Complex-valued time-periodic solutions and breathers of nonlinear wave equations (Q1175624)

This paper is devoted to the nonlinear wave equation \(u_{tt}-\Delta_ xu+g(u)=f(t,x)\) on the exterior domain \(R_ t\times\{x\in \mathbb{R}^ n:| x|>R\}\), with the periodic-boundary conditions \(u(t+T,x)=u(t,x)\), \(u(t,.)\in L_ 2(\{| x|>R\})\). The forcing terms \(f(t,x)\) is assumed to be a complex function, \(T\)-period in \(t\) and radially symmetric in \(x\), decaying to zero in a suitable way as \(| x|\to\infty\), while the nonlinearity \(g(z)\) is an analytic function on the complex disc \(\{| z|<\rho_ 0\}\) with \(g(0)=0\). The author, using some results on the linearized equation recently obtained in a joint paper with \textit{A. M. Fink} [Int. J. Math. Math. Sci. 13, No. 4, 625-644 (1990; Zbl 0731.35060)] gives a sufficient condition on the number \(g'(0)\) for the existence and the multiplicity of complex radial solutions to the above problem.