An Application of Category Theory to a Class of the Systems of the Superquadratic Wave Equations

We investigate the nontrivial solutions for a class of the systems of the superquadratic nonlinear wave equations with Dirichlet boundary condition and periodic condition with a superquadratic nonlinear terms at infinity which have continuous derivatives. We approach the variational method and use the critical point theory on the manifold, in terms of the limit relative category of the sublevel subsets of the corresponding functional.


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 F : −π/2, π/2 × R × R × R → R is a superquadratic function at infinity which has continuous derivatives F r x, t, r, s , F s x, t, r, s with respect to r, s, for almost any 2
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 u 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 u, v . In this paper we show by the limit relative category theory that system 1.1 has at least two nontrivial solutions u, v .
Let us set Then system 1.1 can be rewritten by We note that √ ab, − √ ab are two eigenvalues of the matrix A Let λ mn be the eigenvalues of the eigenvalue problem u tt − u xx λu in −π/2, π/2 × R, u ±π/2, t 0, u x, t π u x, t u −x, t u x, −t . Our main result is the following. In Section 2, we obtain some results on the nonlinear term F. 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.

2.7
Let us set E D × D. We endow the Hilbert space E with the norm We are looking for the weak solutions of 1.
Since |λ mn | ≥ 1 for all m, n, we have the following lemma.
We state the lemmas. For the proofs of Lemmas 2.1, 2.2, and 2.3, we refer 4 . By F1 and F3 , we obtain the lower bound for F x, t, u, v in the term of |u| μ |v| μ .

Lemma 2.3. Assume that F satisfies the conditions (F1) and (F3). Then there exist
then there exists u h n , v h n n and w ∈ E such that Proof. i Follows from F1 and F2 , since 1 < ν. ii iii Is easily obtained with standard arguments. iv Is implied by F3 and the fact that and suitable constants C , C . To get the conclusion it suffices to estimate |U| ν / U E L r in terms of U μ L μ / U E . If μ ≥ rν, then this is a consequence of Hölder inequality. If μ < rν, by the standard interpolation arguments, it follows that where l is such that l −1 ν/μ. Thus we prove v .
Journal of Inequalities and Applications

2.18
Thus ii follows.

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 I is strongly indefinite functional, it is convenient to use P.S. * c condition and the limit relative category which is a suitable version of P.S. c condition and the relative category, respectively. Now, we consider the critical point theory on the manifold with boundary. Let E be a Hilbert space and let M be the closure of an open subset of E such that M can be endowed with the structure of C 2 manifold with boundary. Let f : W → R be a C 1,1 functional, where W is an open set containing M. For applying the usual topological methods of critical points theory we need a suitable notion of critical point for f on M. We recall the following notions: lower gradient of f on M, P.S. * c condition, and the limit relative category see 4 .
where we denote by ν u the unit normal vector to ∂M at the point u, pointing outwards. We say that u is a lower critical for f on M, if grad − M f u 0. Since the functional I u which is introduced in Section 4 is strongly indefinite, the notion of the P.S. * c condition and the limit relative category is a very useful tool for the proof of the main theorems.
Let E − , E 0 , E be the subspace of E on which the functional U → 1/2 Q LU · U is positive definite, null, negative definite, and E − , E 0 , and E are mutually orthogonal. Let P be the projection for E onto E , P 0 the one from E onto E 0 , and P − the one from E onto E − . Let E n n be a sequence of closed subspaces of E with the conditions: E n and E − n are subspaces of E , dim E n < ∞, E n ⊂ E n 1 , n∈N E n is dense in E. Let P E n be the orthogonal projections from E onto E n . M n M ∩ E n , for any n, and let be the closure of an open subset of E n and have the structure of a C 2 manifold with boundary in E n . We assume that for any n there exists a retraction r n : M → M n . For given B ⊂ E, we will write B n B ∩ E n .
We say that f satisfies the P.S. * c condition with respect to M n n , on the manifold with boundary M, if for any sequence k n n in N and any sequence u n n in M such that k n → ∞, u n ∈ M k n , for all n, f u n → c, grad − M kn f u n → 0, there exists a subsequence of u n n which converges to a point u ∈ M such that grad − M f u 0.
Let Y be a closed subspace of M.

3.5
If such an h does not exist, we say that cat M,Y B ∞.
Definition 3.4. Let X, Y be a topological pair and let X n n be a sequence of subsets of X. For any subset B of X we define the limit relative category of B in X, Y , with respect to X n n , 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 Y be a fixed subset of M. We set We have the following multiplicity theorem.
Proof. Let c c i ; using the P.S. * c condition, with respect to M n n , one can prove that, for any neighbourhood N of Journal of Inequalities and Applications 9 there exist n 0 in N and δ > 0 such that grad − M ≥ δ for all n ≥ n 0 and all x ∈ E n \ N with c − δ ≤ f x ≤ c δ. Moreover it is not difficult to see that, for all n, the function f n : E n → R ∪ { ∞} defined by f n f x , if x ∈ M n , f n x ∞, otherwise, is φ-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 h : H → R ∪ { ∞} be a lower semicontinuous function and assume h to be φ-convex of order 2 (see [3]). Let c ∈ R, δ > 0, and D be a closed set in H such that Then there exists > 0 and a continuous deformation η : Proof. See 6, Lemmas 4.5 and 4.6 .
Now we state the following multiplicity result for the proof see 7, Theorem 4.6 which will be used in the proofs of our main theorems. Theorem 3.7. Let H be a Hilbert space and let H X 1 ⊕ X 2 ⊕ X 3 , where X 1 , X 2 , X 3 are three closed subspaces of H with X 1 , X 2 of finite dimension. For a given subspace X of H, let P X be the orthogonal projection from H onto X. Set 3.12 and let f : W → R be a C 1,1 function defined on a neighborhood W of C. Let 1 < ρ < R, R 1 > 0. We define

