Bifurcation from infinity and multiplicity of solutions for an elliptic system

In this paper, we are concerned with the bifurcation from infinity and multiplicity of solutions of the semilinear elliptic system

$$\begin{aligned}&-\Delta u=\lambda u+f(x,u)-w,\\&-\Delta w=\kappa u-\zeta w, \end{aligned}$$

which can be considered as the stationary problem of reaction–diffusion equations. We treat this problem in the framework of dynamical systems, and deal with it via the approach of a pure dynamical nature, which is different from those in the literature. By using the Shape theory of attractors and the Poincaré–Lefschetz duality theory of Conley index, we establish some new multiplicity results of solutions of the system on bifurcations from infinity under an appropriate Landesman–Lazer type condition, improving the earlier works in the literature.