Existence and asymptotic behavior of ground states for Schrödinger systems with Hardy potential

Abstract

In this paper we study the following Schrödinger systems with Hardy potential −Δu+u−μ|x|2v=f(x,|z|)v,x∈RN−Δv+v−μ|x|2u=f(x,|z|)u,x∈RNwhere z=(u,v)∈R2 and μ∈R is a positive parameter. This problem is related to coupled nonlinear Schrödinger equations for Bose–Einstein condensate. Under some suitable conditions on the parameter μ and nonlinearity f, we first prove the existence, exponential decay and convergence of ground state solutions via variational methods. Moreover, we prove the monotonicity and convergence property of the energy of ground state solutions. Finally, we also give the asymptotic behavior of ground state solutions as parameter μ tends to 0.