Klein-Gordon-Maxwell System in a bounded domain

This paper is concerned with the Klein-Gordon-Maxwell system in a bounded spatial domain. We discuss the existence of standing waves $\psi=u(x)e^{-i\omega t}$ in equilibrium with a purely electrostatic field $\mathbf{E}=-\nabla\phi(x)$. We assume an homogeneous Dirichlet boundary condition on $u$ and an inhomogeneous Neumann boundary condition on $\phi$. In the"linear"case we characterize the existence of nontrivial solutions for small boundary data. With a suitable nonlinear perturbation in the matter equation, we get the existence of infinitely many solutions.


Introduction
Many recent papers show the application of global variational methods to the study of the interaction between matter and electromagnetic fields. A typical example is given by the Klein-Gordon-Maxwell (KGM for short) system.
We consider a matter field ψ, whose free Lagrangian density is given by with m > 0. The field is charged and in equilibrium with its own electromagnetic field (E, B), represented by means of the gauge potentials (A, φ), Abelian gauge theories provide a model for the interaction; formally we replace the ordinary derivatives (∂ t , ∇) in (1) with the so-called gauge covariant derivatives where q is a nonzero coupling constant (see e.g. [1]). Moreover, we add the Lagrangian density associated with the electromagnetic field The KGM system is given by the Euler-Lagrange equations corresponding to the total Lagrangian density The study of the KGM system is carried out for special classes of solutions (and for suitable classes of lower order nonlinear perturbation in L 0 ). In this paper we consider that is a standing wave in equilibrium with a purely electrostatic field Under this ansatz, the KGM system reduces to −∆u − (qφ − ω) 2 u + m 2 u = 0, (see [2] or [3] where the complete set of equations has been deducted). We shall study (2) in a bounded domain Ω ⊂ R 3 with smooth boundary ∂Ω. The unknowns are the real functions u and φ defined on Ω and the frequency ω ∈ R. Throughout the paper we assume the following boundary conditions The problem (2) has a variational structure and we apply global variational methods. First we notice that the system is symmetric with respect to u, that is, the pair (u, φ) is a solution if and only if (−u, φ) is a solution.
Moreover, due to the Neumann condition (3b), the existence of solutions is independent on the frequency ω. Indeed the pair (u, φ) is a solution of (2)- (3) if and only if the pair (u, φ − ω/q) is a solution of the following problem with the same boundary conditions (3). In other words, for any ω ∈ R, the existence of a standing wave ψ = u(x)e −iωt in equilibrium with a purely electrostatic field is equivalent to the existence of a static matter field u(x), in equilibrium with the same electric field. So we focus our attention on the problem (4). The boundary datum h plays a key role. If h = 0, then it is easy to see that the system (4)-(3) have only the solutions u = 0, φ = const.
If ∂Ω h dσ = 0, then (4)-(3) has infinitely many solutions corresponding to u = 0. Such solutions have the form u = 0, φ = χ+const (see Lemma 2.1 below, where χ is introduced) and we call them trivial. In this case we are interested in finding nontrivial solutions (i.e. solutions with u = 0).
On the other hand, it is well known that the Neumann condition gives rise to a necessary condition for the existence of solutions of the boundary value problem. In our case, from (4b)-(3b), we get Hence, whenever ∂Ω h dσ = 0, solutions of (4)-(3), if any, are nontrivial.
The following theorem characterizes the existence of nontrivial solutions for small boundary data. Our second result is concerned with a nonlinear lower order perturbation in (4a). So we study the following system in Ω, ∆φ = 4πq 2 φu 2 in Ω, again with the boundary conditions (3). The nonlinear term g is usually interpreted as a self-interaction among many particles in the same field ψ. We assume g ∈ C Ω × R, R and (g1) ∃ a 1 , a 2 ≥ 0, ∃ p ∈ (2, 6) such that |g (x, t)| ≤ a 1 + a 2 |t| p−1 ; (g2) g (x, t) = o (|t|) as t → 0 uniformly in x; (g3) ∃ s ∈ (2, p] and r ≥ 0 such that for every |t| ≥ r: with p ∈ (2, 6).
The present paper has been motivated by some results about the system (5) in the case Ω = R 3 . To the best of our knowledge, our results are the first ones in the case of a bounded domain. Under Dirichlet boundary conditions on both u and φ, the existence results for (4) and (5) are analogous and simpler (see [5]).
About the system (2) in R 3 , Theorem 1.1 in [6] shows that there exists only the trivial solution.
A different class of solutions for the KGM system is introduced in the papers [3] and [7], where the authors show the existence of magnetostatic and electromagnetostatic solutions (3-dimensional vortices).

