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We prove the existence of critical points of the $N$-vortex Hamiltonian  $H_\Omega (x_1,\ldots, x_N)  =\sum\limits^N_{i=1}\Gamma^2_i h(x_i)  + \sum\limits_{i,j=1\atop j\not= k}^N\Gamma_i\Gamma_jG(x_i,x_j)+2\sum\limits_{i=1}^N\Gamma_i\psi_0(x_i)$  in a bounded domain $\Omega\subset{\mathbb R}^2$ which may be simply ormultiply connected. Here $G$ denotes the Green function for the DirichletLaplace operator in $\Omega$, more generally a hydrodynamic Green function, and$h$ the Robin function. Moreover $\psi_0\in C^1(\overline\Omega)$ is a harmonicfunction on $\Omega$. We obtain new critical points $x=(x_1,\dots,x_N)$ for$N=3$ or $N=4$ under conditions on the vorticities $\Gamma_i\in{\mathbbR}\setminus\{0\}$. These critical points correspond to point vortex equilibriaof the Euler equation in vorticity form. The case $\Gamma_i=(-1)^i$ ofcounter-rotating vortices with identical vortex strength is included. The pointvortex equilibria can be desingularized to obtain smooth steady state solutionsof the Euler equations for an ideal fluid. The velocity of these steady stateswill be irrotational except for $N$ vorticFity blobs near $x_1,\dots,x_N$.

Critical points of the N-vortex Hamiltonian in bounded planar domains and steady state solutions of the incompressible Euler equations