction method have been developed for such problems such as, among others, the localized energy method see del Pino, Felmer, and Musso [3], Rey and Wei [3], and Wei [6]. Here, we use the Lyapunov Schmidt method with respect to the H -norm, as it is used, for instance, in Esposito, Pistoia, and Vétois [7] for the Yamabe equation. In our case, on the sphere, we restrict ourselves to radial functions in H. We let S 4, g be the round 4-sphere, m 0, m, q > 0 be positive real numbers such that m 2 0 2, and ω m 0, +m 0. We let Φ be the map given by.. Then, see Hebey and Truong [0], Φ is locally Lipschitz and differentiable, and its differential DΦu = V u at u is given by g V u ϕ + m 2 + q 2 u 2 V u ϕ = 2qu qφu ϕ 4.

19 STABLE PHASES FOR THE KGMP SYSTEM 9 for all ϕ H M. There also holds that 0 Φ q in H. Letting I ω : H R be given by I ω u = u 2 g 2 dv g + S 2 m2 0 ω 2 u 2 dv g 4 S 4 u 4 4 +dv g + q 4.2 S 2 ω2 Φuu 2 dv g 4 S 4 for all u H, where u + = maxu, 0, it follows that I ω is C in H and that its critical points u H are such that g u + m 2 0u = u ω 2 qφu 2 u. 4.3 By the maximum principle, since m 2 0 > ω 2 qφu 2, we then get that u 0. In particular, u, Φu solves 0.2. We define ε ω = 2 + ω 2 m so that ε ω 0 if and only if ω 2 m We fix x 0 S 4, and we define the radial Sobolev space H r S 4 = { u H S 4 s.t. u σ = u for all σ G x0 }, where G x0 is the group of rotations about x 0. In particular, Hr H is a closed subspace of H. Similarly, for any p, we define the radial Lebesgue space L p rs 4 = { u L p S 4 s.t. u σ = u for all σ G x0 }. We let L g be the conformal Laplacian on S 4, g given by L g = g + 2. As is easily checked by using standard minimization technics, for any f L 4/3 r, there exists a unique u = L g f, u Hr, such that L g u = f. Given u Hr we define f ω u = u ε ω u qω 2 2 qφu Φuu, 4.5 where Φ is given by., and ε ω is as in 4.4. Now, given δ > 0, we let W δ be the fundamental solution of the conformal critical equation. Then 2δ W δ = δ2 cos d g x 0, x and, as is well known, W δ satisfies that L g W δ = W 3 δ 4.7 in S 4 for all δ > 0. There also holds, see for instance Druet, Hebey and Robert [6], that µ δ W δ = µ 2 δ + + R δ 4.8 dgx0, 2 8 δ for all δ > 0, where µ δ =, and R 2+ +δ2 δ 0 in H as δ 0. We aim to construct solutions of 4.3 of the form u = W δ + ψ δ. We define Z δ H r by Z δ = δ dw δ. 4.9 dδ By Bianchi-Egnell [2], Z δ is, up to the product by a constant, the only solution in Hr of the linearized equation associated to 4.7, and thus the only solution in Hr of L g Z δ = 3Wδ 2 Z δ. 4.0

