PDF - Journal of Inequalities and Applications

Wu and Ge Journal of Inequalities and Applications 2013, 2013:583 http://www.journalofinequalitiesandapplications.com/content/2013/1/583

RESEARCH

Open Access

A multiplicity result for the non-homogeneous Klein-Gordon-Maxwell system in rotationally symmetric bounded domains Yuhu Wu1* and Bin Ge2 * Correspondence: [email protected] 1 Department of Mathematics, Harbin University of Science and Technology, Harbin, 150080, PR China Full list of author information is available at the end of the article

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.

1 Introduction This paper is concerned with the existence and multiplicity of solutions for a class of KleinGordon-Maxwell (KGM for short) systems in rotationally symmetric bounded domains. We fix  ≤ ρ < ∞ and define = ρ := int x ∈ RN (N = ,  or ) : ρ ≤ |x| < ρ +  , when ρ = , =  is a ball and when ρ > , = ρ is an annulus in RN (N = , , ). We consider the KGM system in this type symmetric bounded domains (the model in a bounded domain was introduced by D’Avenia et al. [])

–u – (qφ – ω) u + m u = f (|x|, u) + g(x),

φ = πq(qφ – ω)u

in

()

with Dirichlet boundary conditions

u = , φ=ζ

on ∂,

()

where m, q and ω are real constants and ζ ∈ C(∂). 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  Benci and Fortunato [] proposed this couplement, and in their subsequent article [] ©2013 Wu and Ge; licensee Springer. 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.


Page 2 of 12

they proved the existence of infinitely many radially symmetric solutions for the KleinGordon-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 R . They extended the interval of definitions of power in the nonlinearity exhibited in []. Non-existence results of the KGM system in the whole R can be found in [] and []. Positive and ground state solutions for the critical KGM system with potentials in R were obtained in []. For related works, see [–], and []. The KGM system in the case of bounded domains was firstly considered by D’Avenia et al. [, ]. 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 ≡ . However, on the non-homogeneous case g = , 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 R . Candela and Salvatore [, ] 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. In the uncoupled case q = , problem ()-() can be split into ⎧ ⎨–u + (m – ω )u = f (|x|, u) + g(x) in , ⎩u =  on ∂,

()

and ⎧ ⎨φ =  in , ⎩φ = ζ on ∂.

()

The existence and uniqueness of solutions of problem () and () are independent of each other. And it is clear that problem () has a unique solution. In this paper, we mainly consider the coupled case q = . In this case, the change of variables ⎧ ⎨u = √π qu, q ⎩ϕq = qϕ – ω, transforms system ()-() into ⎧ ⎪ –uq + φq u + m uq = fq (|x|, uq ) + g(x) ⎪ ⎪ ⎪ ⎪ ⎨φ = φ u q q q ⎪ ⎪uq =  ⎪ ⎪ ⎪ ⎩ φq = qζ – ω with

√ u . fq |x|, uq = π qf |x|, √ π q

in , in , on ∂, on ∂

()


Page 3 of 12

Based on the observation above, we will focus on studying the existence and multiplicity of solutions for problem (), instead of problem ()-(). For the sake of simplicity, from now on, we shall omit the subscript q in problem (). We assume that the nonlinearity term f in () satisfies the following conditions. N ) such that |f (r, u)| ≤ a + a |u|p– . (f ) There exist a , a ≥  and p ∈ (, N– (f ) f is odd with respect to the second variable, that is, for all r ≥ , u ∈ R, f (r, –u) = –f (r, u). (f ) There exists a constant s ∈ (, p] such that for all r ≥ , u > ,  < sF(r, u) ≤ uf (r, u),

t with F(r, u) =  f (r, τ ) dτ .

And we also assume that the non-homogeneous term g satisfies (g ) g ∈ L () and g(|x|) = g(x), ∀x ∈ . Condition (f ) is known in the literature as the Ambrosetti-Rabinowitz type condition. The main result we provide in this paper is the following theorem. Theorem . Assume that conditions (f )-(f ) and (g ) hold and –

s– |m| < qζ – ω ≤  s

a.e. ∂.

