Nonlinearities and nontrivial solutions for the nonlinear hyperbolic system Original Research Article

We study the uniqueness and the existence of multiple nontrivial solutions u(x,t)u(x,t) for a perturbation [(u+v+1)+−1][(u+v+1)+−1] of the hyperbolic system with Dirichlet boundary condition equation(0.1) View the MathML sourceutt−uxx=μ[(u+2v+1)+−1]in (−π2,π2)×R, Turn MathJax on View the MathML sourcevtt−vxx=ν[(u+2v+1)+−1]in (−π2,π2)×R, Turn MathJax on where u+=max{u,0}u+=max{u,0}, μ,νμ,ν are nonzero constants. Here the nonlinearity (μ+2ν)[(w+1)+−1](μ+2ν)[(w+1)+−1] crosses the eigenvalues of the wave operator.