MULTIPLICITY AND NONLINEARITY IN THE NONLINEAR ELLIPTIC SYSTEM

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