  In this paper we investigate the effect of nonlinear damping on the
Lugiato-Lefever equation $$ \i \partial_t a = -(\i-\zeta) a - da_{xx}
-(1+\i\kappa)|a|^2a +\i f $$ on the torus or the real line. For the case of the
torus it is shown that for small nonlinear damping $\kappa>0$ stationary
spatially periodic solutions exist on branches that bifurcate from constant
solutions whereas all nonconstant solutions disappear when the damping
parameter $\kappa$ exceeds a critical value. These results apply both for
normal ($d<0$) and anomalous ($d>0$) dispersion. For the case of the real line
we show by the Implicit Function Theorem that for small nonlinear damping
$\kappa>0$ and large detuning $\zeta\gg 1$ and large forcing $f\gg 1$ strongly
localized, bright solitary stationary solutions exists in the case of anomalous
dispersion $d>0$. These results are achieved by using techniques from
bifurcation and continuation theory and by proving a convergence result for
solutions of the time-dependent Lugiato-Lefever equation.