()

If ζ is a radial function on ∂, then system () has infinitely many radial solutions (ui , φi ) ∈  () × Hr () with ∇ui  → +∞ and {φi } bounded in L∞ (). H,r Remark . Theorem . can be viewed as a reasonable extension of Theorem . of [], where the multiplicity of solutions to the non-homogeneous KGM system with homogenous boundary conditions in a bounded ball of R can be found. This paper is organized as follows. In Section  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 [, ], useful for stating our multiplicity results. Finally, in Section  we prove our main theorem.

2 Preliminary results In this section, we present the variational framework to deal with problem () and also give some preliminaries which are useful later. To get homogeneous boundary conditions, we change variables as follows: ϕ = φ – φ , where φ is the unique solution of ⎧ ⎨φ =   ⎩φ = qζ – ω

in , on ∂.

()


Page 4 of 12

Then problem () can be rewritten as ⎧   ⎪ ⎪ ⎨–u – (ϕ + φ ) u + m u = f (|x|, u) + g(x) in , ϕ = (ϕ + φ )u ⎪ ⎪ ⎩ u = , ϕ=

()

in , on ∂.

Remark . By the maximum principle, () implies –

s– |m| < φ ≤  a.e. ∂. s

()

System () contains the Euler-Lagrange equations related to the functional I : H () × H () → R defined as  I(u, ϕ) = 

|∇u| – |∇ϕ| + m – (ϕ + φ ) u dx

–

F |x|, u dx –

gu dx.

()

By the standard argument, I(u, ϕ) is C  on H () × H (). 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 . of [], which deals with the case of the entire domain R , 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 . For any u ∈ H () and for any h ∈ H – (), there exists a unique solution ϕ := ( – u )– [h] ∈ H () of the equation ⎧ ⎨ϕ = u ϕ + h in , ⎩ϕ =  on ∂. Moreover, for every u ∈ H () and for every h, k ∈ H – (),

–

– h, – u [k] = k, – u [h] , where ·, · denotes the duality pairing between H () and H – (). By taking h = φ u , we can deduce the following result. Proposition . For any u ∈ H (), there exists unique ϕ = ϕu ∈ H () which satisfies ⎧ ⎨ϕ = (ϕ + φ )u  ⎩ϕ = 

in , on ∂.

()


Page 5 of 12

Furthermore, the map : u ∈ H () → ϕu ∈ H () is of class C  and for every u, v ∈ H (),

– [u] [v] =  – u (ϕu + φ )uv .

()

From the above proposition, one directly gets the following assertion. Corollary . Let u ∈ H () and set u := ( [u])[u] ∈ H (). Then u is a solution to the integral equation



(φ + ϕu )ϕu u dx

φ u u dx =

and

u ≤ . Since the map : u ∈ H () → ϕu ∈ H () is continuously differentiable, we define the reduced C  functional K(u) := I(u, ϕu ). It is easy to see that the pair (u, ϕ) ∈ H () × H () is a solution of a critical point for I if and only if u ∈ H () is a critical point for K and ϕ = [u] (see []). Hence, we look for critical points of K . Since, by Proposition ., the functional ϕu satisfies

∇ϕu 

ϕu (ϕu + φ )u dx,

=–

we obtain, for any u ∈ H (),  K(u) = I(u, ϕu ) = 

|∇u| + m – φ (ϕu + φ ) u dx

–

F |x|, u dx –

()

gu dx.

According to Lemma . and Proposition ., for every u, v ∈ H (), we have K (v)[u] =

∇v∇u dx + m – φ

∇v∇u dx + m

=



∇v∇u dx +

ϕv φ vu dx

– φ

vu dx –

f |x|, v u dx –

gu dx

ϕv φ vu dx

f |x|, v u dx –

 m – (φ + ϕv ) vu dx –

–  (φ + ϕv )uv – v φ v dx –

– =

vu dx –

– φ v  – v  (φ + ϕv )uv dx –

–

gu dx

f |x|, v u dx –

gu dx. ()

Hence we get, as an operator in H – (), K (v) =

∂I (v, ϕv ) = –v + m – (ϕv + φ ) v – f |x|, v – g. ∂v

()


Page 6 of 12

Lemma . (Lemma  of []) For every u ∈ H (), the function ϕu satisfies the following inequalities: – max{, φ } ≤ ϕu ≤ max{, –φ }

a.e. in .

