An Elliptic Problem Related to Planar Vortex Pairs

Abstract

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 of R-2; u = 0 on partial derivative Omega, where q is a positive harmonic function. We show that the A(lambda) = {x is an element of Omega : u(lambda)( x) > q( x)} of the solution u(lambda) shrinks to a global minimum point of q on the boundary partial derivative Omega as lambda --> +infinity. Furthermore, we show that for each strict local minimum x(0) point of q(x) on the boundary partial derivative Omega, there exists a solution u(lambda) whose vortex core shrinks to this strict local minimum point x(0) as lambda --> +infinity.