Klein–Gordon–Maxwell Systems with Nonconstant Coupling Coefficient

1 Introduction

We are interested in the system of nonautonomous elliptic equations

where Δ is the Laplace operator in ℝ3, Ω⊂ℝ3 is a bounded and smooth domain, m∈ℝ, q∈L6⁢(Ω)∖{0}. We complement these equations with the boundary conditions

(1.2)

where 𝐧 is the unit outward normal vector to ∂⁡Ω and α∈H1/2⁢(∂⁡Ω).

We look for nontrivial solutions, by which we mean pairs (u,ϕ)∈H01⁢(Ω)×H1⁢(Ω) satisfying (1.1)–(1.2) in the usual weak sense, with u≠0. Note that if (u,ϕ) is a nontrivial solution, the pair (-u,ϕ) is a nontrivial solution as well.

System (1.1) arises in connection with the so-called Klein–Gordon–Maxwell equations, which model the interaction of a charged matter field with the electromagnetic field (𝐄,𝐇). They are the Euler–Lagrange equations of the Lagrangian density

where ψ is a complex-valued function representing the matter field, while ϕ and 𝐀 are the gauge potentials, related to the electromagnetic field via the equations 𝐄=-∇⁡ϕ-∂t⁡𝐀, 𝐇=∇×𝐀. For the derivation of the Lagrangian density and details on the physical model, we refer to [5, 6, 11]. Let us point out that, in the physical model, q is a constant which represents the electric charge of the matter field; nonconstant coupling coefficients, however, are worth investigating from a mathematical point of view.

Confining attention to standing waves, in equilibrium with a purely electrostatic field, amounts to imposing ψ⁢(t,x)=ei⁢ω⁢t⁢u⁢(x), where u is a real-valued function and ω is a real number, 𝐀=0, and ϕ depends only on x. With these choices, the Klein–Gordon–Maxwell equations considerably simplify and become

In the special case of static solutions, corresponding to ω=0, system (1.3) reduces to (1.1). In the physical model, the boundary condition (1.2a) means that the matter field is confined to the region Ω, while (1.2b) amounts to prescribing the normal component of the electric field on ∂⁡Ω; up to a sign, the surface integral ∫∂⁡Ωα⁢𝑑σ represents the flux of the electric field through the boundary of Ω, and thus, the total charge contained in Ω.

Problem (1.3)–(1.2) was investigated in [9], for a constant coupling coefficient q. In this case, a degeneracy phenomenon occurs and the existence of solutions to (1.3)–(1.2) does not depend on ω (see [9] and [10, Remark 1.2]). Thus, for autonomous systems, letting ω=0 in (1.3) entails no loss of generality; this is not true if the coupling coefficient is not constant. The existence of infinitely many static solutions in the nonautonomous case was proved in [10], under the assumption that q vanishes at most on a set of measure zero. Our main result generalizes [10, Theorem 1.3] in that we impose no conditions on the zero-level set of q, and it provides additional information on the solutions. We will address problem (1.3)–(1.2) with ω≠0 in a forthcoming paper.

At least for small data, assuming ∫∂⁡Ωα⁢𝑑σ≠0 is necessary for the existence of nontrivial solutions, as the following result shows.

Note that if ∫∂⁡Ωα⁢𝑑σ=0, then every pair (0,ϕ), with ϕ a harmonic function satisfying the Neumann boundary condition (1.2b), is a trivial solution to (1.1)–(1.2).

Our results are obtained by way of variational methods. We follow an approach introduced by Benci and Fortunato (in [5] for Klein–Gordon–Maxwell systems, and earlier in [4] for Schrödinger–Maxwell systems) and subsequently implemented by many authors. Most results in the literature concern systems posed in unbounded spatial domains, possibly featuring lower-order nonlinear perturbations; see, for instance, [3, 7, 8, 14, 17]. We also refer to [13, 12] for recent applications to Klein–Gordon–Maxwell systems with Neumann boundary conditions on Riemannian manifolds.

To prove our multiplicity result, we apply Ljusternik–Schnirelmann theory to a functional J, defined in a subset of H01⁢(Ω), whose critical points correspond with nontrivial solutions to problem (1.1)–(1.2). The definition of J depends on whether a certain Neumann problem is uniquely solvable. The easiest way to guarantee that this occurs is to assume, as in [10], that q vanishes at most on a set of measure zero. Here, instead, we build the solvability requirement into Λq, the domain of J.

The paper is organized as follows: In Section 2, we define the set Λq and address the solvability issue. In Section 3, we define the functional J and investigate its properties. Section 4 is devoted to the proofs of Theorems 1.1 and 1.3.

2 Preliminaries

Throughout the paper, we will use the following notation:

1. For any integrable function f:Ω→ℝ, ∥f∥p is the usual norm in Lp⁢(Ω) (p∈[1,∞]) and f¯ is the average of f in Ω;

2. H01⁢(Ω) is endowed with the norm ∥∇⁡f∥2;

3. H1⁢(Ω) is endowed with the norm ∥f∥:=(∥∇⁡f∥22+|f¯|2)1/2;

4. H′:=L⁢(H1⁢(Ω),ℝ);

5. A:=∫∂⁡Ωα⁢𝑑σ, ∥α∥1/2:=∥α∥H1/2⁢(∂⁡Ω).

3 The Constrained Functional

In view of the observations at the beginning of Section 2.2, the set

is the graph of the map Φ:Λq⟶H1⁢(Ω) defined by

Note that Fφ⁢φ′′⁢(u,φ)=-2⁢𝒜q⁢u for every (u,φ)∈Λq×H1⁢(Ω), therefore Fφ⁢φ′′⁢(u,φ) is an isomorphism by Proposition 2.1 (ii); moreover, Fφ⁢u′′ and Fφ⁢φ′′ are continuous in Λq×H1⁢(Ω). This implies that Φ is continuously differentiable in Λq.

Constraining the functional F on the set Zq amounts to considering the functional J:Λq⟶ℝ defined by

The following assertions are a straightforward consequence of the construction of J.

On account of Proposition 3.1, nontrivial solutions to problem (1.1)–(1.2) are in one-to-one correspondence with critical points of J in Λq.

Before investigating further properties of J, we note that

with ηu:=ℒq⁢u⁢(A/|Ω|) and ξu:=-ℒq⁢u⁢((q⁢u)2⁢χ) for every u∈Λq. By Remark 2.2, ηu and ξu satisfy the equations

respectively, with homogeneous Neumann boundary conditions.

Recall that χ is the unique solution of (2.1); by elliptic regularity theory and Sobolev’s inequalities, there exists κ∈(0,∞) such that

Let σ∈(0,∞) be such that ∥u∥3≤σ⁢∥∇⁡u∥2 for every u∈H01⁢(Ω). Let

4 Proof of the Main Results