We discuss the existence of a blow-up solution for a multi-component parabolic-elliptic drift-diffusion model in higher space dimensions. We show that the local existence, uniqueness and well-posedness of a solution in the weighted spaces. Moreover we prove that if the initial data satisfies certain conditions, then the corresponding solution blows up in a finite time. This is a system case for the blow up result of the chemotactic and drift-diffusion equation proved by Nagai (J Inequal Appl 6:37-55, 2001) and Nagai et al. (Hiroshima J Math 30:463-497, 2000) and gravitational interaction of particles by Biler (Colloq Math 68:229-239, 1995), Biler and Nadzieja (Colloq Math 66:319-334, 1994, Adv Differ Equ 3:177-197, 1998). We generalize the result in Kurokiba and Ogawa (Differ Integral Equ 16:427-452, 2003, Differ Integral Equ 28:441-472, 2015) and Kurokiba (Differ Integral Equ 27(5-6):425-446, 2014) for the multi-component problem and give a sufficient condition for the finite time blow up of the solution. The condition is different from the one obtained by Corrias et al. (Milan J Math 72:1-28, 2004).