A multiplicity result for the non-homogeneous Klein-Gordon-Maxwell system in rotationally symmetric bounded domains
© Wu and Ge; licensee Springer. 2013
Received: 17 May 2013
Accepted: 11 November 2013
Published: 13 December 2013
This paper is concerned with the Klein-Gordon equation coupled with the Maxwell equation in the rotationally symmetric bounded domains when a non-homogeneous term breaks the symmetry of the associated functional. Under some suitable assumption on nonlinear perturbation, we obtain infinitely many radially symmetric solutions to the non-homogeneous Klein-Gordon-Maxwell system.
when , is a ball and when , is an annulus in ().
where m, q and ω are real constants and .
This system appears as a model which describes the nonlinear Klein-Gordon field in a three-dimensional space interacting with the electromagnetic field. Specifically, in 2002 Benci and Fortunato  proposed this couplement, and in their subsequent article  they proved the existence of infinitely many radially symmetric solutions for the Klein-Gordon-Maxwell system when the nonlinearity exhibits subcritical behavior. D’Aprile and Mugnai  established the existence of infinitely many radially symmetric solutions for the subcritical KGM system in . They extended the interval of definitions of power in the nonlinearity exhibited in . Non-existence results of the KGM system in the whole can be found in  and . Positive and ground state solutions for the critical KGM system with potentials in were obtained in . For related works, see [7–9], and .
The KGM system in the case of bounded domains was firstly considered by D’Avenia et al. [11, 12]. They established the existence and multiplicity of solutions to the KGM system in bounded domains under suitable Dirichlet or mixed boundary conditions.
All the above mentioned results are mainly focused on the homogeneous case . However, on the non-homogeneous case , only a few results are known for KGM systems. In , Chen and Tang proved two different solutions for the non-homogeneous KGM equations in the whole . Candela and Salvatore [14, 15] dealt with multiplicity of solutions to non-homogeneous Schrödinger-Maxwell and Klein-Gordon-Maxwell systems with homogenous boundary conditions in a bounded ball, respectively. Inspired by Candela and Salvatore’s results, in this paper we deal with the non-homogeneous KGM system in the special bounded symmetric domains - balls or annuli - and obtain infinitely many radially symmetric solutions by using the variational method.
Based on the observation above, we will focus on studying the existence and multiplicity of solutions for problem (5), instead of problem (1)-(2). For the sake of simplicity, from now on, we shall omit the subscript q in problem (5). We assume that the nonlinearity term f in (5) satisfies the following conditions.
(f1) There exist and such that .
(f2) f is odd with respect to the second variable, that is, for all , , .
(f3) There exists a constant such that for all , , , with .
And we also assume that the non-homogeneous term g satisfies
(g1) and , .
Condition (f3) is known in the literature as the Ambrosetti-Rabinowitz type condition. The main result we provide in this paper is the following theorem.
If ζ is a radial function on ∂ Ω, then system (5) has infinitely many radial solutions with and bounded in .
Remark 1.1 Theorem 1.1 can be viewed as a reasonable extension of Theorem 1.1 of , where the multiplicity of solutions to the non-homogeneous KGM system with homogenous boundary conditions in a bounded ball of can be found.
This paper is organized as follows. In Section 2 we introduce the variational tools: a variational principle, stated as in , which allows us to reduce the previous systems to a semilinear elliptic equation in the only variable u, and a perturbation method, introduced by Bolle [16, 17], useful for stating our multiplicity results. Finally, in Section 3 we prove our main theorem.
2 Preliminary results
In this section, we present the variational framework to deal with problem (5) and also give some preliminaries which are useful later.
By the standard argument, is on .
The functional I is strongly indefinite, that is, unbounded from below and from above on an infinite dimensional subspace. So we apply a well-known reduction argument (see, e.g., ) to avoid this indefiniteness.
Similar to Lemma 2.1 of , which deals with the case of the entire domain , one easily obtains the following auxiliary result. To avoid repetition, the details of the proof, which is mainly based on the Lax-Milgram lemma, are omitted.
where denotes the duality pairing between and .
By taking , we can deduce the following result.
From the above proposition, one directly gets the following assertion.
Since the map is continuously differentiable, we define the reduced functional . It is easy to see that the pair is a solution of a critical point for I if and only if is a critical point for K and (see ). Hence, we look for critical points of K.
Lemma 2.2 (Lemma 9 of )
Now we recall Bolle’s method for dealing with problems with broken symmetry.
So is an increasing sequence of a finite dimensional subspace of E.
We make the following hypotheses:
there is a subsequence converging in .
In this framework, the following result of Bolle can be proved (for more details and the proof, see [, Theorem 2.2]).
either has a critical level such that ,
Remark 2.3 Note that if in , then for all , the functional is non-decreasing on . Hence, in case (1) of the above theorem, we have .
3 Proof of Theorem 1.1
for all and .
The following lemma allows us to prove that functional J verifies the assumption of Bolle’s abstract theorem.
Now, we choose δ satisfying . Then and , which implies equation (22). □
In the sequel, denotes some suitable positive constants. Now, we give the following simple fact.
Lemma 3.2 For any , if the boundary condition ζ is radial on ∂ Ω in equation (7), then is also radial in Ω.
is the unique solution of equation (27). □
Proof of Theorem 1.1 We first prove that any critical point of is also a critical point of K in .
Then, according to the principle of symmetric criticality , any critical point of is also a critical point of K in . Then we want to apply Bolle’s method to K restricted to .
That is, are bounded in . Moreover, by Remark 2.2, are bounded in . Hence these two sequences and , up to subsequences, converge in (the dual space of ). Thus, as −Δ is an isomorphism from to its dual space , there exists a subsequence of converging strongly in by the standard argument.
Hence Lemma 3.1 implies that J satisfies assumption (H2) of Theorem 2.1.
using inequalities (25) and (33). Then, as in a finite dimensional space all the norms are equivalent and , we get that J satisfies assumption (H4) of Theorem 2.1.
which contradicts with (36).
That means case (1) in the result of Theorem 2.1 occurs for infinitely many indexes, and a sequence of the critical level of exists such that by Remark 2.3 and (38). Thus, if is the corresponding sequence of critical points, by (34) for , it follows that . We complete the proof of Theorem 1.1. □
This work is supported by the National Natural Science Foundation of China (Grant 11226128).
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