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Bounded solutions for the nonlinear wave equation
Boundary Value Problemsvolume 2013, Article number: 257 (2013)
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.
1 Introduction and statement of the main result
Let be a function from 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:
is T-periodic in t,
, uniformly with respect to and ,
there exists such that , , .
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 (), 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 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 and satisfies the Palais-Smale condition. By a critical point theorem for indefinite functionals (cf. ), we prove that there exists at least one solution of (1.1) which is bounded, and we prove Theorem 1.1.
2 Invariant Banach space
Let ; T is a rational multiple of π
where a and b are coprime integers. Let be the operator defined by
Let A be the adjoint of in . We investigate the weak solutions of
We note that the eigenvalues of A are , and , and the corresponding eigenfunctions are
We also note that the set of functions , is an orthogonal base for . Let u be a function of . Then there exists one and only one function of 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 , by
We assume that b is even and a is odd. Let H be the closed subspace of defined by
Then H is invariant under shift: Let and τ be a real number. If , then . H is invariant under g: Let such that . Then . Let . Then
Let be a linear operator of H defined by
Then is self-adjoint in H and , where is the kernel of A. In fact, let . Then
Let j and k be such that
Since b is even and a is odd, k is even. Using (2.1), we have , and therefore . The eigenvalues of are , where j is odd and k is even and . Given , we write
where is a scalar product on E. With this scalar product, E is a Hilbert space with a norm
By the classical theorem of Riesz (cf. [, p.525]), we have
Since for every
it follows that for every , there is such that
Then . Let be the orthogonal projection from E onto and be the orthogonal projection from E onto . We can write for .
3 Critical point theorem and the proof of Theorem 1.1
Now, we are going to seek a function u in E such that
We consider the associated functional of (3.1),
By (g1), I is well defined.
We recall the critical point theorem for the indefinite functional (cf. ).
Theorem 3.1 (Critical point theorem for the indefinite functional)
Let E be a real Hilbert space with and . Suppose that satisfies (PS), and that
, where and is bounded and self-adjoint, ,
is compact, and
there exist a subspace and sets , and constants such that
Q is bounded and ,
S and ∂Q link.
Then I possesses a critical value .
The eigenvalues of are , where j is odd and k is even, and . Thus is a compact operator.
It is convenient for the following to rearrange the eigenvalues , where j is odd and k is even, by increasing magnitude: from now on we denote by the sequence of negative eigenvalues, by the sequence of positive ones, so that
We note that each eigenvalue has a finite multiplicity and that , as .
We shall show that the functional I satisfies the geometrical assumptions of the critical point theorem for the indefinite functional.
By the following lemma, and the solutions of (3.1) coincide with the nonzero critical points of .
Lemma 3.1 Assume that g satisfies (g1)-(g3). Then is continuous and Fréchet differentiable in E with the Fréchet derivative
for all . Moreover, if we set
then is continuous with respect to weak convergence, is compact, and
This implies that and is weakly continuous.
For the proof of Lemma 3.1, refer to .
Lemma 3.2 Assume that g satisfies (g1)-(g3). The problem
has only a trivial solution.
Proof Let . Since , . Thus (3.4) has only a trivial solution. □
Lemma 3.3 Assume that g satisfies (g1)-(g3). Then satisfies the Palais-Smale condition: If for a sequence , is bounded from above and as , then has a convergent subsequence.
Proof Let be a sequence with and as . It suffices to show that is bounded. By contradiction, we suppose that as . Let . Then and converges weakly to an element, say w. Then we have
Since is bounded, letting m∞ in (3.5), we have
On the other hand, we have
Letting in (3.7), we have
By (3.6) and (3.8), and . Thus , so converges strongly to w and by (3.8), w is a weak solution of the problem
By Lemma 3.2, , which is absurd for the fact that . Thus is bounded. □
Lemma 3.4 Assume that g satisfies the conditions (g1)-(g3). There exist a subspace and sets , and constants and such that
there are and such that if , then ,
there exists such that and ,
and ∂Q link.
Proof (i) Let . Since is bounded, there exists a constant such that
for . Then there exist a constant and a constant such that if , then .
(ii) Let us choose an element . Let . Then , , . We note that
Thus we have
for . Then there exists such that if , then .
(iii) Let us choose . By (ii), .
(iv) S and Q link at the set . □
Proof of Theorem 1.1 By Lemma 3.1, is continuous and Fréchet differentiable in E. By Lemma 3.3, the functional I satisfies the (PS) condition. We note that . By Lemma 3.4, there are constants , and a bounded neighborhood of 0 such that , and there are and such that if . Let us set
By Theorem 3.1, I possesses a critical value . Moreover, c can be characterized as
Thus we prove that I has at least one nontrivial critical point, we denote by a critical point of I such that . We claim that c is bounded. In fact, we have
and by (g3),
for some constant . Since , c is bounded:
We claim that is bounded. In fact, by contradiction, and, for any , imply that
is not bounded, which is absurd to the fact that is bounded. Thus is bounded, so (1.1) has at least one bounded weak solution, and hence we prove Theorem 1.1. □
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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).
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