- Research
- Open Access

# A multiplicity result for the non-homogeneous Klein-Gordon-Maxwell system in rotationally symmetric bounded domains

- Yuhu Wu
^{1}Email author and - Bin Ge
^{2}

**2013**:583

https://doi.org/10.1186/1029-242X-2013-583

© Wu and Ge; licensee Springer. 2013

**Received:**17 May 2013**Accepted:**11 November 2013**Published:**13 December 2013

## Abstract

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.

## Keywords

- Symmetric Solution
- Radial Symmetry
- Finite Dimensional Space
- Multiplicity Result
- Finite Dimensional Subspace

## 1 Introduction

when $\rho =0$, $\mathrm{\Omega}={\mathrm{\Omega}}_{0}$ is a ball and when $\rho >0$, $\mathrm{\Omega}={\mathrm{\Omega}}_{\rho}$ is an *annulus* in ${R}^{N}$ ($N=3,4,5$).

*et al.*[1])

where *m*, *q* and *ω* are real constants and $\zeta \in C(\partial \mathrm{\Omega})$.

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 [2] proposed this couplement, and in their subsequent article [3] 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 [1] established the existence of infinitely many radially symmetric solutions for the subcritical KGM system in ${R}^{3}$. They extended the interval of definitions of power in the nonlinearity exhibited in [3]. Non-existence results of the KGM system in the whole ${R}^{3}$ can be found in [4] and [5]. Positive and ground state solutions for the critical KGM system with potentials in ${R}^{3}$ were obtained in [6]. For related works, see [7–9], and [10].

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 $g\equiv 0$. However, on the non-homogeneous case $g\ne 0$, only a few results are known for KGM systems. In [13], Chen and Tang proved two different solutions for the non-homogeneous KGM equations in the whole ${R}^{3}$. 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.

(f_{1}) There exist ${a}_{1},{a}_{2}\ge 0$ and $p\in (2,\frac{2N}{N-2})$ such that $|f(r,u)|\le {a}_{1}+{a}_{2}{|u|}^{p-1}$.

(f_{2}) *f* is odd with respect to the second variable, that is, for all $r\ge 0$, $u\in R$, $f(r,-u)=-f(r,u)$.

(f_{3}) There exists a constant $s\in (2,p]$ such that for all $r\ge 0$, $u>0$, $0<sF(r,u)\le uf(r,u)$, with $F(r,u)={\int}_{0}^{t}f(r,\tau )\phantom{\rule{0.2em}{0ex}}d\tau $.

And we also assume that the non-homogeneous term *g* satisfies

(g_{1}) $g\in {L}^{2}(\mathrm{\Omega})$ and $g(|x|)=g(x)$, $\mathrm{\forall}x\in \mathrm{\Omega}$.

Condition (f_{3}) is known in the literature as the Ambrosetti-Rabinowitz type condition. The main result we provide in this paper is the following theorem.

**Theorem 1.1**

*Assume that conditions*(f

_{1})-(f

_{3})

*and*(g

_{1})

*hold and*

*If* *ζ* *is a radial function on* *∂* Ω, *then system* (5) *has infinitely many radial solutions* $({u}_{i},{\varphi}_{i})\in {H}_{0,r}^{1}(\mathrm{\Omega})\times {H}_{r}^{1}(\mathrm{\Omega})$ *with* ${\parallel \mathrm{\nabla}{u}_{i}\parallel}_{2}\to +\mathrm{\infty}$ *and* $\{{\varphi}_{i}\}$ *bounded in* ${L}^{\mathrm{\infty}}(\mathrm{\Omega})$.

**Remark 1.1** Theorem 1.1 can be viewed as a reasonable extension of Theorem 1.1 of [14], where the multiplicity of solutions to the non-homogeneous KGM system with homogenous boundary conditions in a bounded ball of ${R}^{3}$ can be found.

This paper is organized as follows. In Section 2 we introduce the variational tools: a variational principle, stated as in [3], 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.

**Remark 2.1**By the maximum principle, (6) implies

By the standard argument, $I(u,\phi )$ is ${C}^{1}$ on ${H}_{0}^{1}(\mathrm{\Omega})\times {H}_{0}^{1}(\mathrm{\Omega})$.

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.*, [1]) to avoid this indefiniteness.

Similar to Lemma 2.1 of [4], which deals with the case of the entire domain ${R}^{3}$, 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.