20 20 OLIVIER DRUET, EMMANUEL HEBEY, AND JÉRÔME VÉTOIS We let, Lg be the scalar product associated with L g so that u, v Lg = u v + 2uv dv g 4. S 4 for all u, v Hr, and let Lg be the corresponding norm. Then we define Z δ = { u H r s.t. u, Z δ Lg = 0 }, 4.2 and let Π δ : H r Zδ be the projection onto Z δ. It follows that Π δ u = u u, Z δ Lg Z δ 2 Z δ L g for all u. By 4.7 and 4.0, W δ Zδ for all δ > 0. At last we define L δ,ω, N δ,ω, and R δ,ω by L δ,ω u = Π δ u L g Df ω W δ.u, N δ,ω = Π δ L g f ω W δ + u f ω W δ Df ω W δ.u, 4.3 R δ,ω = Π δ L g f ω W δ W δ for all u Hr, where f ω is given by 4.5. We split the proof of Theorem 4 in several lemmas. The first lemma consists in the following rewriting of equation 4.3. Lemma 4.. A function ψ Z δ is such that u = W δ + ψ solves 4.3 in Z δ if and only if L δ,ω ψ = N δ,ω ψ + R δ,ω, 4.4 where L δ,ω, N δ,ω, and R δ,ω are as in 4.3. Proof of Lemma 4.. Let u = W δ + ψ δ. As we easily check, ψ satisfies 4.4 if and only if Π δ u L g f ω u = 0, and thus if and only if u L g f ω u = αz δ, where α R. In particular, ψ satisfies 4.4 if and only if g u + m 2 0u = u ω 2 qφu 2 u + αl g Z δ, and thus if and only if u solves 4.3 in Zδ. This ends the proof of Lemma 4.. Now we prove that the following estimate holds true. Lemma 4.2. There exist δ 0 > 0 and ε 0 > 0 such that for any δ 0, δ 0 and any ε 0, ε 0, L δ,ω : Z δ Z δ is invertible. Moreover, there exists C > 0, independent of δ and ω, such that for all δ 0, δ 0, ε 0, ε 0, and ψ Z δ. L δ,ω ψ Lg C ψ Lg 4.5 Proof of Lemma 4.2. In order to apply the Fredholm alternative, we show that for any δ > 0 and ω R fixed, the map L δ,ω : ψ ψ L δ,ω ψ is compact. We let ψ α α be a bounded sequence in Zδ. For any α N, we define the function ϕ α = L g Df ω W δ.ψ α so that L δ,ω ψ α = Π δ ϕ α. Since ψ α α N is bounded in H, we get from 4. that Df ω W δ.ψ α α is bounded in H. By elliptic regularity theory, it follows that ϕ α α N is compact in H, and thus that L δ,ω ψ α α is compact in Zδ. This proves that the map L δ,ω is compact. Now, by the Fredholm alternative, in order to get the invertibility of L δ,ω, it suffices to prove that the

21 STABLE PHASES FOR THE KGMP SYSTEM 2 kernel of L δ,ω is reduced to {0}, which is a consequence of 4.5. We prove 4.5 by contradiction. We assume that there exist δ α α, ω α α, and ψ α α such that δ α 0 and ε ωα 0 as α +, such that ψ α Z δ α and ψ α Lg = for all α, and such that L δα,ω α ψ α Lg as α +. We claim that ψα L g Df ωα W δα.ψ α Lg = o. 4.7 By 4.6, in order to obtain 4.7 it suffices to prove that ψα L g Df ωα W δα.ψ α, Z δα L g = o 4.8 By 4.0 we get that ψα L g Df ωα W δα.ψ α, Z δα L g = S 4 3W 2 δα ψ α Df ωα W δα.