On a two-component Bose-Einstein condensate with steep potential wells

In this paper, we study the following two-component systems of nonlinear Schrodinger equations { Delta u - (lambda a(x) + a(0)(x))u + mu(1)u(3) + beta v(2)u = 0 in R-3, Delta v - (lambda b(x) + b(0)(x))v + mu(2)v(3) + beta u(2)v = 0 in R-3, u, v is an element of H-1 (R-3),R- u, v > 0 in R-3, where lambda, mu(1), mu(2) > 0 and beta < 0 are parameters; a(x), b(x) >= 0 are steep potentials and a(0)(x), b(0)(x) are sign-changing weight functions; a(x), b(x), a(0)(x) and b(0)(x) are not necessarily to be radial symmetric. By the variationalmethod, we obtain a ground state solution and multi-bump solutions for such systems with lambda sufficiently large. The concentration behaviors of solutions as both lambda -> +infinity and beta -> -infinity are also considered. In particular, the phenomenon of phase separations is observed in the whole space R-3. In the Hartree-Fock theory, this provides a theoretical enlightenment of phase separation in R-3 for the 2-mixtures of Bose-Einstein condensates.