**Lemma 2.1**

*For any*$u\in {H}_{0}^{1}(\mathrm{\Omega})$

*and for any*$h\in {H}^{-1}(\mathrm{\Omega})$,

*there exists a unique solution*$\phi :={(\mathrm{\Delta}-{u}^{2})}^{-1}[h]\in {H}_{0}^{1}(\mathrm{\Omega})$

*of the equation*

*Moreover*,

*for every*$u\in {H}_{0}^{1}(\mathrm{\Omega})$

*and for every*$h,k\in {H}^{-1}(\mathrm{\Omega})$,

*where* $\u3008\cdot ,\cdot \u3009$ *denotes the duality pairing between* ${H}_{1}^{0}(\mathrm{\Omega})$ *and* ${H}^{-1}(\mathrm{\Omega})$.

By taking $h={\varphi}_{0}{u}^{2}$, we can deduce the following result.

**Proposition 2.1**

*For any*$u\in {H}_{0}^{1}(\mathrm{\Omega})$,

*there exists unique*$\phi ={\phi}_{u}\in {H}_{0}^{1}(\mathrm{\Omega})$

*which satisfies*

*Furthermore*,

*the map*$\mathrm{\Phi}:u\in {H}_{0}^{1}(\mathrm{\Omega})\to {\phi}_{u}\in {H}_{0}^{1}(\mathrm{\Omega})$

*is of class*${C}^{1}$

*and for every*$u,v\in {H}_{0}^{1}(\mathrm{\Omega})$,

From the above proposition, one directly gets the following assertion.

**Corollary 2.1**

*Let*$u\in {H}_{0}^{1}(\mathrm{\Omega})$

*and set*${\mathrm{\Psi}}_{u}:=({\mathrm{\Phi}}^{\prime}[u])[u]\in {H}_{0}^{1}(\mathrm{\Omega})$.

*Then*${\mathrm{\Psi}}_{u}$

*is a solution to the integral equation*

*and*

Since the map $\mathrm{\Phi}:u\in {H}_{0}^{1}(\mathrm{\Omega})\to {\phi}_{u}\in {H}_{0}^{1}(\mathrm{\Omega})$ is continuously differentiable, we define the reduced ${C}^{1}$ functional $K(u):=I(u,{\phi}_{u})$. It is easy to see that the pair $(u,\phi )\in {H}_{0}^{1}(\mathrm{\Omega})\times {H}_{0}^{1}(\mathrm{\Omega})$ is a solution of a critical point for *I* if and only if $u\in {H}_{0}^{1}(\mathrm{\Omega})$ is a critical point for *K* and $\phi =\mathrm{\Phi}[u]$ (see [2]). Hence, we look for critical points of *K*.

**Lemma 2.2** (Lemma 9 of [11])

*For every*$u\in {H}_{0}^{1}(\mathrm{\Omega})$,

*the function*${\phi}_{u}$

*satisfies the following inequalities*:

**Remark 2.2**In fact, according to (9), we observe that

Now we recall Bolle’s method for dealing with problems with broken symmetry.

*E*be a Hilbert space equipped with the norm $\parallel \cdot \parallel $. Assume that $E={E}_{-}\oplus {E}_{+}$, where $dim({E}_{-})<+\mathrm{\infty}$, and let ${({e}_{k})}_{k\ge 1}$ be an orthonormal base of ${E}_{+}$. Consider

So ${\{{E}_{k}\}}_{k}$ is an increasing sequence of a finite dimensional subspace of *E*.

with $\mathrm{\Gamma}=\{\gamma \in C(E,E):\gamma \text{is odd and}\mathrm{\exists}R0\text{s.t.}\gamma (v)=v\text{for}\parallel v\parallel \ge R\}$.

We make the following hypotheses:

_{1})

*I*satisfies the following weaker form of the Palais-Smale condition: for every sequence ${\{({\theta}_{n},{v}_{n})\}}_{n}\subset [0,1]\times H$ such that

there is a subsequence converging in $[0,1]\times E$.

