  In this paper we solve the exit problems for (reflected) spectrally negative
L\'evy processes, which are exponentially killed with a killing intensity
dependent on the present state of the process and analyze respective
resolvents. All identities are given in terms of new generalizations of scale
functions. For the particular cases $\omega(x)=q$ and $\omega(x)=q
\mathbf{1}_{(a,b)}(x)$, we obtain results for the classical exit problems and
the Laplace transforms of the occupation times in a given interval, until first
passage times, respectively. Our results can also be applied to find the
bankruptcy probability in the so-called Omega model, where bankruptcy occurs at
rate $\omega(x)$ when the L\'evy surplus process is at level $x<0$. Finally, we
apply the these results to obtain some exit identities for a spectrally
positive self-similar Markov processes. The main method throughout all the
proofs relies on the classical fluctuation identities for L\'evy processes, the
Markov property and some basic properties of a Poisson process.
