  We have used XMM-Newton to observe six Lyman Break Analogs (LBAs): members of
the rare population of local galaxies that have properties that are very
similar to distant Lyman Break Galaxies. Our six targets were specifically
selected because they have optical emission-line properties that are
intermediate between starbursts and Type 2 (obscured) AGN. Our new X-ray data
provide an important diagnostic of the presence of an AGN. We find X-ray
luminosities of order 10^{42} erg/s and ratios of X-ray to far-IR luminosities
that are higher than values in pure starburst galaxies by factors ranging from
~ 3 to 30. This strongly suggests the presence of an AGN in at least some of
the galaxies. The ratios of the luminosities of the hard (2-10 keV) X-ray to [O
III]\lambda 5007 emission-line are low by about an order-of-magnitude compared
to Type 1 AGN, but are consistent with the broad range seen in Type 2 AGN.
Either the AGN hard X-rays are significantly obscured or the [O III] emission
is dominated by the starburst. We searched for an iron emission line at ~ 6.4
keV, which is a key feature of obscured AGN, but only detected emission at the
~ 2\sigma level. Finally, we find that the ratios of the mid-infrared (24\mu m)
continuum to [O III]\lambda 5007 luminosities in these LBAs are higher than the
values for Type 2 AGN by an average of 0.8 dex. Combining all these clues, we
conclude that an AGN is likely to be present, but that the bolometric
luminosity is produced primarily by an intense starburst. If these black holes
are radiating at the Eddington limit, their masses would lie in the range of
10^5 to 10^6 M_{sun}. These objects may offer ideal local laboratories to
investigate the processes by which black holes grew in the early universe.
