N-Vortex Equilibria for Ideal Fluids in Bounded Planar Domains and New Nodal Solutions of the sinh-Poisson and the Lane-Emden-Fowler Equations

Abstract

We prove the existence of equilibria of the N-vortex Hamiltonian in a bounded domain ${\Omega\subset\mathbb{R}^2}$ , which is not necessarily simply connected. On an arbitrary bounded domain we obtain new equilibria for N = 3 or N = 4. If Ω has an axial symmetry we obtain a symmetric equilibrium for each ${N\in\mathbb{N}}$ . We also obtain new stream functions solving the sinh-Poisson equation ${-\Delta\psi=\rho\sinh\psi}$ in Ω with Dirichlet boundary conditions for ρ > 0 small. The stream function ${\psi_\rho}$ induces a stationary velocity field ${v_\rho}$ solving the Euler equation in Ω. On an arbitrary bounded domain we obtain velocitiy fields having three or four counter-rotating vortices. If Ω has an axial symmetry we obtain for each N a velocity field ${v_\rho}$ that has a chain of N counter-rotating vortices, analogous to the Mallier-Maslowe row of counter-rotating vortices in the plane. Our methods also yield new nodal solutions for other semilinear Dirichlet problems, in particular for the Lane-Emden-Fowler equation ${-\Delta u=|u|^{p-1}u}$ in Ω with p large.