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A multiplicity result for the nonhomogeneous KleinGordonMaxwell system in rotationally symmetric bounded domains
Journal of Inequalities and Applications volume 2013, Article number: 583 (2013)
Abstract
This paper is concerned with the KleinGordon equation coupled with the Maxwell equation in the rotationally symmetric bounded domains when a nonhomogeneous term breaks the symmetry of the associated functional. Under some suitable assumption on nonlinear perturbation, we obtain infinitely many radially symmetric solutions to the nonhomogeneous KleinGordonMaxwell system.
1 Introduction
This paper is concerned with the existence and multiplicity of solutions for a class of KleinGordonMaxwell (KGM for short) systems in rotationally symmetric bounded domains. We fix 0\le \rho <\mathrm{\infty} and define
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).
We consider the KGM system in this type symmetric bounded domains (the model in a bounded domain was introduced by D’Avenia et al. [1])
with Dirichlet boundary conditions
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 KleinGordon field in a threedimensional 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 KleinGordonMaxwell 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]. Nonexistence 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 nonhomogeneous case g\ne 0, only a few results are known for KGM systems. In [13], Chen and Tang proved two different solutions for the nonhomogeneous KGM equations in the whole {R}^{3}. Candela and Salvatore [14, 15] dealt with multiplicity of solutions to nonhomogeneous SchrödingerMaxwell and KleinGordonMaxwell systems with homogenous boundary conditions in a bounded ball, respectively. Inspired by Candela and Salvatore’s results, in this paper we deal with the nonhomogeneous 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=0, problem (1)(2) can be split into
and
The existence and uniqueness of solutions of problem (3) and (4) are independent of each other. And it is clear that problem (4) has a unique solution. In this paper, we mainly consider the coupled case q\ne 0. In this case, the change of variables
transforms system (1)(2) into
with
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}{N2}) such that f(r,u)\le {a}_{1}+{a}_{2}{u}^{p1}.
(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 nonhomogeneous 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 AmbrosettiRabinowitz 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 nonhomogeneous 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.
To get homogeneous boundary conditions, we change variables as follows:
where {\varphi}_{0} is the unique solution of
Then problem (5) can be rewritten as
Remark 2.1 By the maximum principle, (6) implies
System (8) contains the EulerLagrange equations related to the functional
defined as
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 wellknown 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 LaxMilgram 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.
Since, by Proposition 2.1, the functional {\phi}_{u} satisfies
we obtain, for any u\in {H}_{0}^{1}(\mathrm{\Omega}),
According to Lemma 2.1 and Proposition 2.1, for every u,v\in {H}_{0}^{1}(\mathrm{\Omega}), we have
Hence we get, as an operator in {H}^{1}(\mathrm{\Omega}),
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.
Let 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.
Let I:[0,1]\times E\to R be a {C}^{1}functional, and set
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:
(H_{1}) I satisfies the following weaker form of the PalaisSmale 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.
(H_{2}) For all b>0, there exists a constant {C}_{b}>0 such that
(H_{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 ),
(H_{4}) I(0,v) is even and for any finite dimensional space W of E, we have
Let {\chi}_{i}:[0,1]\times R\to R (i=1,2) be the flow associated to {\eta}_{i} that is the solution of problem
Note that {\chi}_{1} and {\chi}_{2} are continuous and that for all \theta \in [0,1], {\chi}_{1}(\theta ,\cdot ) and {\chi}_{2}(\theta ,\cdot ) they are nondecreasing on 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 nondecreasing 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
Now, we consider the family of functionals J(\theta ,u):[0,1]\times {H}_{0,r}^{1}(\mathrm{\Omega})\to R defined by
with
Clearly, 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,
with
By Remark 2.2, we get {\varphi}_{0}\le {\phi}_{u}+{\varphi}_{0}\le 0 and \frac{1}{2}{\varphi}_{0}\le (\frac{1}{2}\delta ){\varphi}_{0}\delta {\phi}_{u}\le (\frac{1}{2}\delta ){\varphi}_{0}. Hence,
By assumption (f_{3}), there exist two constants {b}_{1},{b}_{2}>0 such that for every t\in R,
Hence, we get
According to Remark 2.1, there exists {\epsilon}_{0}>0 such that
Using the CauchySchwarz inequality, we get
Then equation (23) becomes
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
is the unique solution of the equation
(2) Case \rho >0, 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}).
Let O(N)=\{{A}_{N\times N}:\text{orthogonal matrices}\}. Consider the O(N) group action {T}_{g} on {L}^{2}(\mathrm{\Omega}) defined by
Since for each u\in {H}_{0}^{1}(\mathrm{\Omega}), {\phi}_{u} is the unique solution of
so, for any g\in O(N), we have {T}_{g}(\mathrm{\Delta}{\phi}_{u})={T}_{g}({\phi}_{u}){T}_{g}({u}^{2})+{T}_{g}({\varphi}_{0}){T}_{g}({u}_{0}^{2}) in Ω. By radial symmetry of {\varphi}_{0} (Lemma 3.2), we get
Then, by the definition of \mathrm{\Phi}:u\to {\phi}_{u}, we obtain
Using (30) and the {T}_{g} invariance of the norm in {H}_{r}^{1}(\mathrm{\Omega}), {L}^{p}(\mathrm{\Omega}), we deduce that 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}).
Firstly, we prove that 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
Let {\phi}_{n}=\mathrm{\Phi}[{u}_{n}]. Then the expression of \frac{\partial J}{\partial u} and (15) implies
with \epsilon \to 0. Clearly, Lemma 3.1 implies {\{\parallel \mathrm{\nabla}{u}_{n}\parallel \}}_{n} is bounded. By assumption (f_{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}{p1}}(\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.
By equation (20), for all (\theta ,u)\in [0,1]\times {H}_{0,r}^{1}(\mathrm{\Omega}),
Hence Lemma 3.1 implies that J satisfies assumption (H_{2}) of Theorem 2.1.
If (\theta ,u)\in [0,1]\times {H}_{0,r}^{1}(\mathrm{\Omega}) satisfies \frac{\partial J}{\partial u}(\theta ,u)=0, then
with {\alpha}_{4}=\sqrt{({\alpha}_{1}+{\alpha}_{3})max\{{\alpha}_{1},{\alpha}_{3}\}}. So, there exists a positive constant {C}_{3} such that
by (33). Hence, 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.
Now we apply Theorem 2.1 to 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
where {c}_{k} is the critical level of 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
On the other hand, assumption (f_{1}) implies
So, by applying the arguments developed in [20], the radial symmetry of the problem 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. □
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This work is supported by the National Natural Science Foundation of China (Grant 11226128).
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Wu, Y., Ge, B. A multiplicity result for the nonhomogeneous KleinGordonMaxwell system in rotationally symmetric bounded domains. J Inequal Appl 2013, 583 (2013). https://doi.org/10.1186/1029242X2013583
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DOI: https://doi.org/10.1186/1029242X2013583