An application of a variational reduction method to a nonlinear wave equation
type: Item
instance of: scholarly article
title: An application of a variational reduction method to a nonlinear wave equation
zbMATH Open document ID: 0830.35070
author: Tacksun Jung
author: Q-Heung Choi
published in: Journal of Differential Equations
publication date: 1996-02-06T00:00:00Z
review text: The authors study the existence of solutions to the boundary value problem \[ u_{tt} - u_{xx} + au^+ - bu^- = f(x,t),\;(x,t) \in \left( -{ \pi \over 2}, {\pi \over 2} \right) \times \mathbb{R}, \] \[ u \left( - {\pi \over 2}, t \right) = u \left( {\pi \over 2}, t \right) = 0, \;t \in \mathbb{R}, \quad u (x,t + \pi) = u(x,t),\;x \in \left( -{ \pi \over 2} ,{\pi \over 2} \right). \] The solutions are thus periodic in \(t\) with period \(\pi\). \(a\) and \(b\) are real constants. Several existence and uniqueness results are proved. The main result is as follows: Let \(- 1 < b < 3 < a < 7\) with \(1/ \sqrt {a + 1} + 1/ \sqrt{ b + 1 } > 1\), and let \(f(x,t) = (s \sqrt 2/ \pi) \cos (x)\). Then the problem has at least three solutions for \(s > 0\), one of which is positive. For \(s < 0\) it has at least the solution \(u = f(x,t)/(b + 1)\). The condition for \(a\) and \(b\) means that the nonlinearity crosses the first negative eigenvalue \(\lambda = - 3\) of the problem \[ u_{tt} - u_{xx} = 0,\;u \left( - {\pi \over 2}, t \right) = u \left( {\pi \over 2}, t \right) = 0,\;u(x,t + \pi) = u(x,t). \] That is, the interval \([-a, - b]\) contains this eigenvalue and no other one. For the proof, the problem is reduced to a variational problem. The mountain pass theorem is used.
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 35L70
zbMATH DE Number: 756568
zbMATH Keywords: mountain pass theorem
zbMATH Keywords: multiple solutions
reviewed by: Hans-Dieter Alber
MaRDI profile type: Publication
full work available at URL: jdeq.1995.1058
OpenAlex ID: W2035757069
DOI: 10.1006/JDEQ.1995.1058
description: scientific article; zbMATH DE number 756568