Boundary layers to a singularly perturbed Klein–Gordon–Maxwell–Proca system on a compact Riemannian manifold with boundary

We study the semiclassical limit to a singularly perturbed nonlinear Klein–Gordon–Maxwell–Proca system, with Neumann boundary conditions, on a Riemannian manifold M with boundary. We exhibit examples of manifolds, of arbitrary dimension, on which these systems have a solution which concentrates at a closed submanifold of the boundary ofM, forming a positive layer, as the singular perturbation parameter goes to zero. Our results allow supercritical nonlinearities and apply, in particular, to bounded domains in RN . Similar results are obtained for the more classical electrostatic Klein–Gordon–Maxwell system with appropriate boundary conditions.


Introduction
On a compact smooth Riemannian manifold (M, g) with boundary, we consider the system where ∆ g = div g ∇ g is the Laplace-Beltrami operator (without a sign), ε > 0, q > 0, ω ∈ ℝ, α ∈ C 2 (M) is a realvalued function which satisfies α(x) > ω 2 on M, p ∈ (2, ∞), and Λ is given by We are interested in studying the semiclassical limit to this system, i.e., the existence of positive solutions and their asymptotic profile, as ε → 0.
The seminal paper [3] by Benci and Fortunato attracted the attention of the mathematical community, and motivated much of the recent activity towards the study of this type of systems. For ε = 1, existence and nonexistence results for subcritical nonlinear terms have been obtained, e.g., in [1,3,6,10,[13][14][15]27] for systems in the entire space ℝ 3 , or in a bounded domain in ℝ 3 with Dirichlet or Neumann boundary conditions. KGMP-systems on a closed (i.e., compact and without boundary) Riemannian manifold of dimension 3 or 4 have been recently investigated in [17,24,25] for subcritical or critical nonlinearities.
The existence and asymptotic behavior of semiclassical states in flat domains have been investigated, e.g., in [11,12,31]. In [11], D'Aprile and Wei constructed a family of positive radial solutions (u ε , v e ) to a KGM-system in a 3-dimensional ball, with Dirichlet boundary conditions, such that u ε concentrates around a sphere which lies in the interior of the ball. For compact manifolds of dimensions 2 and 3, with or without boundary, the existence and multiplicity of positive semiclassical states, such that u ε concentrates at a point, have been exhibited, e.g., in [20,21,23], for subcritical nonlinearities. The concentration at a positive-dimensional submanifold for a KGMP-system on closed manifolds of arbitrary dimension, and for nonlinearities which include supercritical ones, was recently exhibited in [7].
Our aim is to extend the results in [7,8] to manifolds with boundary, i.e., we will establish the existence of positive semiclassical states (u ε , v e ) to system (1.1), on some compact Riemannian manifolds M with boundary, such that u ε concentrates at a positive-dimensional submanifold as ε → 0. Our results apply, in particular, to systems with supercritical nonlinearities in bounded smooth domains Ω of ℝ N of any dimension.
The Neumann boundary condition ∂v ∂ν = 0 on v seems to be more meaningful from a physical point of view, as it gives a condition on the electric field on ∂M. However, if the Proca mass is 0, i.e., if Λ(u) = qu 2 , and we set ∂v ∂ν = 0, then the second equation in system (1.1) admits the trivial solution v = 1 q and the first equation reduces to a Schrödinger equation, making the coupling effect unnoticeable. This is why we impose a Dirichlet boundary condition on v when Λ(u) = qu 2 .
The Neumann boundary condition ∂u ∂ν = 0 on u produces an effect of the boundary of M on the existence and concentration of solutions to system (1.1). In fact, the solutions that we obtain form a positive layer which concentrates around a submanifold of ∂M as ε → 0.
As in [7], our approach consists in reducing system (1.1) to a similar system, with the same power nonlinearity, on a manifold of lower dimension. Solutions to the new system which concentrate at a point will give rise to solutions to the original system concentrating at a positive-dimensional submanifold. This approach was introduced by Ruf and Srikanth in [29] and has been used, for instance, in [9,28,30]. We begin by describing some of the reductions that we will use.

