Bounded solutions for the nonlinear wave equation
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Bounded solutions for the nonlinear wave equation Tacksun Jung1 and Q-Heung Choi2* * Correspondence: [email protected] 2 Department of Mathematics Education, Inha University, Incheon, 402-751, Korea Full list of author information is available at the end of the article
Abstract 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 Keywords: wave equation; critical point theory; invariant function; invariant subspace; (P.S.)c condition; eigenvalue problem
1 Introduction and statement of the main result Let g(x, t, u) be a C function from [, π] × R × 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: utt – uxx = g(x, t, u), u(, t) = u(π, t) = , (.) u(x, t + T) = u(x, t), ut (x, ) = ut (x, T) ∀x ∈ [, π], where T is a rational multiple of π . We assume that g satisfies the following conditions: (g) g ∈ C ([, π] × R × R, R) is T-periodic in t, (g) g(x, t, ) = , g(x, t, ξ ) = o(|ξ |) uniformly with respect to x ∈ [, π] and t ∈ R, (g) there exists C > such that |g(x, t, ξ )| < C ∀x ∈ [, π], t ∈ R, ξ ∈ R. Nonlinear problem of this type has been considered by many authors (cf. [–]). The purpose of this paper is to show the existence of T-periodic weak solutions of problem (.). Our main result is as follows. Theorem . Assume that g satisfies (g)-(g). Then problem (.) 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. ©2013 Jung and Choi; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Jung and Choi Boundary Value Problems 2013, 2013:257 http://www.boundaryvalueproblems.com/content/2013/1/257
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The outline of the proof of Theorem . is as follows. In Section , we introduce two Banach spaces H and E of functions satisfying some symmetry properties, stable by A (Au = utt – uxx ), g such that the intersection of H with the kernel of A is reduced to . The search of a solution of problem (.) in the space H reduces the problem to a situation where A– is a compact operator. In Section , we introduce a functional I defined on E whose critical points and weak solutions of (.) possess one-to-one correspondence. We prove that I ∈ C (E, R) and satisfies the Palais-Smale condition. By a critical
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