Bounded solutions for the nonlinear wave equation

We investigate the number of periodic weak solutions for the wave equation with nonlinearity decaying at the origin. We get a theorem which shows the existence of a bounded weak solution for this problem. We obtain this result by approaching the variational method and applying the critical point theory for the indefinite functional induced from the invariant subspaces and the invariant functional.

MSC:35L05, 35L70.

Let g(x,t,u) be a C^1 function from [0,\pi ]\times R\times R to R and T-periodic in t. In this paper we are concerned with the number of weak periodic solutions of the following wave equation with boundary and periodic conditions:

where T is a rational multiple of π. We assume that g satisfies the following conditions:

1. (g1)

   g\in C^1([0,\pi ]\times R\times R,R) is T-periodic in t,

2. (g2)

   g(x,t,0)=0, g(x,t,\xi )=o(|\xi |) uniformly with respect to x\in [0,\pi ] and t\in R,

3. (g3)

   there exists C>0 such that \mathrm{\forall }x\in [0,\pi ], t\in R, \xi \in R.

Nonlinear problem of this type has been considered by many authors (cf. [1–5]).

The purpose of this paper is to show the existence of T-periodic weak solutions of problem (1.1).

Our main result is as follows.

Theorem 1.1 Assume that g satisfies (g1)-(g3). Then problem (1.1) has at least one bounded solution.

For the proof of our main result, we approach the variational method and apply the critical point theory induced from the invariant subspaces and the invariant functional. The outline of the proof of Theorem 1.1 is as follows. In Section 2, we introduce two Banach spaces H and E of functions satisfying some symmetry properties, stable by A (Au=u_{tt}-u_{xx}), g such that the intersection of H with the kernel of A is reduced to 0. The search of a solution of problem (1.1) in the space H reduces the problem to a situation where A^{-1} is a compact operator. In Section 3, we introduce a functional I defined on E whose critical points and weak solutions of (1.1) possess one-to-one correspondence. We prove that I\in C^1(E,R) and satisfies the Palais-Smale condition. By a critical point theorem for indefinite functionals (cf. [6]), we prove that there exists at least one solution of (1.1) which is bounded, and we prove Theorem 1.1.

Let \mathrm{\Omega }=(0,\pi )\times (0,T); T is a rational multiple of π

where a and b are coprime integers. Let \mathcal{A} be the operator defined by

Let A be the adjoint of \mathcal{A} in L^2(\mathrm{\Omega }). We investigate the weak solutions of

We note that the eigenvalues of A are {\lambda }_{jk}=j^2-{(\frac{2\pi k}{T})}^2, j=1,2,\dots and k=0,1,2,\dots , and the corresponding eigenfunctions are

We also note that the set of functions sinjxsin\frac{2\pi kt}{T}, sinjxcos\frac{2\pi kt}{T} is an orthogonal base for L^2(\mathrm{\Omega }). Let u be a function of L^2(\mathrm{\Omega }). Then there exists one and only one function of L^2([0,\pi ]\times R) which is T-periodic in t and equals u on Ω. We shall denote this function by u. Let us denote an element u, in L^2(\mathrm{\Omega }), by

with

We assume that b is even and a is odd. Let H be the closed subspace of L^2(\mathrm{\Omega }) defined by

Then H is invariant under shift: Let h\in H and τ be a real number. If v(x,t)=u(x,t+\tau ), then v\in H. H is invariant under g: Let u\in H such that g(u)\in L^2(\mathrm{\Omega }). Then g(u)\in H. Let \tilde{u}(x,t)=u(x,t+\frac{T}{2}). Then

Therefore

Let A_1 be a linear operator of H defined by

Then A_1 is self-adjoint in H and H\cap N(A)=\{0\}, where N(A) is the kernel of A. In fact, let u\in H\cap N(A). Then

Let j and k be such that

Since b is even and a is odd, k is even. Using (2.1), we have u_{j,k}=0, and therefore H\cap N(A)=\{0\}. The eigenvalues of A_1 are j^2-{(\frac{2\pi k}{T})}^2, where j is odd and k is even and H\cap N(A)=\{0\}. Given u\in H, we write

with

Let

where (,) is a scalar product on E. With this scalar product, E is a Hilbert space with a norm

Let

By the classical theorem of Riesz (cf. [[7], p.525]), we have

Since for every ϵ>o

it follows that for every r\in [2,+\mathrm{\infty }), there is c_r\in R such that

Let

Then E=E_{+}\oplus E_{-}. Let P_{+} be the orthogonal projection from E onto E_{+} and P_{-} be the orthogonal projection from E onto E_{-}. We can write u=P_{+}u+P_{-}u for u\in E.

