The multiplicity of solutions and geometry in a wave equation (Q1424086)

Using a variational reduction method and degree theory, the authors study the multiplicity of the solutions of the periodic-Dirichlet problem (with the solution even in \(t\) and \(x,\) on \([-\pi/2,\pi/2] \times [-\pi/2,\pi/2],\) for the wave equation \[ u_{tt} - u_{xx} + bu^+ - au^- = f(x,t). \] The source term \(f\) is generated by the eigenfunctions of the one-dimensional wave operator. The authors investigate the multiplicity of solutions when the nonlinearity crosses an eigenvalue and \(f\) is generated by two or three eigenfunctions.