  Semilinear stochastic evolution equations with L\'evy noise and monotone
nonlinear drift are considered. The existence and uniqueness of the mild
solutions in $L^p$ for these equations is proved and a sufficient condition for
exponential asymptotic stability of the solutions is derived. The main tool in
our study is an It\^o type inequality for the $p$th power of stochastic
convolution integrals in Hilbert spaces.
