  We propose a non-linear extension of U(1) \times U(1) (abelian) ABJM model
including T_{M2} (higher derivative) corrections. The action proposed here is
expected to describe a single M2-brane proving C^4/Z_k target space. The model
includes couplings with the 3-form background in the eleven-dimensional
supergravity which is consistent with the orbifold projection. We show that the
novel higgs mechanism proposed by Mukhi and Papageorgakis does work even in the
presence of higher derivative corrections and couplings with the background
field, giving the correct structure of the Dirac-Born-Infeld action with
Wess-Zumino term for a D2-brane. We also find half BPS solutions in the full
non-linear theory which is interpreted as an another M2-brane intersecting with
the original M2-brane. A possible generalization to U(N) \times U(N) gauge
group is briefly discussed.
