  We investigate a class of quadratic-exponential growth BSDEs with jumps. The
quadratic structure introduced by Barrieu & El Karoui (2013) yields the
universal bounds on the possible solutions. With local Lipschitz continuity and
the so-called A_gamma-condition for the comparison principle to hold, we prove
the existence of a unique solution under the general quadratic-exponential
structure. We have also shown that the strong convergence occurs under more
general (not necessarily monotone) sequence of drivers, which is then applied
to give the sufficient conditions for the Malliavin's differentiability.
