The role of the scalar curvature in some singularly perturbed coupled elliptic systems on Riemannian manifolds

Given a 3-dimensional Riemannian manifold (M,g), we investigate the existence of positive solutions of singularly perturbed Klein-Gordon-Maxwell systems and Schroedinger-Maxwell systems on M, with a subcritical nonlinearity. We prove that when the perturbation parameter epsilon is small enough, any stable critical point x_0 of the scalar curvature of the manifold (M,g) generates a positive solution (u_eps,v_eps) to both the systems such that u_eps concentrates at xi_0 as epsilon goes to zero.

Given real numbers ε > 0, a > 0, q > 0, ω ∈ (− √ a, √ a) and 2 < p < 6, we consider the following singularly perturbed electrostatic Klein-Gordon-Maxwell system (1)    −ε 2 ∆ g u + au = u p−1 + ω 2 (qv − 1) 2 u in M −∆ g v + (1 + q 2 u 2 )v = qu 2 in M u, v > 0 and the Schrödinger-Maxwell system KGM systems and SM systems provide a model for the description of the interaction between a charged particle of matter u constrained to move on M and its own electrostatic field v.
The Schrödinger-Maxwell and the Klein-Gordon-Maxwell systems have been object of interest for many authors.
In the pioneering paper [6] Benci-Fortunato studied the following Schrödinger-Maxwell system −∆u + u + ωuv = 0 in Ω ⊂ R 3 or in R 3 −∆v = γu 2 u = v = 0 on ∂Ω Regarding the system in a semiclassical regime (here ε is a positive parameter small enough) Ruiz [34] and D'Aprile-Wei [13] showed the existence of a family of radially symmetric solutions respectively for Ω = R 3 or a ball. D'Aprile-Wei [14] also proved the existence of clustered solutions in the case of a bounded domain Ω in R 3 . Ghimenti-Micheletti [21] give an estimate on the number of solutions of (3). Moreover, when ε = 1 we have results of existence and nonexistence of solutions for pure power nonlinearities f (v) = |v| p−2 v, 2 < p < 6 or in presence of a more general nonlinearity (see [1,2,3,5,11,24,25,33,36]). In particular, Siciliano [35] proves an estimate on the number of solution for a pure power nonlinearity when p is close to the critical exponent.
As far as we know, the first result concerning the Klein-Gordon systems on manifold is due to Druet-Hebey [19]. They prove uniform bounds and the existence of a solution for the system (1) when ε = 1, a is positive function and the exponent p is either subcritical or critical, i.e. p ∈ (2,6]. In particular, the existence of a solution in the critical case, i.e. p = 6, is obtained provided the function a is suitable small with respect to the scalar curvature of the metric g. Recently, Ghimenti-Micheletti [20] give an estimate on the number of low energy solution for the system (1) in terms of the topology of the manifold.
In this paper, we show that the existence and the multiplicity of solutions of both systems (1) and (2) in the subcritical case when ε is small enough is strictly related to the geometry of the manifold (M, g). More precisely, we prove that the number of solutions to (1) or (2) is affected by the number of stable critical points of the scalar curvature S g of the metric g. Indeed, our result reads as follows. Theorem 1. Assume K is a C 1 -stable critical set of S g . Then there existsε > 0 such that for any ε ∈ (0,ε) the KGM system (1) and the SM system (2) have a solution (u ε , v ε ) such that u ε concentrates at a point ξ 0 ∈ K as ε goes to 0. More precisely, there exists a point ξ ε ∈ M such that if ε goes to zero where the function W ε,ξε is defined in (17).
We recall the the definition of C 1 -stable critical set.
Definition 2. Let f ∈ C 1 (M, R). We say that K ⊂ M is a C 1 -stable critical set of f if K ⊂ {x ∈ M : ∇ g f (x) = 0} and for any µ > 0 there exists δ > 0 such that, then h has a critical point ξ with d g (ξ, K) ≤ µ. Here d g denotes the geodesic distance associated to the Riemannian metric g.
It is easy to see that if K is the set of the strict local minimum (or maximum) points of f , then K is a C 1 -stable critical set of f . Moreover, if K consists of nondegenerate critical points, then K is a C 1 -stable critical set of f .
By Theorem 1 we deduce that multiplicity of solutions of (1) and (2) is strictly related to stable critical points of the scalar curvature. At this aim, it is useful to point out that Micheletti-Pistoia [31] proved that, generically with respect to the metric g, the scalar curvature S g is a Morse function on the manifold M. More precisely, they proved Then generically with respect to the metric g, the critical points of the scalar curvature S g are nondegenerate, in a finite number and at least P 1 (M ) where P t (M ) is the Poincaré polynomial of M in the t variable. Therefore, we can conclude as follows.
Corollary 4. Generically with respect to the metric g, if ε is small enough the KGM system (1) and the SM system (2) have at least P 1 (M ) positive solutions (u ε , v ε ) such that u ε concentrates at one nondegenerate critical point of the scalar curvature S g as ε goes to 0.
The proof of our results relies on a very well known Ljapunov-Schmidt reduction. In Section 2 we recall some known results, we write the approximate solution, we sketch the proof of the Ljapunov Schmidt procedure and we prove Theorem 1. In Section 3 we reduce the problem to a finite dimensional one, while in Section 4 we study the reduced problem. In Appendix A we give some important estimates. All the proofs are given for the system (1), but it is clear that up some minor modifications they also hold true for the system (2).

