Multiple solutions for an elliptic problem related to vortex pairs

Let Ω be a bounded domain in (N 2), φ a harmonic function in . In this paper we study the existence of solutions to the following problem arising in the study of vortex pairs The set is called “vortex core”. Existence of solutions whose “vortex core” consists of one component and asymptotic behavior of “vortex core” were studied by many authors for large λ recently. Under the condition that φ has k strictly local minimum points on the boundary ∂Ω, we obtain in this paper that for λ large enough, (Pλ) has a solution with “vortex core” consisting of k components by a constructive way.