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$$N$$-vortex equilibria for ideal fluids in bounded planar domains and new nodal solutions of the sinh-Poisson and the Lane-Emden-Fowler equations. (English) Zbl 1195.35250
Commun. Math. Phys. 297, No. 3, 653-686 (2010); erratum ibid. 333, No. 2, 1107 (2015).
Summary: 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 $$\Omega$$ 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 $$\Omega$$ with Dirichlet boundary conditions for $$\rho > 0$$ small. The stream function $${\psi_\rho}$$ induces a stationary velocity field $${v_\rho}$$ solving the Euler equation in $$\Omega$$. On an arbitrary bounded domain we obtain velocitiy fields having three or four counter-rotating vortices. If $$\Omega$$ 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 $$\Omega$$ with $$p$$ large.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35Q31 Euler equations 35J25 Boundary value problems for second-order elliptic equations 76D17 Viscous vortex flows
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