2.
Preliminaries and scheme of the proof of Theorem 1 2.1. The function Ψ. First of all, we reduce the system to a single equation. In order to overcome the problems given by the competition between u and v, using an idea of Benci and Fortunato [7], we introduce the map Ψ : It follows from standard variational arguments that Ψ is well-defined in H 1 g (M ) as soon as λ := a − ω 2 > 0, i.e. ω ∈] − √ a, + √ a[. By the maximum principle and by regularity theory is not difficult to prove that (5) 0 < Ψ(u) < 1/q for all u in H 1 g (M ). Moreover, it holds true that Lemma 5. The map Ψ : .
Also, we have For the proofs of these results we refer to [19]. Now, we introduce the functionals I ε , J ε , G ε : Here F (u) := 1 p (u + ) p , so that F ′ (u) = f (u) := (u + ) p−1 . By Lemma 6 we deduce Therefore, if u is a critical point of the functional I ε we have In particular, if u = 0 by the maximum principle and by the regularity theory we have that u > 0. Thus the pair (u, Ψ(u)) is a solution of Problem (1). Finally, the problem is reduced to find a solution to the single equation (10).

2.2.
Setting of the problem. In the following we denote by B g (ξ, r) the geodesic ball in M centered in ξ with radius r and by B(x, r) the ball in R 3 centered in x with radius r. It is possible to define a system of coordinates on M called normal coordinates. We denote by g ξ the Riemannian metric read in B(0, r) ⊂ IR 3 through the normal coordinates defined by the exponential map exp ξ at ξ. We denote |g ξ (z)| := det (g ij (z)) and g ij ξ (z) is the inverse matrix of g ξ (z). In particular, it holds (11) g ij ξ (0) = δ ij and ∂g ij ξ ∂z k (0) = 0 for any i, j, k.
Here δ ij denotes the Kronecker symbol. We denote by u 2 g :=M |∇ g u| 2 + u 2 dµ g and |u| q g :=M |u| q dµ g the standard norms in the spaces H 1 g (M ) and L q (M ). Let H ε be the Hilbert space H 1 g (M ) equipped with the inner product which induces the norm Let L q ε be the Banach space L q g (M ) equipped the norm It is clear that for any q ∈ [2, 6) the embedding H ε ֒→ L q ε is a continuous map. It is not difficult to check that where the constant c does not depend on ε.
In particular, the embedding i ε : We can rewrite problem (10) in the equivalent way where we set

2.
3. An approximation for the solution. It is well known (see [22,27]) that there exists a unique positive spherically symmetric function U ∈ H 1 (IR N ) such that Moreover, the function U and its derivatives are exponentially decaying at infinity, namely Let χ r be a smooth cut-off function such that where we set U ε (z) := U z ε . We will look for a solution to (13) or equivalently to (10) as u ε := W ε,ξ + φ, where the rest term φ belongs to a suitable space which will be introduced in the following.
It is well known that every solution to the linear equation is a linear combination of the functions Let us define on M the functions ξ be the orthogonal projections. In order to solve problem (13) we will solve the couple of equations 2.4. Scheme of the proof of Theorem 1. The first step is to solve equation (19). More precisely, if ε is small enough for any fixed ξ ∈ M , we will find a function φ ∈ Π ⊥ ε,ξ such that (19) holds. First of all, we define the linear operator L ε,ξ : [30] we proved the invertibility of L ε,ξ .
Finally, we use a contraction mapping argument to solve equation (19). This is done in Section 3 Proposition 9. There exists ε 0 > 0 and c > 0 such that for any ξ ∈ M and for any ε ∈ (0, ε 0 ) there exists a unique φ ε,ξ = φ(ε, ξ) which solves equation (19). Moreover The second step is to solve equation (20). More precisely, for ε small enough we will find the point ξ in M such that equation (20) is satisfied.
Let us introduce the reduced energy I ε : M → IR defined by where the energy I ε whose critical points are solution to problem (10) is defined in (7). First of all, arguing exactly as in Lemma 4.1 of [30] we get Proposition 10. ξ ε is a critical point of I ε if and only if the function u ε = W ε,ξε + φ ε,ξε is a solution to problem (10).
Thus, the problem is reduced to search for critical points of I ε whose asymptotic expansion is given in Section 4 and reads as follows.
Proposition 11. It holds true that C 1 −uniformly with respect to ξ as ε goes to zero. Here S g (ξ) is the scalar curvature of M at ξ and c i 's are constants.
Finally, we can prove Theorem 1 by showing that I ε has a critical point in M .
Proof of Theorem 1. If K is a C 1 -stable critical set of the scalar curvature of M (see Definition 2), by Proposition 11, we deduce that if ε is small enough the function I ε has a critical point ξ ε such that ξ ε → ξ 0 as ε goes to zero. The claim follows by Proposition 10.