Remark . In fact, according to (), we observe that  ≤ ϕu ≤ –φ

a.e. in .

()

Now we recall Bolle’s method for dealing with problems with broken symmetry. Let E be a Hilbert space equipped with the norm · . Assume that E = E– ⊕ E+ , where dim(E– ) < +∞, and let (ek )k≥ be an orthonormal base of E+ . Consider Ek+ = Ek ⊕ Rek+ ,

E = E – ,

k ≥ .

So {Ek }k is an increasing sequence of a finite dimensional subspace of E. Let I : [, ] × E → R be a C  -functional, and set

ck = inf sup I , γ (v) , γ ∈ v∈E

()

k

with = {γ ∈ C(E, E) : γ is odd and ∃R >  s.t. γ (v) = v for v ≥ R}. We make the following hypotheses: (H ) I satisfies the following weaker form of the Palais-Smale condition: for every sequence {(θn , vn )}n ⊂ [, ] × H such that

I(θn , vn )

n

are bounded and

lim

n→+∞

∂I (θn , vn ) = , ∂v

there is a subsequence converging in [, ] × E. (H ) For all b > , there exists a constant Cb >  such that I(θ , v) < b

implies

∂I

(θ , v) < Cb ∂I (θ , v) +  v +  . ∂v ∂θ

(H ) There exist two continuous maps η , η : [, ] × R → R, which are Lipschitz continuous with respect to the second variable, such that η (θ , ·) ≤ η (θ , ·) and such that, for all critical points v of I(θ, ·),

∂ I(θ , v) ≤ η θ , I(θ, v) . η θ , I(θ, v) ≤ ∂θ (H ) I(, v) is even and for any finite dimensional space W of E, we have lim

sup I(θ , v) = –∞.

|v|→+∞ θ ∈[,] v∈A


Page 7 of 12

Let χi : [, ] × R → R (i = , ) be the flow associated to ηi that is the solution of problem ⎧ ⎨ ∂ χ (θ, s) = η (θ, χ (θ, s)), i i ∂θ i ⎩ηi (, s) = s.

()

Note that χ and χ are continuous and that for all θ ∈ [, ], χ (θ , ·) and χ (θ , ·) they are non-decreasing on R. Moreover, since η ≤ η , we have χ ≤ χ . Set η (s) = sup η (θ, s), θ ∈[,]

η (s) = sup η (θ , s). θ ∈[,]

In this framework, the following result of Bolle can be proved (for more details and the proof, see [, Theorem .]). Theorem . Assume that I : [, ] × E → R is C  and satisfies (H )-(H ). Then there is C >  such that for every k, () either I(, x) has a critical level ck such that χ (, ck ) ≤ χ (, ck+ ) ≤ ck , () or ck+ – ck ≤ C(η (ck+ ) + η (ck ) + ). Remark . Note that if η ≤  in [, ] × R, then for all s ∈ R, the functional χ (·, s) is non-decreasing on [, ]. Hence, in case () of the above theorem, we have ck ≤ ck .

3 Proof of Theorem 1.1  Now, we consider the family of functionals J(θ, u) : [, ] × H,r () → R defined by J(θ, u) = J (u) – θ

()

gu dx,

with J (u) =

 

|∇u| + m – φ (ϕu + φ ) u dx –

F |x|, u dx.

