  We explore the compact U(1) lattice gauge theory with staggered fermions and
gauge field action -\sum_P [\beta \cos(\Theta_P) + \gamma \cos(2\Theta_P)],
both for dynamical fermions and in the quenched approximation. (\Theta_P
denotes the plaquette angle.) In simulations with dynamical fermions at various
\gamma \le -0.2 on 6^4 lattices we find the energy gap at the phase transition
of a size comparable to the pure gauge theory for \gamma \le 0 on the same
lattice, diminishing with decreasing \gamma. This suggests a second order
transition in the thermodynamic limit of the theory with fermions for \gamma
below some finite negative value. Studying the theory on large lattices at
\gamma = -0.2 in the quenched approximation by means of the equation of state
we find non-Gaussian values of the critical exponents associated with the
chiral condensate, \beta \simeq 0.32 and \delta \simeq 1.8, and determine the
scaling function. Furthermore, we evaluate the meson spectrum and study the
PCAC relation.
