It is known through numerical simulations and physical experiments, generic solutions of the 2D Euler equations should develop coherent structures as time goes to infinity. Based on the above observations,  Šverák and Shnirelman have made several conjectures concerning global behaviors of solutions to 2D Euler equations. A key part in solving the above conjectures is to understand the topological behaviors of steady states. In this talk, I will mainly discuss the connection between the steady states and semi-linear elliptic equations, both in the rigidity and flexibility aspects.