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

In this paper we study the following Schrodinger systems with Hardy potential { -Delta u + u - mu/vertical bar x vertical bar(2)v = f(x, vertical bar z vertical bar)v, x is an element of R-N -Delta v + v - mu/vertical bar x vertical bar(2)u = f(x, vertical bar z vertical bar)u, x is an element of R-N where z = (u, v) is an element of R-2 and mu is an element of R is a positive parameter. This problem is related to coupled nonlinear Schrodinger equations for Bose-Einstein condensate. Under some suitable conditions on the parameter mu 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 mu tends to 0. (C) 2019 Elsevier Ltd. All rights reserved.