ψ α Zδα dv g, and by Hölder s inequality and Lemma 4.3 it follows that ψ α L g Df ωα W δα.ψ α, Z δα L g = O 3Wδ 2 α ψ α Df ωα W δα.ψ α L 4/3 Z δα L = O 3Wδ 2 α ψ α Df ωα W δα.ψ α L 4/3. By the definition 4.5 of f ωα there holds that 3W 2 δ α ψ α Df ωα W δα.ψ α = ε ωα ψ α + qω 2 α 2 qφw δα ΦW δα ψ α + 2qω 2 α qφw δα W δα DΦW δα.ψ α and since 0 ΦW δα q, ε ω α 0 as α +, ψ α Lg = for all α, and W δ L 2 0 as δ 0, it follows that 3W 2 δα ψ α Df ωα W δα.ψ L α 4/3 By. and 4., and = o ψ α L 4/3 + O ΦW δα H ψ α H + O W δα DΦW δα.ψ α L 4/3 = o ψ α H + O ΦW δα H ψ α H + O W δα L 2 DΦW δα.ψ α H = o + O ΦW δα H + o DΦW δα.ψ α H ΦW δα H = O W δα 2 L 8/3 = o 4.2 DΦW δα.ψ α H = O W δα L 2 ψ α H = o ψ α H = o By 4.2 and 4.22, coming back to 4.20, we obtain that 3W 2 δ α ψ α Df ωα W δα.ψ α L 4/3 = o. 4.23

22 22 OLIVIER DRUET, EMMANUEL HEBEY, AND JÉRÔME VÉTOIS Then, by 4.9 and 4.23, we deduce 4.8, and thus also 4.7. Then, by 4.7 and 4.23, for any sequence ϕ α α in H, ψ α, ϕ α Lg = 3 Wδ 2 α ψ α ϕ α dv g + o ϕ α H S 4 Given ϕ Cc R 4, we apply 4.24 with ϕ α given by ϕ α x = ϕ exp x δ α δ 0 x α for all x S 4. In particular, ϕ α H = O, and by using 4.8, we obtain from 4.24 that ψ ψα ϕ α ϕdx = 24 R 4 R + x 2 dx + o, where ψ α is given by ψ α x = δ α ψ α expx0 δ α x. Since ψ α H = O, there holds that ψ α L 2 = O, and we deduce that, up to a subsequence, ψα ψ in Ḣ as α +. Passing into the limit in 4.25, and since 4.25 holds for all ϕ Cc R 4, we obtain that ψ solves the equation ψ 24 = + x 2 ψ in R 4. We have that ψ α Hr by the definition of Zδ α in 4.2, and thus ψ is radially symmetric. By Bianchi and Egnell, the sole radially symmetric solutions of 4.26 are of the form ψ x = λ x 2 + x for some λ R. At this point we claim that λ = 0. Since ψ α Zδ α, and Z δα satisfies 4.0, we have that Wδ 2 α Z δα ψ α dv g = S 4 for all α. Rescaling 4.28, splitting the integral over B 0 R and R 4 \B 0 R, using Lemma 4.3 for the noncompact part of the integral, we obtain that lim R + B 0R x 2 ψ x + x 2 4 dx = Then, combining 4.27 and 4.29, it follows that λ = 0 and then that ψ 0. Finally we apply 4.24 with ϕ α = ψ α. We obtain that ψ α 2 L g = Wδ 2 α ψαdv 2 g + o. S 4 We may assume that ψ α ψ in L 2 of any compact subset of R 4. Here again, by rescaling and splitting the integral over B 0 R and R 4 \B 0 R, and since we just got that ψ 0, we obtain that ψ α 2 L g 0 as α +, in obvious contradiction with the fact that ψ α Lg = for all α. This proves 4.5 and Lemma 4.2. The following estimate on the L g -norm of Z δ was used in the proof of Lemma 4.2. It will be used again in the concluding argument in the proof of Theorem 0.2. We state it as a lemma.