Now, we are going to seek a function u in E such that

We consider the associated functional of (3.1),

where

By (g1), I is well defined.

We recall the critical point theorem for the indefinite functional (cf. [6]).

Let

Theorem 3.1 (Critical point theorem for the indefinite functional)

Let E be a real Hilbert space with E=E_1\oplus E_2 and E_2=E_1^{\perp }. Suppose that I\in C^1(E,R) satisfies (PS), and that

1. (I1)

   I(u)=\frac{1}{2}(Lu,u)+bu, where Lu=L_1{P}_{1}u+L_2{P}_{2}u and L_i:E_i\to E_i is bounded and self-adjoint, i=1,2,

2. (I2)

   b^{\prime } is compact, and

3. (I3)

   there exist a subspace \tilde{E}\subset E and sets S\subset E, Q\subset \tilde{E} and constants \alpha >\omega such that

   1. (i)

      S\subset E_1 and I{|}_S\ge \alpha,

   2. (ii)

      Q is bounded and I{|}_{\partial Q}\le \omega,

   3. (iii)

      S and ∂Q link.

Then I possesses a critical value c\ge \alpha.

The eigenvalues of A_1 are {\lambda }_{jk}=j^2-{(\frac{2\pi k}{T})}^2, where j is odd and k is even, j=1,2,\dots and k=0,1,2,\dots . Thus A_1^{-1} is a compact operator.

It is convenient for the following to rearrange the eigenvalues {\lambda }_{jk}, where j is odd and k is even, by increasing magnitude: from now on we denote by the sequence of negative eigenvalues, by {({\eta }_i^{+})}_{i\ge 1}={\lambda }_{jk}>0 the sequence of positive ones, so that

We note that each eigenvalue has a finite multiplicity and that {\eta }_i^{-}\to -\mathrm{\infty }, {\eta }_i^{+}\to +\mathrm{\infty } as i\to \mathrm{\infty }.

We shall show that the functional I satisfies the geometrical assumptions of the critical point theorem for the indefinite functional.

By the following lemma, I(u)\in C^1(E,R) and the solutions of (3.1) coincide with the nonzero critical points of I(u).

Lemma 3.1 Assume that g satisfies (g1)-(g3). Then I(u) is continuous and Fréchet differentiable in E with the Fréchet derivative

for all h\in E. Moreover, if we set

then F^{\prime }(u) is continuous with respect to weak convergence, F^{\prime }(u) is compact, and

This implies that I\in C^1(E,R) and F(u) is weakly continuous.

For the proof of Lemma 3.1, refer to [6].

Lemma 3.2 Assume that g satisfies (g1)-(g3). The problem

has only a trivial solution.

Proof Let Au=u_{tt}-u_{xx}. Since H\cap N(A)=\{0\}, E\cap N(A)=\{0\}. Thus (3.4) has only a trivial solution. □

Lemma 3.3 Assume that g satisfies (g1)-(g3). Then I(u) satisfies the Palais-Smale condition: If for a sequence (u_m), I(u_m) is bounded from above and I^{\prime }(u_m)\to 0 as m\to \mathrm{\infty }, then (u_m) has a convergent subsequence.

