An Elliptic Problem Related to Planar Vortex Pairs.

In this paper, we study the existence and limiting behavior of the mountain pass solutions of the elliptic problem $-\Delta u = \lambda f(u-q(x))$ in $\Omega\subset R^2; u= 0$ on $\partial\Omega$, where $q$ is a positive harmonic function. We show that the "vortex core" $A_\lambda=\{x\in\Omega:u_\lambda(x)>q(x)\}$ of the solution $u_\lambda$ shrinks to a global minimum point of $q$ on the boundary $\partial\Omega$ as $\lambda\to+\infty$. Furthermore, we show that for each strict local minimum $x_0$ point of $q(x)$ on the boundary $\partial\Omega$, there exists a solution $u_\lambda$ whose vortex core shrinks to this strict local minimum point $x_0$ as $\lambda\to+\infty$. [ABSTRACT FROM AUTHOR]