We investigate the existence of solutions u(x, t) for perturbations of the elliptic system with Dirichlet boundary condition $$array {L{xi}+{mu}g({xi}+2{eta})=f;in;{Omega}}\{L{eta}+{ u}g({xi}+2{eta})=f;in;{Omega}}$$ (0.1) where $g(u)=Bu^+-Au^-$, $u^+=max{u,;0}$, $u^-=max{-u,;0}$, ${mu}$, ${ u}$ are nonzero constants and the nonlinearity $({mu}+2{ u})g(u)$ crosses the eigenvalues of the elliptic operator L.