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We assume that Ω is a bounded domain inℝ   2    ${{\mathbb {R}}^2}$    with  N  holes. We study the steady Navier–Stokes problem with inhomogeneous Navier-type boundary conditions in Ω (problem BVP-1) and the two other related problems BVP-2 and BVP-3. We prove the existence of strong solutions of BVP-2 and BVP-3 and the uniqueness of solutions for “small data”. The existence of a strong solution of BVP-1 is proven only if the body force   ${\mathbf {f}}$    belongs to a certain set  (    ℝ   N   )    ${\mathfrak {M}({\mathbb {R}}^N)}$   . We show that problem BVP-1 is not solvable for some configurations of given boundary data and body forces.

The steady Navier–Stokes problem with the inhomogeneous Navier-type boundary conditions in a 2D multiply-connected bounded domain