Functional setting
The first step to study problems (4) and (5) is to reduce to homogeneous boundary conditions. For the sake of simplicity, up to a simple rescaling, we can omit the constant 4π.
Then, there exists a unique χ ∈ H 2 (Ω) solution of Remark 2.2. It is well known that the solution of (7) satisfies where c is a positive constant. So we obtain then (4) becomes Let us consider on H 1 0 (Ω) the norm ∇u 2 and on H 1 (Ω) whereφ denotes the average of a function ϕ on Ω, i.e.
Standard computations show that the solutions of (9) are critical points of the Unfortunately it is strongly unbounded. We adapt a reduction argument introduced in [14]. Let Proof. Let u ∈ Λ and ρ ∈ L 6/5 (Ω) be fixed. We shall apply the Lax-Milgram Lemma.
We consider the bilinear form on H 1 (Ω). By the Hölder and Sobolev inequalities, we get and so a is continuous. Moreover, Indeed, if ϕ → +∞, we distinguish two cases.
On the other hand, by the Sobolev imbedding, we can consider the linear and continuous map The Lax-Milgram Lemma gives the assertion.
So our reduction argument is based on the following result.
Hence the set coincides with the graph of the map u ∈ Λ → ϕ u ∈ H 1 (Ω).
Proof. Since the graph of the map u → ϕ u is given by (11), we refer to the Implicit Function Theorem. Straightforward calculations show that for every ξ, η ∈ H 1 (Ω) and w ∈ H 1 0 (Ω) Then it is easy to see that F ′′ ϕϕ and F ′′ ϕu are continuous. On the other hand we have already seen that, for every (u, ϕ) ∈ Λ × H 1 (Ω), the operator associated to F ′′ ϕϕ (u, ϕ) is invertible (Lemma 2.3). Hence the claim immediately follows.
We can define on Λ the reduced functional It is C 1 and it is easy to see that (u, ϕ) ∈ Λ × H 1 (Ω) is a critical point of F if and only if u is a critical point of J and ϕ = ϕ u . So, to get nontrivial solutions of (4), we look for critical points of the functional J.

With the same change of variable (8), problem (5) becomes
The solutions of (12) are the critical points of the C 1 -functional and, as above, we can consider the reduced C 1 -functional To get nontrivial solution of (12) we look for critical points of J g .
has a unique solution η u ∈ H 1 (Ω). Of course, since the solution of (10) is unique, we have Lemma 3.1 (Properties of ξ u ). For every u ∈ Λ, and a.e. in Ω.
Proof. Multiplying (14) by ξ u and integrating on Ω, we get immediately (17). Moreover, if ξ u is the solution of (14), then ξ u + min χ is the unique solution of and minimizes the functional On the other hand and so ξ u + min χ = − |ξ u + min χ| , a.e. in Ω. Hence ξ u ≤ − min χ, a.e. in Ω. Analogously, ξ u + max χ is the unique solution of and, arguing as before, we get ξ u ≥ − max χ a.e. in Ω.
Finally, multiplying the equation in (15) by η u −η u and integrating, we get Then, by the Hölder and Poincaré-Wirtinger inequalities, we obtain which implies (23).
Finally we have the following relation between ξ u and η u .
Lemma 3.5. For every u ∈ Λ, Proof. Fixed u ∈ Λ, multiplying the equation of (14) by η u and integrating on Ω, we get Multiplying the equation of (15) by ξ u and integrating on Ω, we obtain The claim immediately follows.

Proof of Theorem 1.1
Taking into account Remark 2.2, in this section we assume that h H 1/2 (∂Ω) is sufficiently small in order to get

Existence of nontrivial solutions
In this subsection we assume that ∂Ω h dσ = 0. We give the explicit expression of the functional J(u) = F (u, ϕ u ). If u ∈ Λ, multiplying (10) by ϕ u and integrating on Ω, we have Then, taking into account (16) and (24), we obtain Moreover, for every v ∈ H 1 0 (Ω), Proposition 4.1. The functional J has the following properties: Proof. Assume u → 0. Since the first four terms in (26) are bounded from below, we study the last term. By (22), We claim that |η u | → +∞. Arguing by contradiction, assume that there exists a sequence u n → 0 such that {η n } is bounded (where we mean η n = η un ). Hence, by (23), we have ∇η n 2 → 0. Then, using the Poincaré-Wirtinger inequality, we deduce that {η n } is bounded. On the other hand (21) yields lim n η n 2 = +∞, so we get a contradiction and (a) is proved.
By (17), (25) and (28), we obtain Then, by (19), we deduce (b) and (c). Proof. Let {u n } ⊂ Λ be a Palais-Smale sequence, i.e. and ¿From (29) and (b) of Proposition 4.1 we deduce that {u n } is bounded, hence it converges weakly to u ∈ H 1 0 (Ω). It remains to prove that the convergence is strong and that u = 0. As before, for the sake of simplicity, we set ϕ n = ϕ un , ξ n = ξ un and η n = η un .
So it is sufficient to prove that the right hand side of (30) is bounded in H −1 (Ω). Since u n ⇀ u and J ′ (u n ) → 0, we have only to study {(ξ n + η n + χ) 2 u n }.
¿From (29) we deduce that {κ |Ω|η n /2} is bounded, the same being true for the first four terms in J(u n ). Then, using (23), we conclude that {η n } is bounded, as well as {ξ n } by (19). The claim easily follows. Finally (a) of Proposition 4.1 and (29) show that u cannot be zero. The proof is thereby complete.
Using again (a) of Proposition (4.1), we can see that the sublevels of J are complete. Then, by a standard tool in critical point theory (Deformation Lemma, see e.g. [18]), we conclude that the minimum of J is achieved.

The only if part
In this subsection we show that if ∂Ω h dσ = 0, then problem (9) has only trivial solutions.
Let (u, ϕ) be a solution of (9) with κ = 0. By the first equation we have By the second equation we have − ∇ϕ Then, substituting Ω χϕu 2 dx in (31), we obtain Therefore, taking into account (25), we deduce u = 0.

Proof of Theorem 1.3
In this section we assume κ = 0, so we have Since ϕ u satisfies (32), substituting in (13), we find, for every u = 0, (34) for v ∈ H 1 0 (Ω). About the nonlinear term, we recall that (g 1 ) − (g 3 ) imply that: (G 1 ) for every ε > 0 there exists A ≥ 0 such that for every t ∈ R |G (x, t)| ≤ ε 2 t 2 + A |t| p ; (G 2 ) there exist two constants b 1 , b 2 > 0 such that for every t ∈ R This time the functional has not a singularity in 0, but it can be extended according to the following proposition.