Vortex patch problem for steady lake equation

We study the vortex patch problem for the steady lake equation in a bounded domain and construct three different kinds of solutions where the vorticity concentrates in the domain or near the boundary. Our approach is based on the Lyapunov–Schmidt reduction, which transforms the construction into a problem of seeking critical points for a function related to the kinetic energy. The method in this paper has a wide applicability and can be used to deal with general elliptic equations in divergence form with Heaviside nonlinearity.