Clearly, J is a C  functional such that J(, u) = J (u) and J(, u) = K(u). Notice that the functional J (u) is even by assumption (f ). According to (), ∂J (θ , u) = – ∂θ

∂J (θ , u)[v] = ∂u

()

gu dx,

∇v∇u dx +

 m – (φ + ϕu ) uv dx

f |x|, u v dx – θ

–

gv dx

()

 for all θ ∈ [, ] and u, v ∈ H,r (). The following lemma allows us to prove that functional J verifies the assumption of Bolle’s abstract theorem.


Page 8 of 12

Lemma . There exist three strictly positive constants α , α and α such that for all  (θ , u) ∈ [, ] × H,r (),

∇  + u ss ≤ α J(θ , u) – α

∂J (θ , u)[u] + α . ∂u

()

Proof For any δ > , ∂J (θ , u)[u] = ∂u

J(θ, u) – δ

 –δ 

|∇u| dx +

+ I + I – ( – δ)θ

 – δ m u dx 

gu dx

()

with

 φ (ϕu + φ ) – δ(ϕu + φ ) u dx, 

I = δf |x|, u u dx – F |x|, u dx. I = –

By Remark ., we get φ ≤ ϕu + φ ≤  and –  φ ≤ (  – δ)φ – δϕu ≤ (  – δ)φ . Hence,

I = –

(ϕu + φ )

≥–

 

 – δ φ – δϕu u dx 

φ u dx.

()

By assumption (f ), there exist two constants b , b >  such that for every t ∈ R, F(r, u) ≥ b |u|s – b .

()

Hence, we get

δf |x|, u u dx –

I =

F |x|, u dx

F |x|, u dx

≥ (sδ – )

≥ (sδ – ) b u ss – m()b . According to Remark ., there exists ε >  such that

φ ∞

≤

s– – ε m . s

Using the Cauchy-Schwarz inequality, we get gu dx ≤

ε m 

u  +

g  .  ε m

()


Page 9 of 12

Then equation () becomes ∂J J(θ , u) – δ (θ , u)[u] ∂u   ε  –δ – δ m – φ – m u dx ≥ |∇u| dx +    



g  . + (sδ – ) b u ss – m()b – ε m Now, we choose δ satisfying s < δ < s + ε . Then sδ –  >  and (  – which implies equation ().

ε 

– δ)m –  φ > ,

In the sequel, Ci denotes some suitable positive constants. Now, we give the following simple fact. Lemma . For any  ≤ ρ < +∞, if the boundary condition ζ is radial on ∂ in equation (), then φ is also radial in . Proof () Case ρ = , i.e., =  is a ball. We denote h = ζ (x) for |x| = . Then it is obvious that φ ≡ qh – ω is the unique solution of the equation ⎧ ⎨φ =   ⎩φ = qζ – ω

in ,

()

on ∂.

() Case ρ > , i.e., = ρ is an annulus. We denote h = ζ (x) for |x| = ρ and h = ζ (x) for |x| = ρ + . By calculating using general ordinary differential equation theory and maximal principle, we obtain that the radial function φ (x) =

q(h – h )ρ N– (ρ + )N– N– q(h – h )(ρ + )N– |x| + qh – ω –  (ρ + )N– – ρ N– (ρ + )N– – ρ N–

is the unique solution of equation ().

 () of K|H  () is also a Proof of Theorem . We first prove that any critical point u ∈ H,r ,r  critical point of K in H (). Let O(N) = {AN×N : orthogonal matrices}. Consider the O(N) group action Tg on L () defined by

Tg u(x) := u(gx) for any g ∈ O(N) and u ∈ L ().

()

Since for each u ∈ H (), ϕu is the unique solution of ⎧ ⎨ϕ = (ϕ + φ )u u u   ⎩ ϕu = 

in , on ∂,

()


Page 10 of 12

so, for any g ∈ O(N), we have Tg (ϕu ) = Tg (ϕu )Tg (u ) + Tg (φ )Tg (u ) in . By radial symmetry of φ (Lemma .), we get ⎧ ⎨(T ϕ ) = T (ϕ )(T u) + φ T (u ) g u g u g  g  ⎩Tg ϕu = 

in , on ∂.

Then, by the definition of : u → ϕu , we obtain

Tg (u) = (Tg u).