Reducing the dimension of the system
Let (M, g) be a compact smooth n-dimensional Riemannian manifold with boundary, let f : M → (0, ∞) be a C 1 -function, and let (N, h) be a compact smooth Riemannian manifold without boundary of dimension k ≥ 1. The warped product M × f 2 N is the cartesian product M × N endowed with the Riemannian metric g := g + f 2 h. It is a smooth Riemannian manifold of dimension n + k with boundary ∂M × f 2 N.
For example, if Θ is a bounded smooth domain in ℝ n whose closure is contained in ℝ n−1 × (0, ∞), f(x 1 , . . . , x n ) = x n and k is the standard k-sphere, then, up to isometry, the warped product Θ × f 2 k is which is a bounded smooth domain in ℝ n+k .
if and only if the functions We stress that the exponent p is the same in both systems. Since k ≥ 1, we have that 2 Note also that u ε and v ε are positive if u ε and v ε are positive. Another type of reduction is obtained from the Hopf maps. For N = 2, 4, 8, 16, we write ℝ N ≡ × , where is either the real numbers ℝ, the complex numbers ℂ, the quaternions ℍ, or the Cayley numbers . The Hopf map h is defined by This map is horizontally conformal with dilation λ(z) = 2|z|. It is also invariant under the action of the units Let Ω be a bounded smooth domain in ℝ 2 dim \ {0} such that ζz ∈ Ω for all ζ ∈ S , z ∈ Ω. Then Θ := h (Ω) is a bounded smooth domain in ℝ dim +1 \ {0}. The main property of Hopf maps, for our purposes, is that they locally preserve the Laplace operator up to a factor, i.e., in Ω for every u ∈ C 2 (Θ).

The main results
Let (M, g) be a smooth compact Riemannian manifold with boundary of dimension n = 2, 3, 4. We consider the subcritical system where ε, q > 0, ω ∈ ℝ, a, b, c ∈ C 1 (M) are strictly positive functions such that a(x) > ω 2 b(x) on M, and p ∈ (2, 2 * n ). As before, 2 * n := ∞ if n = 2 and 2 * n := 2n n−2 if n = 3, 4. Theorem 1.3. Let K ⊂ ∂M be a nonempty C 1 -stable critical set for the function Γ : ∂M → ℝ, which is given by Then, for ε small enough, system (1.6) has a positive solution (u ε , v ε ) such that u ε concentrates at a point ξ 0 ∈ K as ε goes to zero.
A C 1 -stable critical set is defined as follows.
The rest of the paper is devoted to the proof of Theorem 1.3.

Reducing system (1.6) 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 [3], we introduce the map Φ : for system (1.6) with Dirichlet boundary conditions, or to the problem for system (1.6) with Neumann boundary conditions. It follows from standard variational arguments that Φ is well defined in H 1 g (M). The proofs of the following two lemmas are contained in [17].
, in case of Dirichlet boundary conditions, or by , in case of Neumann boundary conditions. Moreover, is of class C 1 , and its differential is given by Now, we introduce the functionals I ε , J ε , G ε : H 1 g (M) → ℝ given by From Lemma 2.2 we deduce that Therefore, if u is a critical point of the functional I ε , we have that In particular, if u ̸ = 0, by the maximum principle and regularity arguments we have that u > 0. Thus, the pair (u, Φ(u)) is a positive solution to system (1.6). This reduces solving system (1.6) to finding a solution u ε ∈ H 1 g (M) to the single equation (2.4). Some useful estimates involving the function Φ are contained in the appendix.

