Semiclassical limits of ground states for Hamiltonian elliptic system with gradient term. (April 2018) - Zhang, Jian; Zhang, Wen; Tang, Xianhua
Abstract: In this paper, we study the following Hamiltonian elliptic system with gradient term − ϵ 2 Δ ψ + ϵ b → ⋅ ∇ ψ + ψ + V ( x ) φ = ∑ i = 1 I K i ( x ) | η | p i − 2 φ i n R N, − ϵ 2 Δ φ − ϵ b → ⋅ ∇ φ + φ + V ( x ) ψ = ∑ i = 1 I K i ( x ) | η | p i − 2 ψ i n R N, where η = ( ψ, φ ) : R N → R 2, V, K i ∈ C ( R N, 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 K i 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 K i . Moreover, the convergence and exponential decay of semiclassical solutions are also explored.