_{2}) For all $b>0$, there exists a constant ${C}_{b}>0$ such that

_{3}) There exist two continuous maps ${\eta}_{1},{\eta}_{2}:[0,1]\times R\to R$, which are Lipschitz continuous with respect to the second variable, such that ${\eta}_{1}(\theta ,\cdot )\le {\eta}_{2}(\theta ,\cdot )$ and such that, for all critical points

*v*of $I(\theta ,\cdot )$,

_{4}) $I(0,v)$ is even and for any finite dimensional space

*W*of

*E*, we have

*R*. Moreover, since ${\eta}_{1}\le {\eta}_{2}$, we have ${\chi}_{1}\le {\chi}_{2}$. Set

In this framework, the following result of Bolle can be proved (for more details and the proof, see [[17], Theorem 2.2]).

**Theorem 2.1**

*Assume that*$I:[0,1]\times E\to R$

*is*${C}^{1}$

*and satisfies*(H

_{1})-(H

_{4}).

*Then there is*$C>0$

*such that for every*

*k*,

- (1)
*either*$I(1,x)$*has a critical level*${\overline{c}}_{k}$*such that*${\chi}_{2}(1,{c}_{k})\le {\chi}_{1}(1,{c}_{k+1})\le {\overline{c}}_{k}$, - (2)
*or*${c}_{k+1}-{c}_{k}\le C({\overline{\eta}}_{1}({c}_{k+1})+{\overline{\eta}}_{2}({c}_{k})+1)$.

**Remark 2.3** Note that if ${\eta}_{2}\le 0$ in $[0,1]\times R$, then for all $s\in R$, the functional ${\chi}_{2}(\cdot ,s)$ is non-decreasing on $[0,1]$. Hence, in case (1) of the above theorem, we have ${c}_{k}\le {\overline{c}}_{k}$.

## 3 Proof of Theorem 1.1

*J*is a ${C}^{1}$ functional such that $J(0,u)={J}_{0}(u)$ and $J(1,u)=K(u)$. Notice that the functional ${J}_{0}(u)$ is even by assumption (f

_{2}). According to (14),

for all $\theta \in [0,1]$ and $u,v\in {H}_{0,r}^{1}(\mathrm{\Omega})$.

The following lemma allows us to prove that functional *J* verifies the assumption of Bolle’s abstract theorem.

**Lemma 3.1**

*There exist three strictly positive constants*${\alpha}_{1}$, ${\alpha}_{2}$

*and*${\alpha}_{3}$

*such that for all*$(\theta ,u)\in [0,1]\times {H}_{0,r}^{1}(\mathrm{\Omega})$,

*Proof*For any $\delta >0$,

_{3}), there exist two constants ${b}_{1},{b}_{2}>0$ such that for every $t\in R$,

Now, we choose *δ* satisfying $\frac{1}{s}<\delta <\frac{1}{s}+{\epsilon}_{0}$. Then $s\delta -1>0$ and $(\frac{1}{2}-\frac{{\epsilon}_{0}}{2}-\delta ){m}^{2}-\frac{1}{2}{\varphi}_{0}^{2}>0$, which implies equation (22). □

In the sequel, ${C}_{i}$ denotes some suitable positive constants. Now, we give the following simple fact.

**Lemma 3.2** *For any* $0\le \rho <+\mathrm{\infty}$, *if the boundary condition* *ζ* *is radial on* *∂* Ω *in equation* (7), *then* ${\varphi}_{0}$ *is also radial in* Ω.

*Proof*(1) Case $\rho =0$,

*i.e.*, $\mathrm{\Omega}={\mathrm{\Omega}}_{0}$ is a ball. We denote $h=\zeta (x)$ for $|x|=1$. Then it is obvious that

*i.e.*, $\mathrm{\Omega}={\mathrm{\Omega}}_{\rho}$ is an

*annulus*. We denote ${h}_{1}=\zeta (x)$ for $|x|=\rho $ and ${h}_{2}=\zeta (x)$ for $|x|=\rho +1$. By calculating using general ordinary differential equation theory and maximal principle, we obtain that the radial function

is the unique solution of equation (27). □

*Proof of Theorem 1.1* We first prove that any critical point $u\in {H}_{0,r}^{1}(\mathrm{\Omega})$ of $K{|}_{{H}_{0,r}^{1}(\mathrm{\Omega})}$ is also a critical point of *K* in ${H}_{0}^{1}(\mathrm{\Omega})$.