23 STABLE PHASES FOR THE KGMP SYSTEM 23 Lemma 4.3. There holds that Z δ Lg = 6ω 3 + o 5 as δ 0, where Z δ is given by 4.9, and ω 3 is the volume of the round 3-sphere. Proof of Lemma 4.3. Let β = + δ 2. We compute 2δ β cos r Z δ = ββ cos r 2, where r = d g x 0,. Independently, by 4.0, we obtain that Z δ 2 L g = 3 Wδ 2 Zδ 2 dv g. S 4 The result follows from direct computations noting that in geodesic normal coordinates r 3 dv g = sin r 3 dx. This ends the proof of Lemma 4.3. Now we define the map T δ,ω : Z δ Z δ given by for all ψ Zδ. By Lemma 4., solving 4.3 in Z δ of T δ,ω. T δ,ω ψ = L δ,ω N δ,ωψ + R δ,ω 4.30 amounts to finding a fixed point Lemma 4.4. There exist δ 0 > 0 and ε 0 > 0 such that for any δ 0, δ 0 and any ε 0, ε 0, the map T δ,ω in 4.30 has a fixed point ψ δ,ω Zδ. Moreover, there exists C > 0 such that ψ δ,ω H C ε ω δ + δ 2 ln δ 4.3 for all δ 0, δ 0 and all ε 0, ε 0. In addition, the map δ, ω ψ δ,ω, with values in H, is C with respect to δ and ω. Proof of Lemma 4.4. First we claim that there exists C > 0 such that R δ,ω H C ε ω δ + δ 2 ln δ for all δ > 0 and all ε > 0. It is easily checked that there exists C > 0 such that ϕ H C L g ϕ L 4/ for all ϕ H. Since W δ solves 4.7, Π δ, and 0 ΦW δ q,we then get that R δ,ω H C f ω W δ L g W δ L 4/ C ε ω W δ L 4/3 + ΦW δ W δ L 4/3 for all δ > 0 and ε ω > 0, where C > 0 is independent of δ and ε ω. There holds that ΦW δ W δ L 4/3 = O ΦW δ H W δ L 2 = O W δ 2 L W 8/3 δ L by 4.2. Direct computations give that W δ L 4/3 = Oδ, W δ L 2 = O δ ln δ Then 4.32 follows from theorem to the map T δ,ω in the set B δ,ω Λ = { ψ Z δ s.t. ψ H Λ, W δ L 8/3 = O δ Now, we aim to apply the fixed point ε ω δ + δ 2 } ln δ

24 24 OLIVIER DRUET, EMMANUEL HEBEY, AND JÉRÔME VÉTOIS for some Λ > 0. By Lemma 4.2, there exist δ 0 > 0, ε 0 > 0, and C > 0 such that T δ,ω ψ H C N δ,ω ψ H + R δ,ω H 4.36 for all δ 0, δ 0, ε ω 0, ε 0, and all ψ Zδ, and such that T δ,ω ψ T δ,ω ψ 2 H C N δ,ω ψ N δ,ω ψ 2 H 4.37 for all δ 0, δ 0, ε 0, ε 0, and all ψ, ψ 2 Zδ. Direct computations, using Hölder s inequalities, and estimates like in 4.2 and 4.22, then give that there exists ε δ > 0, ε δ = ε δ Λ, such that ε δ 0 as δ 0, and such that N δ,ω ψ N δ,ω ψ 2 H ε δ ψ ψ 2 H 4.38 for all δ 0, δ 0, ε ω 0, ε 0, and ψ, ψ 2 B δ,ω Λ. By 4.32, 4.36, and 4.38, taking ψ 2 0 and ψ = ψ in 4.38, we get there exists δ 0 > 0 small, ε 0 > 0 small, and Λ > 0 sufficiently large, such that T δ,ω : B δ,ω Λ B δ,ω Λ for all δ 0, δ 0 and ε ω 0, ε 0. In addition, 4.37 and 4.38 give that T δ,ω is a contraction map in B δ,ω Λ. By the fixed point