On the nonexistence of periodic radial solutions for semilinear wave equations in unbounded domain

Abstract

We consider nonlinear periodic radial wave equations of the form $$ u_{tt}-u_{rr}-\frac{N-1}r u_r+g(t,r,u)=0,\qquad t\in\mathbb{R},\quad 0<r<R. $$ The main purpose of the paper is to characterize the nonlinearities $g$ such that the equation has no nontrivial periodic solutions in $\mathbb{R}^N$ within a given natural class of functions $u$. The proofs are based on a new integral identity which we introduce for the solutions of the wave equation.