*K*is $O(N)$-invariant,

*i.e.*,

Then, according to the principle of symmetric criticality [18], any critical point $u\in {H}_{0,r}^{1}(\mathrm{\Omega})$ of $K{|}_{{H}_{0,r}^{1}(\mathrm{\Omega})}$ is also a critical point of *K* in ${H}_{0}^{1}(\mathrm{\Omega})$. Then we want to apply Bolle’s method to *K* restricted to ${H}_{0,r}^{1}(\mathrm{\Omega})$.

*J*satisfies assumption (H

_{1}) of Theorem 2.1. So let us consider a sequence $\{({\theta}_{n},{u}_{n})\}\subset [0,1]\times {H}_{0,r}^{1}(\mathrm{\Omega})$ such that

_{1}) and ${L}^{p}(\mathrm{\Omega})\hookrightarrow {H}_{0,r}^{1}(\mathrm{\Omega})$, we get

That is, ${\{f(r,{u}_{n})\}}_{n}$ are bounded in ${L}^{\frac{p}{p-1}}(\mathrm{\Omega})$. Moreover, by Remark 2.2, ${\{[{m}^{2}-{({\phi}_{n}+{\varphi}_{0})}^{2}]{u}_{n}\}}_{n}$ are bounded in ${L}^{2}(\mathrm{\Omega})$. Hence these two sequences ${\{f(r,{u}_{n})\}}_{n}$ and ${\{[{m}^{2}-{({\phi}_{n}+{\varphi}_{0})}^{2}]{u}_{n}\}}_{n}$, up to subsequences, converge in ${H}_{r}^{-1}(\mathrm{\Omega})$ (the dual space of ${H}_{0,r}^{1}(\mathrm{\Omega})$). Thus, as −Δ is an isomorphism from ${H}_{0,r}^{1}(\mathrm{\Omega})$ to its dual space ${H}_{r}^{-1}(\mathrm{\Omega})$, there exists a subsequence of $\{{u}_{n}\}$ converging strongly in ${H}_{0,r}^{1}(\mathrm{\Omega})$ by the standard argument.

Hence Lemma 3.1 implies that *J* satisfies assumption (H_{2}) of Theorem 2.1.

*J*verifies assumption (H

_{3}) of Theorem 2.1 with ${\eta}_{2}(\theta ,t)=-{\eta}_{1}(\theta ,t)={C}_{3}{(1+{t}^{2})}^{\frac{1}{2s}}$. Notice that ${\overline{\eta}}_{1}(t)={\overline{\eta}}_{2}(t)={C}_{3}{(1+{t}^{2})}^{\frac{1}{2s}}$. So, by the definition of

*J*,

using inequalities (25) and (33). Then, as in a finite dimensional space all the norms are equivalent and $s>2$, we get that *J* satisfies assumption (H_{4}) of Theorem 2.1.

*J*and assume that case (1) in the results of Theorem 2.1 occurs for all

*k*large enough. Then, according to the definition of ${\overline{\eta}}_{1}$ and ${\overline{\eta}}_{2}$, we obtain

*J*defined by (17). Hence, using a similar argument as in Lemma 5.3 of [19], there exists an integer ${k}_{0}\in N$ such that

_{1}) implies

which contradicts with (36).

That means case (1) in the result of Theorem 2.1 occurs for infinitely many indexes, and a sequence $\{{\overline{c}}_{n}\}$ of the critical level of $J(1,\cdot )=K(\cdot )$ exists such that ${\overline{c}}_{k}\to +\mathrm{\infty}$ by Remark 2.3 and (38). Thus, if ${\{{u}_{n}\}}_{n}$ is the corresponding sequence of critical points, by (34) for $\theta =1$, it follows that ${\parallel \mathrm{\nabla}{u}_{n}\parallel}_{2}\to +\mathrm{\infty}$. We complete the proof of Theorem 1.1. □

## Declarations

### Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant 11226128).