theorem it follows that T δ,ω admits a fixed point ψ δ,ω B δ,ω Λ. This proves both the existence of the fixed point and 4.3. It remains to prove that ψ δ,ω is C with respect to δ and ω. To be more precise, we prove the regularity of ψ δ,ω with respect to δ and ε ω by applying the implicit function theorem to the map G : 0, + 2 H H given by Gδ, ε ω, ψ = ψ Π δ ψ Π δ L g fω Π δ W δ + Π δ ψ for all ψ H. It is easily checked that G is C with respect to its three variables, while we just proved that Gδ, ε ω, W δ + ψ δ,ω = 0 when δ > 0 and ε ω > 0 are sufficiently small. There holds that D ψ Gδ, ε ω, W δ + ψ δ,ω.v = v Π δ L g Dfω W δ + ψ δ,ω.π δ v for all v H. Using the Fredholm alternative, we obtain the invertibility of the differential D ψ Gδ, ε ω, W δ + ψ δ,ω if we prove that its kernel is reduced to {0}. Direct computations, similar to the ones needed to get 4.38, give that there exists C > 0 such that D ψ Gδ, ε ω, W δ + ψ δ,ω.v H C v H for all v H. In particular, KerD ψ Gδ, ε ω, W δ + ψ δ,ω = {0}, and by the implicit function theorem, this ends the proof Lemma 4.4. Now we define the reduced functional I ω by I ω δ = I ω W δ + ψ δ,ω, 4.39 where I ω is as in 4.2 and ψ δ,ω is given by Lemma 4.4. We prove that the following holds true. Lemma 4.5. There holds that for all δ > 0 small and ε ω > 0 small. I ω δ = I ω W δ + O ε 2 ωδ 2 + δ 4 ln δ 4.40

25 STABLE PHASES FOR THE KGMP SYSTEM 25 Proof of Lemma 4.5. Using that W δ is a solution of 4.7, we get that I ω δ I ω W δ = ψ δ,ω 2 dv g + 2 S 2 2 ε ω ψδ,ωdv 2 g 4 S 4 ε ω W δ ψ δ,ω dv g S 4 Wδ + ψ δ,ω 4 + Wδ 4 4W 3 δ ψ δ,ω dvg 4 S q 2 S ω2 ΦW δ + ψ δ,ω ΦW δ W δ + ψ δ,ω 2 dv g 4 + q 2 ω2 ΦW δ ψδ,ωdv 2 g + qω 2 ΦW δ W δ ψ δ,ω dv g. S 4 S 4 For any a > 0 and b real numbers, a + b 4 + a 4 4a 3 b Ca 2 + b 2 b 2 for some C > 0 independent of a and b. By Hölder s and Sobolev inequalities, it follows that S 4 Wδ + ψ δ,ω 4 + W 4 δ 4W 3 δ ψ δ,ω dvg = O W δ 2 H + ψ δ,ε 2 H ψ δ,ω 2 H = O ψ δ,ω 2 H We also have that ψ δ,ω 2 dv g + 2 S 2 2 ε ω ψδ,ωdv 2 g = O ψ δ,ω 2 H, S 4 and by Hölder s and Sobolev inequalities, we get that that and that By., S 4 W δ ψ δ,ω dv g = O W δ L 4/3 ψ δ,ω H, 4.44 ΦW δ + ψ δ,ω ΦW δ W δ + ψ δ,ω 2 dv g S 4 = O 4.45 ΦW δ + ψ δ,ω ΦW δ H Wδ 2 L + ψ 8/3 δ,ω 2 H, S 4 ΦWδ ψ 2 δ,ω + qω 2 ΦW δ W δ ψ δ,ω dvg = O ΦW δ H ψ δ,ω H ψ δ,ω L 2 + W δ L ΦW δ + ψ δ,ω ΦW δ H = O W δ L 2 + ψ δ,ω L 2 ψ δ,ω H, 4.47 and it follows from 4.2, 4.35, and that I ω δ I ω W δ = O ψ δ,ω 2 H + ε ω δ + δ 2 ln δ ψ δ,ω H Then 4.40 follows from 4.3 and This ends the proof of Lemma 4.5. Now we compute I ω W δ.

