| Contents |
| We consider the problem
\[
-\Delta u=\lambda(u-\phi)_{+}^{p-1},\quad x\in\Omega,\: u|_{\partial\Omega}=0
\]
where $\Omega$ is a bounded domain in $R^{N}$, $\phi$ is a positive
harmonic function in $\bar{\Omega}$.
This problem is related to steady vortex pairs in an ideal fluid.
Under the following condition: $\phi$ has $k$ ($k\geq1$) strictly
local minimum points $\bar{z}_{1},\cdots,\bar{z}_{k}\in\partial\Omega$,
we are able to prove the existence of a solution pair $(u_{\lambda},A_{\lambda})$
satisfying that the 'vortex core' (where $u_{\lambda}>\phi$) $A_{\lambda}$
has exactly $k$ components $A_{\lambda,j}$, $j=1,2,\cdots,k$ which
shrink to the points $\bar{z}_{1},\cdots,\bar{z}_{k}$ respectively
as $\lambda\rightarrow+\infty$. Moreover, $A_{\lambda,j}$ is approximately
a ball with very precise estimates of $z_{j}-\bar{z}_{j}$ and $diam(A_{\lambda,j})$.
(Joint work with Shuangjie Peng). |
|