  We propose a detailed stability analysis of the Lugiato-Lefever model for
Kerr optical frequency combs in whispering gallery mode resonators pumped in
the normal dispersion regime. We analyze the spatial bifurcation structure of
the stationary states depending on two parameters that are experimentally
tunable, namely the pump power and the cavity detuning. Our study demonstrates
that the non-trivial equilibria play an important role in this bifurcation map,
as their associated eigenvalues undergo critical bifurcations that are
foreshadowing the existence of localized spatial structures. In particular, we
show that in the normal dispersion regime, dark cavity solitons can emerge in
the system, and thereby generate a Kerr comb. We also show how these solitons
can coexist in the resonator as long as they do not interact with each other.
The Kerr combs created by these (sets of) dark solitons are also analyzed, and
their stability is discussed as well.
