Bounded solutions for the nonlinear wave equation
© Jung and Choi; licensee Springer. 2013
Received: 27 June 2013
Accepted: 25 October 2013
Published: 25 November 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
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
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
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)
, where and is bounded and self-adjoint, ,
is compact, and
- (I3)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.
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 .
This implies that and is weakly continuous.
For the proof of Lemma 3.1, refer to .
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.
By Lemma 3.2, , which is absurd for the fact that . Thus is bounded. □
there are and such that if , then ,
there exists such that and ,
and ∂Q link.
for . Then there exist a constant and a constant such that if , then .
for . Then there exists such that if , then .
(iii) Let us choose . By (ii), .
(iv) S and Q link at the set . □
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. □
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).
- Amann H, Zehnder E: Multiple periodic solutions for a class of nonlinear-autonomous wave equations. Houst. J. Math. 1981, 7: 147-174.MATHMathSciNetGoogle Scholar
- Brezis H, Coron JM: Periodic solutions of nonlinear wave equations and Hamiltonian systems. Am. J. Math. 1981, 103: 559-570. 10.2307/2374104MATHMathSciNetView ArticleGoogle Scholar
- Brezis H, Nirenberg L: Forced vibrations for a nonlinear wave equation. Commun. Pure Appl. Math. 1978, 31: 1-30.MATHMathSciNetView ArticleGoogle Scholar
- Choi QH, Jung T: An application of a variational reduction method to a nonlinear wave equation. J. Differ. Equ. 1995, 7: 390-410.MathSciNetView ArticleGoogle Scholar
- Rabinowitz PH: Free vibrations for a semilinear wave equation. Commun. Pure Appl. Math. 1968, 31: 31-68.MathSciNetView ArticleGoogle Scholar
- Benci V, Rabinowitz PH: Critical point theorems for indefinite functionals. Invent. Math. 1979, 52: 241-273. 10.1007/BF01389883MATHMathSciNetView ArticleGoogle Scholar
- Fadell ER, Rabinowitz PH: Generalized cohomological index theories for the group actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math. 1978, 45: 139-174. 10.1007/BF01390270MATHMathSciNetView ArticleGoogle Scholar
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