()

Using () and the Tg invariance of the norm in Hr (), Lp (), we deduce that K is O(N)invariant, i.e., K(Tg u) = K(u),

∀u ∈ H (), g ∈ O(N).

Then, according to the principle of symmetric criticality [], any critical point u ∈  () of K|H  () is also a critical point of K in H (). Then we want to apply Bolle’s H,r ,r  (). method to K restricted to H,r Firstly, we prove that J satisfies assumption (H ) of Theorem .. So let us consider a  () such that sequence {(θn , un )} ⊂ [, ] × H,r

J(θn , un )

n

is bounded and

lim

n→∞

Let ϕn = [un ]. Then the expression of εn =

∂J ∂u

∂J (θn , un ) = . ∂u

()

and () implies

∂J (θn , un ) = –un + m – (ϕn + φ ) un – f |x|, un – θ g ∂u

()

with ε → . Clearly, Lemma . implies { ∇un }n is bounded. By assumption (f ) and  (), we get Lp () → H,r

p

f (r, un ) p– dx ≤ C un p +  ≤ C ∇u p +  .  p p

That is, {f (r, un )}n are bounded in L p– (). Moreover, by Remark ., {[m – (ϕn + φ ) ]un }n are bounded in L (). Hence these two sequences {f (r, un )}n and {[m – (ϕn + φ ) ]un }n ,  ()). Thus, as – is an up to subsequences, converge in Hr– () (the dual space of H,r  () to its dual space Hr– (), there exists a subsequence of {un } isomorphism from H,r  () by the standard argument. converging strongly in H,r  (), By equation (), for all (θ , u) ∈ [, ] × H,r ∂J (θ, u) ≤ g  u  ≤ g  ∇u  . ∂θ Hence Lemma . implies that J satisfies assumption (H ) of Theorem .. ∂J  () satisfies ∂u (θ , u) = , then If (θ , u) ∈ [, ] × H,r



u ss ≤ α J(θ, u) + α ≤ α  + J  (θ , u)  ,

()


with α =

Page 11 of 12

(α + α ) max{α , α }. So, there exists a positive constant C such that

∂J

 (θ, u) ≤ C  + J  (θ, u) s ∂θ by (). Hence, J verifies assumption (H ) of Theorem . with η (θ , t) = –η (θ , t) = C ( +   t  ) s . Notice that η (t) = η (t) = C ( + t  ) s . So, by the definition of J,  J(θ, u) ≤ ∇u  + C u  –  ≤

F |x|, u dx – θ

gu dx



∇u  + C u  – b u ss – b m() – g  , 

()

using inequalities () and (). Then, as in a finite dimensional space all the norms are equivalent and s > , we get that J satisfies assumption (H ) of Theorem .. Now we apply Theorem . to J and assume that case () in the results of Theorem . occurs for all k large enough. Then, according to the definition of η and η , we obtain 

s + cks +  , ck+ – ck ≤ C ck+

()

where ck is the critical level of J defined by (). Hence, using a similar argument as in Lemma . of [], there exists an integer k ∈ N such that s

ck ≤ C k s–

for all k ≥ k .

()

On the other hand, assumption (f ) implies  J(, u) ≥ ∇u  – C u pp – C . 

()

So, by applying the arguments developed in [], the radial symmetry of the problem implies p

ck ≥ C k p– ,

if k is large enough,

()

which contradicts with (). That means case () in the result of Theorem . occurs for infinitely many indexes, and a sequence {cn } of the critical level of J(, ·) = K(·) exists such that ck → +∞ by Remark . and (). Thus, if {un }n is the corresponding sequence of critical points, by () for θ = , it follows that ∇un  → +∞. We complete the proof of Theorem ..

Competing interests We declare that we have no competing interests. Authors’ contributions All authors contributed equally to the manuscript and read and approved the final manuscript. Author details 1 Department of Mathematics, Harbin University of Science and Technology, Harbin, 150080, PR China. 2 Department of Mathematics, Harbin Engineering University, Harbin, 150001, PR China.


Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant 11226128). Received: 17 May 2013 Accepted: 11 November 2013 Published: 13 Dec 2013 References 1. D’Aprile, T, Mugnai, D: Solitary waves for nonlinear Klein-Gordon-maxwell and Schrodinger-maxwell equations. Proc. R. Soc. Edinb., Sect. A 134, 893-906 (2004) 2. Benci, V, Fortunato, D: The nonlinear Klein-Gordon equation coupled with the Maxwell equations. Nonlinear Anal. 47, 6065-6072 (2001) 3. Benci, V, Fortunato, D: Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations. Rev. Math. Phys. 14, 409-420 (2002) 4. D’Aprile, T, Mugnai, D: Non-existence results for the coupled Klein-Gordon-Maxwell equations. Adv. Nonlinear Stud. 4, 307-322 (2004) 5. Cassani, D: Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell’s equations. Nonlinear Anal. 58, 733-747 (2004) 6. Carrião, P, Cunha, P, Miyagaki, O: Positive and ground state solutions for the critical Klein-Gordon-Maxwell system with potentials. Nonlinear Anal. TMA 75, 4068-4078 (2012) 7. Azzollini, A, Pomponio, A: Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations. Topol. Methods Nonlinear Anal. 35, 33-42 (2010) 8. Carrião, P, Cunha, P, Miyagaki, O: Existence results for the Klein-Gordon-Maxwell equations in higher dimensions with critical exponents. Commun. Pure Appl. Anal. 10, 709-718 (2011) 9. Georgiev, V, Visciglia, N: Solitary waves for Klein-Gordon-Maxwell system with external Coulomb potential. J. Math. Pures Appl. 84, 957-983 (2005) 10. Mugnai, D: Coupled Klein-Gordon and Born-Infeld-type equations: looking for solitary waves. Proc. R. Soc. Lond. Ser. A 460, 1519-1527 (2004) 11. D’Avenia, P, Pisani, L, Siciliano, G: Dirichlet and Neumann problems for Klein-Gordon-Maxwell systems. Nonlinear Anal. 71, 1985-1995 (2009) 12. D’Avenia, P, Pisani, L, Siciliano, G: Klein-Gordon-Maxwell systems in a bounded domain. Discrete Contin. Dyn. Syst. 26, 135-149 (2010) 13. Chen, SJ, Tang, CL: Multiple solutions for nonhomogeneous Schrödinger-Maxwell and Klein-Gordon-Maxwell equations on R3 . NoDEA Nonlinear Differ. Equ. Appl. 17, 559-574 (2010) 14. Candela, AM, Salvatore, A: Multiple solitary waves for non-homogeneous Schrödinger-Maxwell equations. Mediterr. J. Math. 3(3-4), 483-493 (2006) 15. Candela, AM, Salvatore, A: Multiple solitary waves for non-homogeneous Klein-Gordon-Maxwell equations. In: More Progresses in Analysis, pp. 753-762 (2009) 16. Bolle, P: On the Bolza problem. J. Differ. Equ. 152, 274-288 (1999) 17. Bolle, P, Ghoussoub, N, Tehrani, H: The multiplicity of solutions in nonhomogeneous boundary value problems. Manuscr. Math. 101, 325-350 (2000) 18. Palais, RS: The principle of symmetric criticality. Commun. Math. Phys. 69, 19-30 (1979) 19. Bahri, A, Berestycki, H: A perturbation method in critical point theory and applications. Trans. Am. Math. Soc. 267, 1-32 (1981) 20. Candela, AM, Palmieri, G, Salvatore, A: Radial solutions of semilinear elliptic equations with broken symmetry. Topol. Methods Nonlinear Anal. 27, 117-132 (2006) 10.1186/1029-242X-2013-583 Cite this article as: Wu and Ge: A multiplicity result for the non-homogeneous Klein-Gordon-Maxwell system in rotationally symmetric bounded domains. Journal of Inequalities and Applications 2013, 2013:583

Page 12 of 12