Proof Let (u_m) be a sequence with I(u_m)\le M and I^{\prime }(u_m)\to 0 as m\to \mathrm{\infty }. It suffices to show that (u_m) is bounded. By contradiction, we suppose that \parallel u_m\parallel \to \mathrm{\infty } as m\to \mathrm{\infty }. Let w_m=\frac{u_m}{\parallel u_m\parallel }. Then \parallel w_m=1\parallel and w_m converges weakly to an element, say w. Then we have

Since G(x,t,u) is bounded, letting m∞ in (3.5), we have

On the other hand, we have

Letting m\to \mathrm{\infty } in (3.7), we have

By (3.6) and (3.8), {lim}_{m\to \mathrm{\infty }}{\parallel P_{+}{w}_m\parallel }^2={\parallel P_{+}w\parallel }^2 and {lim}_{m\to \mathrm{\infty }}{\parallel P_{-}{w}_m\parallel }^2={\parallel P_{-}w\parallel }^2. Thus {lim}_{m\to \mathrm{\infty }}\parallel w_m\parallel =\parallel w\parallel, so w_m converges strongly to w and by (3.8), w is a weak solution of the problem

By Lemma 3.2, w=0, which is absurd for the fact that \parallel w\parallel =1. Thus (u_m) is bounded. □

Let

Lemma 3.4 Assume that g satisfies the conditions (g1)-(g3). There exist a subspace \tilde{E}\subset E and sets \partial B_{\rho }\subset E, Q\subset \tilde{E} and constants \rho >0 and \alpha >0 such that

1. (i)

   \partial B_{\rho }\subset E_{+} and I{|}_{\partial B_{\rho }}\ge \alpha,

2. (ii)

   there are e\in \partial B_1\cap E_{+} and R>\rho such that if , then I{|}_{\partial Q}\le 0,

3. (iii)

   there exists u_0\in E\mathrm{\setminus }Q such that \parallel u_0\parallel >R and I(u_0)\le 0,

4. (iv)

   \partial B_{\rho } and ∂Q link.

Proof (i) Let u\in E_{+}. Since G(x,t,u) is bounded, there exists a constant C>0 such that

for C>0. Then there exist a constant \rho >0 and a constant \alpha >0 such that if u\in \partial B_{\rho }\cap E_{+}, then I(u)\ge \alpha.

(ii) Let us choose an element e\in B_1\cap E_{+}. Let . Then u=v+w, v\in B_r\cap E_{-}, w=re. We note that

Thus we have

for C_1>0. Then there exists R>\rho >0 such that if , then I(u){|}_{\partial Q}\le 0.

(iii) Let us choose u_0\in E\mathrm{\setminus }Q. By (ii), I(u_0)\le 0.

(iv) S and Q link at the set \{e\}. □

Proof of Theorem 1.1 By Lemma 3.1, I(u) is continuous and Fréchet differentiable in E. By Lemma 3.3, the functional I satisfies the (PS) condition. We note that I(0)=0. By Lemma 3.4, there are constants \rho >0, \alpha >0 and a bounded neighborhood B_{\rho } of 0 such that I{|}_{\partial B_{\rho }\cap E_{+}}\ge \alpha, and there are e\in \partial B_1\cap E_{+} and R>\rho such that if . Let us set

By Theorem 3.1, I possesses a critical value c\ge \alpha. Moreover, c can be characterized as

Thus we prove that I has at least one nontrivial critical point, we denote by \tilde{u} a critical point of I such that I(\tilde{u})=c. We claim that c is bounded. In fact, we have

and by (g3),

for some constant C_1>0. Since 0\le \tau \le 1, c is bounded:

We claim that \tilde{u} is bounded. In fact, by contradiction, {\tilde{u}}_{tt}+{\tilde{u}}_{xx}=g(x,t,\tilde{u}) and, for any K>0, {max}_{\mathrm{\Omega }}|\tilde{u}(x,t)|>K imply that

is not bounded, which is absurd to the fact that c=I(\tilde{u}) is bounded. Thus \tilde{u} is bounded, so (1.1) has at least one bounded weak solution, and hence we prove Theorem 1.1. □

This work (Choi) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (KRF-2013010343).

Affiliations

1. Department of Mathematics, Kunsan National University, Kunsan, 573-701, Korea

   • Tacksun Jung

2. Department of Mathematics Education, Inha University, Incheon, 402-751, Korea

   • Q-Heung Choi

Corresponding author

Correspondence to Q-Heung Choi.

Competing interests

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The authors did not provide this information.

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