3.13
Assume that sup f Σ < inf f S 3.14 and that the P.S. c condition holds for f on C, with respect to the sequence C n n , for all c ∈ α, β , where

3.15
Moreover one assumes β < ∞ and f| X 1 ⊕X 3 has no critical points z in Then there exist two lower critical points z 1 , z 2 for f on C such that α ≤ f z i ≤ β, i 1.2.

Proof of Theorem 1.1
Let I 1,1 loc E, R be the functional defined in 3.2 . Let Y be a closed subspace of E with finite dimension. Let us set Then E is the topological direct sum of the subspaces X 1 , X 2 , and X 3 . Let P X be the orthogonal projection from E onto X. Let us set Then C is the smooth manifold with boundary. Let C n C ∩ E n . Let us define a functional Ψ : We have Let us define the constrained functional I : C → R by Then I ∈ C 1,1 loc . It turns out that

4.6
Journal of Inequalities and Applications

11
We note that if U is the critical point of I and lies in the interior of C, then U Ψ U is the critical point of I. Thus it suffices to find the critical points, which lies in the interior of C, for I. We also note that Let us set

4.8
We will prove the multiplicity result by using Theorem 3.7 for I, C, S 23 ρ , Δ 12 R, R 1 , and Σ 12 R, R 1 . Now we have the following linking geometry for I.

4.12
Since √ ab < 1 λ 00 , there exist a small number ρ > 0 and a small sphere S 23 ρ ⊂ E with radius ρ such that if U ∈ S 23 ρ ⊂ E , then inf I U > 0.
Next we will show that there exist R > ρ, R 1 > 0, and R > 1 such that for some τ 1 > 0, τ 2 > 0. Let us choose a sequence U n n , U n u n , v n such that U n E → ∞. Let us setǓ n U n / U n E . By Lemma 2.3, we have that Since U n E → ∞, two possible cases arise. For the case Ǔ n L μ → 0 it follows thatǓ n 0, and hence P Ǔ n → 0 and P 0Ǔ n → 0. Proof. It suffices to prove that I has no critical point U ψ U such that I U c and U ∈ X 1 ⊕ X 3 . We notice that from Lemma 4.1, for fixed U 1 ∈ X 1 , the functional U 3 → I U 1 U 3 is weakly convex in X 3 , while, for fixed U 3 ∈ X 3 , the functional U 1 → I U 1 U 3 is strictly concave in X 1 . Moreover 0, 0 is a critical point in X 1 ⊕ X 3 with I 0, 0 0. So if U U 1 U 3 is another critical point for I| X 1 ⊕X 3 , then we have We shall prove that the functional I satisfies the P.S. * c condition with respect to C n n for any c ∈ α, β , where α inf W∈ S 23 ρ I W and β sup V ∈ Δ 12 R,R 1 I V .
To prove that I satisfies the P.S. * c condition with respect to C n n for any c ∈ α, β , we first shall prove that I satisfies the P.S. * c condition with respect to E n n for any real number c ∈ R. Proof. Let c ∈ R and h n be a sequence in N such that h n → ∞, and let U n n be a sequence such that U n u n , v n ∈ E h n , ∀n, I U n −→ c, P E hn ∇I U n −→ 0.

4.19
We claim that U n n is bounded. By contradiction we suppose that U n E → ∞ and set U n U n / U n E . Then P E hn ∇I U n , U n ∇I U n , U n 2 I U n U n E − Q ∇F x, t, U n · U n dxdt − 2 Q F x, t, U n dxdt U n E −→ 0.

4.20
Hence Q ∇F x, t, U n · U n dxdt − 2 Q F x, t, U n dxdt U n E −→ 0.

4.21
By v of Lemma 2.4, grad Q F x, t, U n dxdt U n E converges 4.22 Using the properties of the projections we get P E hn P X 1 ⊕X 3 ∇I Z n −→ 0, 4.27 which contradicts to Lemma 4.2. In fact, let Z be the limit point of the subsequence Z k n of Z n , then Z ∈ ∂C and grad − C I Z P X 1 ⊕X 3 grad I Z − grad I Z , P X 2 Z P X 2 Z.

4.28
Proof of Theorem 1.1. We assume that the conditions 1.4 , 1.5 , and 1.6 hold and F satisfies F1 , F2 , F3 , and F4 . We note that I : C → R ∈ C 1,1 loc and by Lemma 4.1, there exist R > ρ > 0, R 1 > 0 and R > 1 such that sup V ∈ Σ 12 R,R 1 I V < inf W∈ S 23 ρ I W . 4.29 By Lemma 4.2, I has no critical point U in X 1 ⊕ X 3 whose critical value is c > 0. By Lemma 4.4, I satisfies the P.S. * c condition with respect to C n n for any c ∈ α, β , α > 0. Thus by Theorem 3.7, I has at least two critical points U i , i 1, 2, in int C with 0 < inf