On a compact smooth Riemannian manifold with boundary, we consider the system
where is the Laplace–Beltrami operator (without a sign), , , , is a real-valued function which satisfies on , , 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 .
Solutions to system (1.1) correspond to standing waves of an electrostatic Klein–Gordon–Maxwell (KGM) system if , and of a Klein–Gordon–Maxwell–Proca (KGMP) system with Proca mass 1 if . For the physical meaning of these systems, we refer to [3, 4, 25].
The seminal paper  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 , 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 , or in a bounded domain in 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 , D’Aprile and Wei constructed a family of positive radial solutions to a KGM-system in a 3-dimensional ball, with Dirichlet boundary conditions, such that 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 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 .
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 to system (1.1), on some compact Riemannian manifolds with boundary, such that concentrates at a positive-dimensional submanifold as . Our results apply, in particular, to systems with supercritical nonlinearities in bounded smooth domains Ω of of any dimension.
The Neumann boundary condition on seems to be more meaningful from a physical point of view, as it gives a condition on the electric field on . However, if the Proca mass is 0, i.e., if , and we set , then the second equation in system (1.1) admits the trivial solution and the first equation reduces to a Schrödinger equation, making the coupling effect unnoticeable. This is why we impose a Dirichlet boundary condition on when .
The Neumann boundary condition on produces an effect of the boundary of 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 as .
As in , 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  and has been used, for instance, in [9, 28, 30]. We begin by describing some of the reductions that we will use.
1.1 Reducing the dimension of the system
Let be a compact smooth n-dimensional Riemannian manifold with boundary, let be a -function, and let be a compact smooth Riemannian manifold without boundary of dimension . The warped product is the cartesian product endowed with the Riemannian metric . It is a smooth Riemannian manifold of dimension with boundary .
For example, if Θ is a bounded smooth domain in whose closure is contained in , and is the standard k-sphere, then, up to isometry, the warped product is
which is a bounded smooth domain in .
Let be the projection, and . A straightforward computation gives the following result; see, e.g., .
The functions solve the system
if and only if the functions , solve the system
We stress that the exponent p is the same in both systems. Since , we have that , where is the critical Sobolev exponent in dimension d, i.e., if and for . So, if , system (1.2) on M is subcritical, whereas system (1.3) on is critical or supercritical. Moreover, if the solution of (1.2) concentrates at a point as , then the function concentrates at the submanifold . Note also that and are positive if and are positive.
Another type of reduction is obtained from the Hopf maps. For , we write , where is either the real numbers , the complex numbers , the quaternions , or the Cayley numbers . The Hopf map is defined by
This map is horizontally conformal with dilation . It is also invariant under the action of the units , i.e., for all , .
Let Ω be a bounded smooth domain in such that for all , . Then is a bounded smooth domain in . 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 . This property allows us to reduce system (1.1) on to a system in Θ. Assume that satisfies for all , . Then the map given by is well defined and of class . Note that for every . The following proposition is an immediate consequence of these facts.
The functions solve the system
if and only if the functions , solve the system
Note again that, if , system (1.4) is subcritical, whereas system (1.5) is critical or supercritical. And if the functions concentrate at a point as , then the functions concentrate at the -dimensional sphere in Ω.
1.2 The main results
Let be a smooth compact Riemannian manifold with boundary of dimension . We consider the subcritical system
where , , are strictly positive functions such that on M, and . As before, if and if .
Let be a nonempty -stable critical set for the function , which is given by
Then, for ε small enough, system (1.6) has a positive solution such that concentrates at a point as ε goes to zero.
A -stable critical set is defined as follows.
Let . A subset of M is called a -stable critical set of f if and if, for any , there exists such that every function which satisfies
has a critical point with . Here denotes the geodesic distance associated to the Riemannian metric g.
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 as with and .
Let Θ be a bounded smooth domain in whose closure is contained in for , and let and be such that . Let
and . If is a nonempty -stable critical set for the function defined by
then, for any , and ε small enough, system (1.1) has a positive solution in such that, for some point , concentrates at the k-dimensional sphere as .
and assume that satisfies for all with , . If is a nonempty -stable critical set for the function defined by
then, for any , and ε small enough, system (1.1) has a positive solution in such that concentrates at the circle , for some , as .
