1. Introduction

We investigate the nontrivial solutions for a class of the systems of the superquadratic nonlinear wave equations with Dirichlet boundary condition and periodic condition: where is a superquadratic function at infinity which has continuous derivatives , with respect to , , for almost any . Moreover we assume that satisfies the following conditions:(F1) =  =  = ,  =  =   = , if , ; for all ,;(F3) for all ;(F4),

where , , , , and , for some .

As the physical model for these systems we can find crossing two beams with travelling waves, which are suspended by cable under a load. The nonlinearity models the fact that cables resist expansion but do not resist compression.

Choi and Jung [1–3] investigate the existence and multiplicity of solutions of the single nonlinear wave equation with Dirichlet boundary condition. In [4] the authors show by critical point theory (Linking Theorem) that system (1.1) has at least one nontrivial solution . In this paper we show by the limit relative category theory that system (1.1) has at least two nontrivial solutions .

Let us set Then system (1.1) can be rewritten by where is the gradient operator, , .

We note that , are two eigenvalues of the matrix

Let be the eigenvalues of the eigenvalue problem in , , .

Our main result is the following.

Theorem 1.1. Assume that Then, for any with (F1), (F2), (F3) and (F4), system (1.3) has at least two nontrivial solutions .

In Section 2, we obtain some results on the nonlinear term . In Section 3, we approach the variational method and recall the abstract results of the critical point theory on the manifold in terms of the limit relative category of the sublevel sets of the corresponding functional of (1.3), which plays a crucial role to prove the multiplicity result. In Section 4, we prove Theorem 1.1.

2. Some Results on the Nonlinear Term

The eigenvalue problem for has infinitely many eigenvalues and corresponding normalized eigenfunctions given by Let be the square and the Hilbert space defined by The set of functions is an orthonormal basis in . Let us denote an element , in , by We define a Hilbert space as follows: Then this space is a Banach space with norm Let us set . We endow the Hilbert space with the norm We are looking for the weak solutions of (1.3) in , that is, such that , , , . Since for all , , we have the following lemma.

We state the lemmas. For the proofs of Lemmas 2.1, 2.2, and 2.3, we refer [4].

Lemma 2.1. (i), where denotes the norm of .
(ii) if and only if .
(iii) implies .

Lemma 2.2. Suppose that is not an eigenvalue of , , and let . Then one has .

By (F1) and (F3), we obtain the lower bound for in the term of .

Lemma 2.3. Assume that satisfies the conditions (F1) and (F3). Then there exist with such that

Lemma 2.4. Assume that satisfies the conditions (F1), (F2), and (F3). Then (i) = ,    if  ,    =  as ;(ii)there exist , and such that (iii) is a compact map;(iv)if   +  = ,  then    = ;(v)if and then there exists and such that

Proof. (i) Follows from (F1) and (F2), since .
(ii) By Lemma 2.3, for , where . Thus (ii) holds.
(iii) Is easily obtained with standard arguments.
(iv) Is implied by (F3) and the fact that for .
(v) By Lemma 2.3 and (F3), for ,
By (F2), and suitable constants , . To get the conclusion it suffices to estimate in terms of . If , then this is a consequence of Hölder inequality. If , by the standard interpolation arguments, it follows that , where is such that . Thus we prove (v).

Lemma 2.5. Assume that satisfies the conditions (F1), (F2), (F3), and (F4). Then there exist , continuous and such that (i), for all ,(ii), for all .

Proof. (i) By (F4), for all , where , , , , , and are constants. Since , (i) follows.
(ii) By (F3), Thus (ii) follows.

3. Abstract Results of Critical Point Theory

Now we are looking for the weak solutions of system (1.3). We shall approach the variational method and recall the abstract results of the critical point theory on the manifold in terms of the limit relative category of the sublevel sets of the functional of (1.3). We observe that the weak solutions of (1.3) coincide with the critical points of the corresponding functional: Now we recall the critical point theory for strongly indefinite functional. Since the functional is strongly indefinite functional, it is convenient to use condition and the limit relative category which is a suitable version of condition and the relative category, respectively.

Now, we consider the critical point theory on the manifold with boundary. Let be a Hilbert space and let be the closure of an open subset of such that can be endowed with the structure of manifold with boundary. Let be a functional, where is an open set containing . For applying the usual topological methods of critical points theory we need a suitable notion of critical point for on . We recall the following notions: lower gradient of on , condition, and the limit relative category (see [4]).

Definition 3.1. If , the lower gradient of on at is defined by where we denote by the unit normal vector to at the point , pointing outwards. We say that is a lower critical for on , if .
Since the functional (which is introduced in Section 4) is strongly indefinite, the notion of the condition and the limit relative category is a very useful tool for the proof of the main theorems.
Let , , be the subspace of on which the functional is positive definite, null, negative definite, and , , and are mutually orthogonal. Let be the projection for onto , the one from onto , and the one from onto . Let be a sequence of closed subspaces of with the conditions: ( and are subspaces of , , , is dense in
Let be the orthogonal projections from onto . , for any , and let be the closure of an open subset of and have the structure of a manifold with boundary in . We assume that for any there exists a retraction . For given , we will write .

Definition 3.2. Let . We say that satisfies the condition with respect to , on the manifold with boundary , if for any sequence in and any sequence in such that , , for all , , , there exists a subsequence of which converges to a point such that .

Let be a closed subspace of .

Definition 3.3. Let be a closed subset of with . We define the relative category of in , as the least integer such that there exist closed subsets , with the following properties:; are contractible in ; and there exists a continuous map such that If such an does not exist, we say that .

Definition 3.4. Let be a topological pair and let be a sequence of subsets of . For any subset of we define the limit relative category of in , with respect to , by

Now we consider a theorem which gives an estimate of the number of critical points of a functional, in terms of the limit relative category of its sublevels. The theorem is proved repeating the classical arguments, using the nonsmooth version of the classical Deformation Lemma for functions on manifolds with boundary.

Let be a fixed subset of . We set We have the following multiplicity theorem.

Theorem 3.5. Let and assume that (1),(2),(3)the condition with respect to holds.Then there exists a lower critical point such that . If then

Proof. Let ; using the condition, with respect to , one can prove that, for any neighbourhood of there exist in and such that for all and all with . Moreover it is not difficult to see that, for all , the function defined by , if , , otherwise, is -convex of order two, according to the definitions of [3]. Then the conclusion follows using the same arguments of [4, 5] and the nonsmooth version of the classical Deformation Lemma.

Lemma 3.6 (Deformation Lemma). Let be a lower semicontinuous function and assume to be -convex of order 2 (see [3]). Let , , and be a closed set in such that Then there exists and a continuous deformation ( is the -neighborhood of and ) such that (i) for all ,(ii) for all for all ,(iii) for all , for all .

Proof. See [6, Lemmas and 4.6].

Now we state the following multiplicity result (for the proof see [7, Theorem ]) which will be used in the proofs of our main theorems.

Theorem 3.7. Let be a Hilbert space and let , where , , are three closed subspaces of with , of finite dimension. For a given subspace of , let be the orthogonal projection from onto . Set and let be a function defined on a neighborhood of . Let , . We define Assume that and that the condition holds for on , with respect to the sequence , for all , where Moreover one assumes and has no critical points in with . Then there exist two lower critical points ,