Semiclassical limits of ground states for Hamiltonian elliptic system with gradient term

Abstract

In this paper, we study the following Hamiltonian elliptic system with gradient term −ϵ2Δψ+ϵb→⋅∇ψ+ψ+V(x)φ=∑i=1IKi(x)|η|pi−2φinRN,−ϵ2Δφ−ϵb→⋅∇φ+φ+V(x)ψ=∑i=1IKi(x)|η|pi−2ψinRN,where η=(ψ,φ):RN→R2, V,Ki∈C(RN,R), ϵ>0 is a small parameter and b→ is a constant vector. Suppose that V is sign-changing and has at least one global minimum, and Ki has at least one global maximum. We prove that there are two families of semiclassical solutions, for sufficiently small ϵ, with the least energy, one concentrating on the set of minimal points of V and the other on the set of maximal points of Ki. Moreover, the convergence and exponential decay of semiclassical solutions are also explored.