The approximate solution
We shall obtain a solution u ε to equation (2.4) using the Lyapunov-Schmidt reduction method. It will be an approximation to a function W ε,ξ , which we introduce next.
If (M, g) is an n-dimensional compact smooth Riemannian manifold with boundary, its boundary ∂M is a closed smooth Riemannian manifold of dimension n − 1, possibly not connected. We fix R > 0, smaller than the injectivity radius of ∂M, such that for each point We write each point x ∈ Q ξ in Fermi coordinates (y 1 , . . . , y n ) at ξ , i.e., (y 1 , . . . , y n−1 ) are normal coordinates for x on ∂M at the point ξ , and y n = dist g (x, x) is the geodesic distance from x to ∂M. We write ψ ∂ ξ : D + → Q ξ for the chart whose inverse is given by (ψ ∂ ξ ) −1 (x) := (y 1 , . . . , y n ), defined on The second fundamental form II(X, Y) of two vector fields X and Y on ∂M is the component of ∇ X Y which is normal to ∂M, where ∇ is the covariant derivative operator in the ambient manifold M. In Fermi coordinates at q it is given by a matrix (h ij ) i,j=1,...,n−1 . One has the well-known formulas where y = (y 1 , . . . , y n ) are the Fermi coordinates, |g| is the determinant of g = (g ij ), g ij are the coefficients of the inverse of (g ij ), and H = 1 n−1 ∑ n−1 i=1 h ii ; see [5,18,19]. Abusing notation, we shall write (h ij ) i,j=1,...,n for the matrix which coincides with the second fundamental form for i, j = 1, . . . , n − 1 and has h i,n = h n,j = 0 for i, j = 1, . . . , n.
By assumption, this function is positive on M. Given ξ ∈ ∂M, we consider the unique positive radial solutionV =V ξ to the equation (2.8) By direct computation, one sees thatV where U is the unique positive radial solution of In the following, we set Here the function χ is a fixed cut-off function of the form χ(ȳ , y n ) :=χ (|ȳ |)χ (y n ) for (ȳ , y n ) ∈ D + , wherẽ Remark 2.3. The following limits hold uniformly with respect to ξ ∈ ∂M, where the constant C does not depend on ξ .
It is well known that the space of solutions to the linearized problem is generated by the functions φ i := ∂V ξ ∂y i for i = 1, . . . , n − 1. The corresponding local functions on the manifold M are given by where φ i ε (y) := φ i ( y ε ) and χ is as above.
In Section 4, we will compute the asymptotic expansion of the reduced functionalĨ ε with respect to the parameter ε. We will show thatĨ If K is a nonempty C 1 -stable critical set for the function Γ, then, by Definition 1.4, there exists a critical point ξ ε ∈ ∂M ofĨ ε such that dist g (ξ ε , K) → 0 as ε → 0. Consequently, u ε = W ε,ξ ε + ϕ ε,ξ ε is a solution of (2.4), and Theorem 1.3 is proved.
Sketch of the proof of Proposition 2.4. Since, by Lemma 3.1, L ε,ξ is invertible, the map and we deduce from Lemmas 3.2 and 3.3 that T ε,ξ is a contraction in the ball centered at 0 with radius Cε in K ⊥ ε,ξ for a suitable constant C. Then T ε,ξ has a unique fixed point. The proof that the map ξ → ϕ ε,ξ is a C 1 -map uses the implicit function theorem. This part of the proof is standard.

The reduced energy
In this section, we obtain the expansion of the functionalĨ ε (ξ) with respect to ε. Recall the notation introduced in Section 2.2.
Proof. As in [7, Lemma 5.1], we obtain the estimates To complete the proof we need the following estimates: The proof of (4.1), (4.2) and (4.3) is technical and it is postponed to the appendix. With these estimates, one can prove the claim following the argument of [7, Lemma 5.1].

Lemma 4.2. The estimate
holds true C 1 -uniformly with respect to ξ ∈ ∂M.
Proof. For y ∈ D + , settingc (y) Using the change of variables y = εζ , from the expansions (2.5), (2.6) and (2.7) we immediately obtain From the definitions of V ξ and U we get C 0 -uniformly with respect to ξ ∈ ∂M. For the sake of readability, the C 1 -convergence is postponed to the appendix, where a proof is given in full detail.

Lemma 4.3. The expression
holds true C 1 -uniformly with respect to ξ ∈ ∂M.
Proof. In Lemma 4.2 we proved that It is enough to show now that G ε (W ε,ξ ) = o(1) holds true C 1 -uniformly with respect to ξ ∈ ∂M. For the C 0 -convergence, by Remark 2.3 and since ‖Φ(W ε,ξ )‖ ε ≤ Cε, we have that For the C 1 -convergence, we estimate

Appendix
We collect a series of technical results that were used previously.

2)
and for n = 3, 4 we have where the constant C 1 does not depend on ε, ξ and φ.
We call the last two integrals I 1 and I 2 , respectively, and we estimate each of them separately. We have, by Remark 2.3, that where t = 2 * n for n = 3, 4 and t > 2 for n = 2.

The pending proofs in Section 4
Conclusion of the proof of Lemma 4.1. To finish the proof of this lemma we need to prove (4.1), (4.2) and (4.3).