## Authors’ Affiliations

## References

- D’Aprile T, Mugnai D: Solitary waves for nonlinear Klein-Gordon-maxwell and Schrodinger-maxwell equations.
*Proc. R. Soc. Edinb., Sect. A*2004, 134: 893–906. 10.1017/S030821050000353XMathSciNetView ArticleMATHGoogle Scholar - Benci V, Fortunato D: The nonlinear Klein-Gordon equation coupled with the Maxwell equations.
*Nonlinear Anal.*2001, 47: 6065–6072. 10.1016/S0362-546X(01)00688-5MathSciNetView ArticleMATHGoogle Scholar - Benci V, Fortunato D: Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations.
*Rev. Math. Phys.*2002, 14: 409–420. 10.1142/S0129055X02001168MathSciNetView ArticleMATHGoogle Scholar - D’Aprile T, Mugnai D: Non-existence results for the coupled Klein-Gordon-Maxwell equations.
*Adv. Nonlinear Stud.*2004, 4: 307–322.MathSciNetMATHGoogle Scholar - Cassani D: Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell’s equations.
*Nonlinear Anal.*2004, 58: 733–747. 10.1016/j.na.2003.05.001MathSciNetView ArticleMATHGoogle Scholar - Carrião P, Cunha P, Miyagaki O: Positive and ground state solutions for the critical Klein-Gordon-Maxwell system with potentials.
*Nonlinear Anal. TMA*2012, 75: 4068–4078. 10.1016/j.na.2012.02.023View ArticleMathSciNetMATHGoogle Scholar - Azzollini A, Pomponio A: Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations.
*Topol. Methods Nonlinear Anal.*2010, 35: 33–42.MathSciNetMATHGoogle Scholar - Carrião P, Cunha P, Miyagaki O: Existence results for the Klein-Gordon-Maxwell equations in higher dimensions with critical exponents.
*Commun. Pure Appl. Anal.*2011, 10: 709–718.MathSciNetMATHGoogle Scholar - Georgiev V, Visciglia N: Solitary waves for Klein-Gordon-Maxwell system with external Coulomb potential.
*J. Math. Pures Appl.*2005, 84: 957–983.MathSciNetView ArticleMATHGoogle Scholar - Mugnai D: Coupled Klein-Gordon and Born-Infeld-type equations: looking for solitary waves.
*Proc. R. Soc. Lond. Ser. A*2004, 460: 1519–1527. 10.1098/rspa.2003.1267MathSciNetView ArticleMATHGoogle Scholar - D’Avenia P, Pisani L, Siciliano G: Dirichlet and Neumann problems for Klein-Gordon-Maxwell systems.
*Nonlinear Anal.*2009, 71: 1985–1995. 10.1016/j.na.2009.01.038MathSciNetView ArticleMATHGoogle Scholar - D’Avenia P, Pisani L, Siciliano G: Klein-Gordon-Maxwell systems in a bounded domain.
*Discrete Contin. Dyn. Syst.*2010, 26: 135–149.MathSciNetMATHGoogle Scholar - Chen SJ, Tang CL:Multiple solutions for nonhomogeneous Schrödinger-Maxwell and Klein-Gordon-Maxwell equations on ${R}^{3}$.
*NoDEA Nonlinear Differ. Equ. Appl.*2010, 17: 559–574. 10.1007/s00030-010-0068-zMathSciNetView ArticleMATHGoogle Scholar - Candela AM, Salvatore A: Multiple solitary waves for non-homogeneous Schrödinger-Maxwell equations.
*Mediterr. J. Math.*2006, 3(3–4):483–493. 10.1007/s00009-006-0092-8MathSciNetView ArticleMATHGoogle Scholar - Candela AM, Salvatore A: Multiple solitary waves for non-homogeneous Klein-Gordon-Maxwell equations.
*More Progresses in Analysis*2009, 753–762.View ArticleGoogle Scholar - Bolle P: On the Bolza problem.
*J. Differ. Equ.*1999, 152: 274–288. 10.1006/jdeq.1998.3484MathSciNetView ArticleMATHGoogle Scholar - Bolle P, Ghoussoub N, Tehrani H: The multiplicity of solutions in nonhomogeneous boundary value problems.
*Manuscr. Math.*2000, 101: 325–350. 10.1007/s002290050219MathSciNetView ArticleMATHGoogle Scholar - Palais RS: The principle of symmetric criticality.
*Commun. Math. Phys.*1979, 69: 19–30. 10.1007/BF01941322MathSciNetView ArticleMATHGoogle Scholar - Bahri A, Berestycki H: A perturbation method in critical point theory and applications.
*Trans. Am. Math. Soc.*1981, 267: 1–32. 10.1090/S0002-9947-1981-0621969-9MathSciNetView ArticleMATHGoogle Scholar - Candela AM, Palmieri G, Salvatore A: Radial solutions of semilinear elliptic equations with broken symmetry.
*Topol. Methods Nonlinear Anal.*2006, 27: 117–132.MathSciNetMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.