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 , we introduce the map which associates to each the solution to the problem
for system (1.6) with Dirichlet boundary conditions, or to the problem
The map is of class and its differential at is the map defined by
for all , in case of Dirichlet boundary conditions, or by
for all , in case of Neumann boundary conditions. Moreover,
The function given by
is of class , and its differential is given by
for any .
Now, we introduce the functionals given by
with , and
From Lemma 2.2 we deduce that
Therefore, if u is a critical point of the functional , we have that
with . In particular, if , by the maximum principle and regularity arguments we have that . Thus, the pair is a positive solution to system (1.6).
Some useful estimates involving the function Φ are contained in the appendix.
2.2 The approximate solution
We shall obtain a solution to equation (2.4) using the Lyapunov–Schmidt reduction method. It will be an approximation to a function , which we introduce next.
If is an n-dimensional compact smooth Riemannian manifold with boundary, its boundary is a closed smooth Riemannian manifold of dimension , possibly not connected. We fix , smaller than the injectivity radius of , such that for each point with there exists a unique for which , where denotes the geodesic distance in . For , we set
We write each point in Fermi coordinates at ξ, i.e., are normal coordinates for on at the point ξ, and is the geodesic distance from x to . We write for the chart whose inverse is given by , defined on
The second fundamental form of two vector fields X and Y on is the component of which is normal to , where is the covariant derivative operator in the ambient manifold M. In Fermi coordinates at q it is given by a matrix . One has the well-known formulas
where are the Fermi coordinates, is the determinant of , are the coefficients of the inverse of , and ; see [5, 18, 19]. Abusing notation, we shall write for the matrix which coincides with the second fundamental form for and has for .
Set . By assumption, this function is positive on M. Given , we consider the unique positive radial solution to the equation
By direct computation, one sees that
where U is the unique positive radial solution of
In the following, we set
The restriction of to the half-space solves the Neumann problem
For and , set . We define the functions by
Here the function χ is a fixed cut-off function of the form for , where is a smooth function such that for , for and .
The following limits hold uniformly with respect to ,
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 for . The corresponding local functions on the manifold M are given by
where and χ is as above.
2.3 Proof of Theorem 1.3
As before, we set . We denote by the space equipped with the scalar product
and the norm . Similarly, we write for the space endowed with the norm
For any , the embedding is compact and there is a positive constant C, independent of ε, such that . The adjoint operator , , is defined by
Note that, for some positive constant C independent of ε,
Using the adjoint operator, we can rewrite equation (2.4) as
For and , let
where the are the functions defined in (2.10). This is an -dimensional subspace of . We denote its orthogonal complement with respect to by
We look for a solution to equation (2.4) of the form with . Thus, solves the equations
where and are the orthogonal projections onto and , respectively.
There exist and such that, for any and any , there is a unique which solves equation (2.12). This function satisfies
Moreover, is a -map.
Now, for each , we introduce the reduced energy , defined by
where is the functional defined in (2.3), whose critical points are the solutions to equation (2.4). It is easy to verify that is a critical point of if and only if the function is a weak solution to problem (2.4).
In Section 4, we will compute the asymptotic expansion of the reduced functional with respect to the parameter ε. We will show that
-uniformly with respect to as , where
3 The finite-dimensional reduction
There exist and such that, for any and ,
We now estimate the remainder term .
There exists such that, for any and , one has
Let be the function such that , i.e.,
Then, for and its Fermi coordinates , setting , and , we have
Moreover, by (2.8), we have
From the definition of we obtain
Using (2.11), we estimate the right-hand side by
By the usual change of variables , we can easily estimate almost all terms in the previous equation. The only term needing more attention is
and by (2.5) we get
since and , so . ∎
There exist and such that, for any , and , we have that
for , with as .
since . Hence, for some if or for if , we have that
and we can chose sufficiently large and sufficiently close to 2 to prove the claim. On the other hand, for , recalling that and using (5.4), we have
In every case, , and we have proved (3.1).
In the light of Remark 2.3, for some we have that
Here, as before, for and for . Notice that, since , we have
By direct computation, one sees that for . Thus, in case , from Lemma 5.2 we obtain that
and, choosing t sufficiently large, we conclude that with . For , again by Lemma 5.2, we have
and since , we have again that with . To estimate we proceed in a similar way, obtaining
For , we have by (5.2) that
and, since t may be chosen arbitrarily large, we have . For , again by (5.2) we conclude that
so with . Collecting the estimates for and , we get (3.2). ∎
Sketch of the proof of Proposition 2.4.
