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

1.1 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 𝕊 k is the standard k-sphere, then, up to isometry, the warped product Θ × f 2 𝕊 k is

which is a bounded smooth domain in ℝ n + k .

Let π M : M × f 2 N → M be the projection, α ^ ∈ 𝒞 2 ⁢ ( M ) and α := α ^ ∘ π M . A straightforward computation gives the following result; see, e.g., [16].

We stress that the exponent p is the same in both systems. Since k ≥ 1 , we have that 2 n + k ∗ < 2 n ∗ , where 2 d ∗ is the critical Sobolev exponent in dimension d, i.e., 2 d ∗ := ∞ if d = 2 and 2 d ∗ := 2 ⁢ d d - 2 for d > 2 . So, if 2 n + k ∗ ≤ p < 2 n ∗ , system (1.2) on M is subcritical, whereas system (1.3) on M × f 2 N is critical or supercritical. Moreover, if the solution u ε of (1.2) concentrates at a point x 0 ∈ M as ε → 0 , then the function 𝔲 ε := u ε ∘ π M concentrates at the submanifold π M - 1 ⁢ ( x 0 ) ≅ ( N , f 2 ⁢ ( x 0 ) ⁢ h ) . 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 ≡ 𝕂 × 𝕂 , where 𝕂 is either the real numbers ℝ , the complex numbers ℂ , the quaternions ℍ , or the Cayley numbers 𝕆 . The Hopf map 𝔥 𝕂 is defined by

This map is horizontally conformal with dilation λ ⁢ ( z ) = 2 ⁢ | z | . It is also invariant under the action of the units S 𝕂 := { ζ ∈ 𝕂 : | ζ | = 1 } , i.e., 𝔥 𝕂 ⁢ ( ζ ⁢ z ) = 𝔥 𝕂 ⁢ ( z ) for all ζ ∈ S 𝕂 , z ∈ 𝕂 × 𝕂 .

Let Ω be a bounded smooth domain in ℝ 2 ⁢ dim ⁡ 𝕂 ∖ { 0 } such that ζ ⁢ z ∈ Ω for all ζ ∈ S 𝕂 , z ∈ Ω . Then Θ := 𝔥 𝕂 ⁢ ( Ω ) is a bounded smooth domain in ℝ dim ⁡ 𝕂 + 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.,

Such maps are called harmonic morphisms; see [2]. This property allows us to reduce system (1.1) on 𝔐 := Ω to a system in Θ. Assume that α ∈ 𝒞 2 ⁢ ( Ω ) satisfies α ⁢ ( ζ ⁢ z ) = α ⁢ ( z ) for all ζ ∈ S 𝕂 , z ∈ Ω . Then the map α ^ : Θ → ℝ given by α ^ ⁢ ( x ) := α ⁢ ( 𝔥 𝕂 - 1 ⁢ ( x ) ) is well defined and of class 𝒞 2 . Note that λ 2 ⁢ ( 𝔥 𝕂 - 1 ⁢ ( x ) ) = 4 ⁢ | x | for every x ∈ ℝ dim ⁡ 𝕂 + 1 . The following proposition is an immediate consequence of these facts.

Note again that, if p ∈ [ 2 2 ⁢ dim ⁡ 𝕂 ∗ , 2 dim ⁡ 𝕂 + 1 ∗ ) , system (1.4) is subcritical, whereas system (1.5) is critical or supercritical. And if the functions u ε concentrate at a point ξ 0 ∈ Θ as ε → 0 , then the functions 𝔲 ε concentrate at the ( dim ⁡ 𝕂 - 1 ) -dimensional sphere 𝔥 𝕂 - 1 ⁢ ( ξ 0 ) in Ω.

Propositions 1.1 and 1.2 lead us to study the following problem.

1.2 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 ∗ := 2 ⁢ n n - 2 if n = 3 , 4 .

A 𝒞 1 -stable critical set is defined as follows.

Theorem 1.3, together with Propositions 1.1 and 1.2, yields the existence of solutions to the KGMP (or the KGM) system (1.1), which concentrate at a submanifold for subcritical, critical and supercritical exponents. The following two results illustrate this fact.

We write the points in ℝ n - 1 × ( 0 , ∞ ) as ( y ¯ , y n ) with y ¯ ∈ ℝ n - 1 and y n ∈ ( 0 , ∞ ) .

The rest of the paper is devoted to the proof of Theorem 1.3.

2.1 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 : H g 1 ⁢ ( M ) → H g 1 ⁢ ( M ) which associates to each u ∈ H g 1 ⁢ ( M ) the solution Φ ⁢ ( u ) to the problem

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 g 1 ⁢ ( M ) . The proofs of the following two lemmas are contained in [17].

Now, we introduce the functionals I ε , J ε , G ε : H g 1 ⁢ ( M ) → ℝ given by

where

with d ⁢ ( x ) := a ⁢ ( x ) - ω 2 ⁢ b ⁢ ( x ) , and

From Lemma 2.2 we deduce that

so

Therefore, if u is a critical point of the functional I ε , we have that

with d ⁢ ( x ) := a ⁢ ( x ) - ω 2 ⁢ b ⁢ ( x ) . 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 g 1 ⁢ ( M ) to the single equation (2.4).

Some useful estimates involving the function Φ are contained in the appendix.

2.2 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 x ∈ M with dist g ⁡ ( x , ∂ ⁡ M ) < R there exists a unique x ¯ ∈ ∂ ⁡ M for which dist g ⁡ ( x , x ¯ ) = dist g ⁡ ( x , ∂ ⁡ M ) , where dist g denotes the geodesic distance in ( M , g ) . For ξ ∈ ∂ ⁡ M , we set

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 i ⁢ j ) 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 i ⁢ j ) , g i ⁢ j are the coefficients of the inverse of ( g i ⁢ j ) , and H = 1 n - 1 ⁢ ∑ i = 1 n - 1 h i ⁢ i ; see [5, 18, 19]. Abusing notation, we shall write ( h i ⁢ j ) 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 .

Set d ⁢ ( x ) := a ⁢ ( x ) - ω 2 ⁢ b ⁢ ( x ) . By assumption, this function is positive on M. Given ξ ∈ ∂ ⁡ M , we consider the unique positive radial solution V ¯ = V ¯ ξ to the equation

By direct computation, one sees that

where U is the unique positive radial solution of

In the following, we set

so

The restriction V ξ ⁢ ( y ) := V ¯ ξ | ℝ + n of V ¯ ξ to the half-space ℝ + n := { y n ≥ 0 } solves the Neumann problem

For ξ ∈ ∂ ⁡ M and ε > 0 , set V ε ξ ⁢ ( y ) := V ξ ⁢ ( y ε ) . We define the functions W ε , ξ ∈ 𝒞 ∞ ⁢ ( M ) by

Here the function χ is a fixed cut-off function of the form χ ⁢ ( y ¯ , y n ) := χ ~ ⁢ ( | y ¯ | ) ⁢ χ ~ ⁢ ( y n ) for ( y ¯ , y n ) ∈ D + , where χ ~ : ℝ + → [ 0 , 1 ] is a smooth function such that χ ⁢ ( s ) ≡ 1 for 0 ≤ s ≤ R 2 , χ ⁢ ( s ) ≡ 0 for s ≥ R and | χ ~ ′ ⁢ ( s ) | ≤ 1 R .

It is well known that the space of solutions to the linearized problem