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abstract: |
    In this paper, we construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure is carried out by constructing solutions to the following elliptic problem $$\begin{cases}
    -{\varepsilon}^2 \Delta u=\sum_{j=1}^m \chi_{{\Omega}_j}(u-q-\frac{\kappa_j}{2\pi}\ln\frac{1}{{\varepsilon}})_+^p, \quad & x\in\Omega, \\
    u=0, \quad & x\in\partial\Omega,
    \end{cases}$$ where $p>1$, $\Omega\subset\mathbb{R}^2$ is a bounded domain, ${\Omega}_i\subset\subset{\Omega}, i=1\cdots,m$ are suitable small domains such that ${\Omega}_i\bigcap{\Omega}_j $ is empty if $i\neq j$ and $q$ is a harmonic function.

    We showed that if $\Omega$ is simply-connected smooth domain, then for any given stable critical point of Kirchhoff-Routh function $\mathcal{W}(z_1,\cdots,z_m)$ with the positive strength $\kappa_i>0$, there is a stationary classical solution approximating stationary $m$ points vortex solution of incompressible Euler equations with vorticity $\sum_{j=1}^m\kappa_i$.

    Existence and asymptotic behavior of single point non-vanishing vortex solutions were studied by D. Smets and J. Van Schaftingen in [@SV].\
    *AMS 2000 Subject Classifications: Primary $35\mathrm{J}60$;  Secondary $35\mathrm{JB}05$; $35\mathrm{J}40$ Keywords: The Euler equation; Multiple non-vanishing vortices; Free boundary problem.*
address:
- 'Institute of Applied Mathematics, Chinese Academy of Science, Beijing 100190, P.R. China'
- 'Institute of Applied Mathematics, Chinese Academy of Science, Beijing 100190, P.R. China'
- 'Department of Mathematics, The Chinese University of Hong Kong, Shatin, 