An application of a variational reduction method to a nonlinear wave equation
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.
scientific article; zbMATH DE number 756568