Regularization of planar vortices for the incompressible flow

In this paper, we continue to construct stationary classical solutions for the incompressible planar flows approximating singular stationary solutions of this problem. This procedure is carried out by constructing solutions for the following elliptic equations

$\{\begin{array}{ll}-\Delta u=\lambda \sum _{j=1}^k{1}_{B\delta \left(x_{O,j}\right)}{\left(u-k_j\right)}_{+}^p,\hfill & in\Omega ,\hfill \\ u=0,\hfill & on?\Omega \hfill \end{array}$

where 0 < p < 1, ? ?² is a bounded simply-connected smooth domain, κ (i, 1, …, k) is prescribed positive constant. The result we prove is that for any given non-degenerate critical point x₀ = (x_0,1, … x_{0, k}) of the Kirchhoff-Routh function defined on Ω^k corresponding to (κ₁, …, κ_k), there exists a stationary classical solution approximating stationary k points vortex solution. Moreover, as λ → + ∞, the vorticity set {y: u_λ > κ_j} ∩ B_δ(x_{0, j}) shrinks to {x_{0, j}}, and the local vorticity strength near each x_{0, j} approaches κ_j, j = 1, …, k. This result makes the study of the above problem with p ≥ 0 complete since the cases p 1, p = 1, p = 0 have already been studied in 11, 12] and 13] respectively.