The finite dimensional reduction
This section is devoted to the proof of Proposition 9. First, we remark that equation (19) is equivalent to and R ε,ξ is defined in (21). In order to solve equation (24), we need to find a fixed point for the operator T ε,ξ : We are going to prove that T ε,ξ is a contraction map on suitable ball of H ε .
In Proposition 8 we estimate the error term R ε,ξ , while in Proposition 3.5 of [30], we estimated the higher order term N ε,ξ (φ).
Lemma 13. There exists ε 0 > 0, c > 0 such that for any ξ ∈ M, ε ∈ (0, ε 0 ) and r > 0 it holds true that Proof. Let us prove (27). By Remark 2.2 in [30] it follows that By Lemma 19 we have By Lemma 19 and the previous estimate we deduce the following and then (27) follows.
Let us prove (28). By Remark 2.2 in [30] it follows that By Remark 18 and Lemma 20 we have for some θ ∈ (0, 1): By Lemma 19 and the estimate of I 1 we have By Lemma 19 and Lemma 20 we have Collecting the estimates of I i 's we get (28).
Proof of Proposition 9 (completed). By Proposition 7, we deduce By Lemma 12 and Lemma 13 together with Proposition 8, we immediately deduce that T ε,ξ is a contraction in the ball centered at 0 with radius cε 2 in K ⊥ ε,ξ for a suitable constant c. Then T ε,ξ has a unique fixed point.
In order to prove that the map ξ → φ ε,ξ is a C 1 −map, we apply the Implicit Function Theorem to the Indeed, at page 246 of [30] we proved that by Lemma 20 we get and by Lemma 19 we get That concludes the proof.

The reduced energy
This section is devoted to the proof of Proposition 11. The first important result is the following one.
Lemma 14. It holds true that (29) uniformly with respect to ξ as ε goes to zero. Moreover, setting ξ(y) = exp ξ (y), y ∈ B(0, r) it holds true that uniformly with respect to ξ as ε goes to zero.
Proof. We argue exactly as in Lemma 5.1 of [30], once we prove the the following estimates: and Let us prove (31). We have (for some θ ∈ [0, 1])

THE ROLE OF THE SCALAR CURVATURE IN SYSTEMS ON RIEMANNIAN MANIFOLDS13
with and Then (31) follows.
By Lemma 20 and the facts that φ ε,ξ(y) H 1 g = O ε 5/2 and Remark 18 we get By the estimate of I 1 we get I 2 = o(ε 2 ), because of Lemma 19 and we also get We use the definition of v ε given in Lemma 21 and we get Here we used the fact that (see (6.3) of [30]) the function φ ε,ξ0 (z) := φ ε,ξ0 exp ξ0 (εz) = φ ε,ξ0 (x) can be estimated as By the estimate of I 3 we get I 5 = o(ε 2 ), because of Lemma 19. Let us prove (33). We have (for some θ ∈ [0, 1]) Arguing as in the proof of (5.10) of [30], the proof of (33) reduces to the proof of the following estimate where the functions Z l ε,ξ(y) are defined in (18). First of all we point out that By Lemma 21 we have that 1 ε 2ṽ ε,ξ we use Lemma 19 and also that φ H 1 g ≤ √ ε φ ε ≤ cε 5/2 and Z l ε,ξ(y) g,3 and (because of Lemma 20) Lemma 15. It holds true that C 1 −uniformly with respect to ξ ∈ M as ε goes to zero. Here Proof. See Lemma (4.2) of [30].
Lemma 16. It holds true that C 1 −uniformly with respect to ξ ∈ M as ε goes to zero. Here Proof.
By the weak convergence of 1 ε 2 n v εn,ξ n in L 6 (R 3 ) we infer We have to prove that the convergence is uniform with respect to ξ ∈ M . By the expansions of |g ξ (εz)| 1/2 and χ(ε|z|), and by (47) we have uniformly with respect to ξ as ε goes to zero. By (48) and by the expansions of |g ξ (εz)| 1/2 and χ(ε|z|) we have uniformly with respect to ξ as ε goes to zero.
We have that We call I 1 (ε, ξ) and I 2 (ε, ξ) respectively the first and the second addendum of the above equation.
Remark 22. We remark that γ is positive radially symmetric and decays exponentially at infinity with its first derivative because it solves −∆γ = qU 2 in R 3 .