Since, by Lemma 3.1, is invertible, the map
is well defined. As
we deduce from Lemmas 3.2 and 3.3 that is a contraction in the ball centered at 0 with radius in for a suitable constant C. Then has a unique fixed point. The proof that the map is a -map uses the implicit function theorem. This part of the proof is standard. ∎
4 The reduced energy
In this section, we obtain the expansion of the functional with respect to ε. Recall the notation introduced in Section 2.2.
holds true -uniformly with respect to ξ as ε goes to zero. Moreover, setting for , we have that
-uniformly with respect to ξ as ε goes to zero, for every .
As in [7, Lemma 5.1], we obtain the estimates
To complete the proof we need the following estimates:
holds true -uniformly with respect to .
For , setting , and with , we have
From the definitions of and U we get
-uniformly with respect to . For the sake of readability, the -convergence is postponed to the appendix, where a proof is given in full detail. ∎
holds true -uniformly with respect to .
In Lemma 4.2 we proved that
It is enough to show now that holds true -uniformly with respect to . For the -convergence, by Remark 2.3 and since , we have that
For the -convergence, we estimate
Now, by Remark 2.3 and since
From Lemma 5.1, choosing if , we get , and for we get
Using Remark 2.3 and choosing , we obtain
Finally, using Lemma 5.2 and noting that , for and we have
while for we get
since . ∎
We collect a series of technical results that were used previously.
5.1 Key estimates for the function Φ
For , and , we have the following estimates:
For and , we have
and for we have
where the constant does not depend on ε, ξ and φ.
where for and for . We recall (see Remark 2.3) that
Thus, we have
For , and , we have the following estimates:
For and , we have
and for we have
where the constant C does not depend on ε, ξ, h and k.
5.2 Change of coordinates along
For , we consider the chart , introduced in Section 2.2, whose inverse expresses a point in Fermi coordinates around ξ.
For and , we consider the change of coordinates map
Since , writing with and , we have that
The derivatives of at are given by
For , we set . The function , defined in (2.9), can now be written as
where and . Thus, we have
If , , then , and we have
where denotes the derivative of the function f with respect to its k-th variable.
5.3 The pending proofs in Section 4
Conclusion of the proof of Lemma 4.1.
Proof of (4.1). For some , we have
Since and , from Remark 2.3 we obtain
Using Lemma 5.2, we conclude that
Proof of (4.2). Recall that for . Since , for some we have
From (5.8) and a straightforward computation we derive that
The term can be estimated in the same way, while for we have
This proves (4.2).
By (2.10) it can be proved easily that . So we have
Now, if , by (5.2) we have that
and if by (5.2) we have that
Conclusion of the proof of Lemma 4.2.
To finish the proof of this lemma we need to prove the -convergence. We do this for the first partial derivative. We set for . Then we have
Next, we estimate each term. Set and . By (5.8), we have
Using the definition (5.6) of , we obtain
Similarly, . Also, we have
Now, an elementary computation yields
We conclude that
The term is more delicate since the factor forces us to expand all factors up to the second order. In the light of (2.5), (2.6) and (2.7), with the convention that the matrix coincides with the second fundamental form when and for , we obtain
By Lemma 5.3, we have
Moreover, since U is radial,
where , and . Thus, we get
Now, by symmetry considerations, for , any term containing to an odd power vanishes, and, since , we get that
Consequently, we obtain
For the second term, setting , we obtain in an analogous way
Expanding , by the exponential decay of U and its derivative, and by (2.7) and the definition of , we get
As before, we obtain that and
and, by (5.9),
due to the symmetry. So,
In a similar way we proceed for , completing the proof. ∎
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About the article
Published Online: 2017-07-21
The first author is supported by CONACYT grant 237661 (México) and PAPIIT-DGAPA-UNAM grant IN104315 (Mexico). The second and third authors are partially supported by the GNAMPA project by INDAM. The second author is partially supported by the PRA project of the university of Pisa.
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 559–582, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0039.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0