Nonlinearities and nontrivial solutions for the nonlinear hyperbolic system

Abstract

We study the uniqueness and the existence of multiple nontrivial solutions u ( x , t ) for a perturbation [ ( u + v + 1 ) + − 1 ] of the hyperbolic system with Dirichlet boundary condition (0.1) u t t − u x x = μ [ ( u + 2 v + 1 ) + − 1 ] in ( − π 2 , π 2 ) × R , v t t − v x x = ν [ ( u + 2 v + 1 ) + − 1 ] in ( − π 2 , π 2 ) × R , where u + = max { u , 0 } , μ , ν are nonzero constants. Here the nonlinearity ( μ + 2 ν ) [ ( w + 1 ) + − 1 ] crosses the eigenvalues of the wave operator.