26 26 OLIVIER DRUET, EMMANUEL HEBEY, AND JÉRÔME VÉTOIS Lemma 4.6. There holds that I ω W δ = 8 3 π2 8π 2 ε ω δ 2 ln δ + q 2 ω2 S 4 ΦW δ W 2 δ dv g + o ε ω δ 2 ln δ 4.49 as δ 0 uniformly with respect to ω, and there exists C > such that C δ2 ΦW δ Wδ 2 dv g Cδ S 4 for all δ > 0 small. Proof of Lemma 4.6. By 4.7, I ω W δ = Wδ 4 dv g 4 S 2 ε ω Wδ 2 dv g + q 4 S 2 ω2 ΦW δ Wδ 2 dv g S 4 Direct computations give that Wδ 4 dv g = 32 S 3 π2 and Wδ 2 dv g = 6π 2 δ 2 ln δ + oδ 2 ln δ S 4 Then 4.49 follows from 4.5 and Independently, testing equation. with u = W δ against ΦW δ, we get by Hölder s and Sobolev inequalities that ΦW δ 2 H S = O ΦW δ Wδ 2 dv g = O W δ 2 L ΦW 8/3 δ H and testing. with u = W δ against W δ, we get that Wδ 3 dv g = O W δ H ΦW δ H + W δ 3 L 4 ΦW δ L 4 S 4 = O ΦW δ H Noting that there exists C > 0 such that W 3 S 4 δ dv g Cδ for all δ > 0 small, it follows from 4.35, 4.53, and 4.54 that 4.50 holds true. This ends the proof of Lemma 4.6. Now, we prove the following lemma establishing the relation between being a critical point of the reduced functional I ω and getting a solution u δ,ω = W δ + ψ δ,ω of 4.3. Lemma 4.7. If δ > 0 and ε ω > 0 are small and I δ = 0, then u δ,ω = W δ + ψ δ,ω is a solution of 4.3. Proof of Lemma 4.7. We let δ α α and ω α α be such that I ω α δ α = 0 and ε ωα > 0 for all α, and such that δ α 0 and ε ωα 0 as α +. By Lemmas 4. and 4.4, if we let u α = W δα + ψ δα,ω α, then u α L g f ωα u α = λ α Z δα for some λ α R, and u α solves 4.3 if and only if λ α = 0 the inequality u α 0 follows from the maximum principle when λ α = 0. There holds that I ω α δ α = λ α Zδα, du δ,ω α δ=δα dδ L g In particular, by 4.55, we get that either u α solves 4.3, or Zδα, du δ,ω α δ=δα = dδ L g

27 STABLE PHASES FOR THE KGMP SYSTEM 27 We assume 4.56 by contradiction. Since ψ δ,ωα Zδ for all δ > 0, we get by 4.3 that Zδα, dψ δ,ω α δ=δα dδ L g = dz δ dδ δ=δ α, ψ δα,ω α L g = O dz 4.57 δ dδ δ=δ α H ψ δα,ω α H. Direct computations give that dz δ dδ δ=δ α = O H Noting that Zδα, du δ,ω α δ=δα dδ L g = Z δα 2 H δ + Z δα, dψ δ,ω α α dδ we then get by 4.3, and by , that Z δα 2 H = O ψ δα,ω δ α H α The contradiction follows from Lemma 4.3 and Lemma 4.7. δ 2 α δ=δα L g, = o This ends the proof of At this point we are ready to prove Theorem 0.2. This is the subject of what follows. Proof of Theorem 0.2. We may assume that m 2 0 > 2. In case m 2 0 = 2, then ω = 0, and letting ω α = 0 we get the desired sequence of solutions with u α = W δα for any sequence δ α α of positive real numbers converging to zero. Now, for any t > 0 and ω R such that ε ω > 0, where ε ω is as in 4.4, we let δ ω t = e εω t In particular, δ ω t 0 uniformly in compact subsets of 0, + as ε ω 0. By 4.40 and 4.49 in Lemmas 4.5 and 4.6 there holds that I ω δ ω t = 8 3 π2 + E ω t + o e 2t εω 4.6 uniformly in compact subsets of 0, + as ε ω 0, where E ω t = 8π 2 te 2t εω + q 2 ω2 S 4 ΦW δt W 2 δt dv g By continuity, for any K >, there exists t K,ω [ K, K] such that I ω δ ω t K,ω = min I ω δ ω t, 4.63 t [ K,K] and, by the definition of ε ω, we have that ε ω 0 if and only if ω 2 m By 4.50 in Lemma 4.6, 8π 2 t + qω2 e 2t εω E ω t 8π 2 t + qcω2 e 2t εω C 2

