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In this letter we study the self-energy of a point-like charge for theelectromagnetic field in a non minimal Lorentz symmetry breaking scenario in a$n+1$ dimensional space time. We consider two variations of a model where theLorentz violation is caused by a background vector $d^{\nu}$ that appears in ahigher 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 spatialdimension $n$ and diverges for even $n$. We also make some comments regardingobstacles in the quantization of the proposed model.

The point-charge self-energy in a nonminimal Lorentz violating Maxwell Electrodynamics