  In this letter we study the self-energy of a point-like charge for the
electromagnetic field in a non minimal Lorentz symmetry breaking scenario in a
$n+1$ dimensional space time. We consider two variations of a model where the
Lorentz violation is caused by a background vector $d^{\nu}$ that appears in a
higher derivative interaction. We restrict our attention to the case where
$d^{\mu}$ is a time-like background vector, namely $d^{2}=d^{\mu}d_{\mu}>0$,
and we verify that the classical self-energy is finite for any odd spatial
dimension $n$ and diverges for even $n$. We also make some comments regarding
obstacles in the quantization of the proposed model.