28 28 OLIVIER DRUET, EMMANUEL HEBEY, AND JÉRÔME VÉTOIS In particular, we get by 4.64 that there exists 0 < t < t 2 < t 3, depending only on C and m 0, such that I ω δ ω t > 8 3 π2 for all t t, I ω δ ω t 2 < 8 3 π2, and I ω δ ω t > I ω δ ω t 2 for all t t 3, 4.65 uniformly with respect to ω for 0 < ε ω sufficiently small. Letting K > be such that [t, t 3 ] K, K, we get with 4.65 that t ω = t K,ω satisfies that K < t ω < K for all ω such that 0 < ε ω. Then, by Lemma 4.7, u δωt ω,ω = W δωt ω + ψ δωt ω,ω 4.66 is a solution of 4.3. We know from 4.3 in Lemma 4.4 that ψ δωt ω,ω H 0 as ε ω 0, and by combining 4.8 and 4.66, we get that u δωt ω,ω has one bubble in its H -decomposition as ε ω 0. By elliptic theory, this automatically implies that u δωt ω,ω L + as ε ω 0. Theorem 0.2 is proved. References [] Benci, V., and Fortunato, D., Solitary waves of the nonlinear Klein-Gordon field equation coupled with the Maxwell equations, Rev. Math. Phys , [2] Bianchi, G., and Egnell, H., A note on the Sobolev inequality, J. Funct. Anal., 00, 8 24, 99. [3] del Pino, M., Felmer, P., and Musso, M., Two-bubble solutions in the super-critical Bahri- Coron s problem, Calc. Var. Partial Differential Equations, 6, 3 45, [4] Druet, O., and Hebey, E., Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium, Analysis and PDEs, 2, 2009, [5], Existence and a priori bounds for electrostatic Klein-Gordon-Maxwell systems in fully inhomogeneous spaces, Commun. Contemp. Math., 2, , 200. [6] Druet, O., Hebey, E., and Robert, F., Blow-up theory for elliptic PDEs in Riemannian geometry, Mathematical Notes, Princeton University Press, vol. 45, [7] Esposito, P., Pistoia, A., and Vétois, J., The effect of linear perturbations on the Yamabe problem. Preprint at arxiv: [8] Floer, A., Weinstein, A., Nonspreading wave packets for the cubic Schrodinger equation with a bounded potential, J. Funct. Anal., 69 no. 3, , 986. [9] Hebey, E., Compactness and stability for nonlinear elliptic equations, Zürich Lectures in Advanced Mathematics, European Mathematical Society, To appear. [0] Hebey, E., and Truong, T., Static Klein-Gordon-Maxwell-Proca systems in 4-dimensional closed manifolds, J. reine angew. Math., 667, 202, [] Hebey, E., and Wei, J., Resonant states for the static Klein-Gordon-Maxwell-Proca system, Math. Res. Lett., To appear. [2] Rey, O., The role of the Green s function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89, no., 52, 990. [3] Rey, O., and Wei, J., Blowing up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. I. N = 3, J. Funct. Anal., 22, , [4] Robert, F., Green s Functions estimates for elliptic type operators, Preprint, [5] M. Struwe, A global compactness result for elliptic boundary problems involving limiting nonlinearities, Math. Z., 87, 984, [6] Wei, J., Existence and stability of spikes for the Gierer-Meinhardt system, Handbook of differential equations: stationary partial differential equations, Vol. V, , Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008.