  In this paper, we consider two different issues, stability and strong
coupling, raised lately in the newly-proposed Horava-Lifshitz (HL) theory of
quantum gravity with projectability condition. We find that all the scalar
modes are stable in the de Sitter background, due to two different kinds of
effects, one from high-order derivatives of the spacetime curvature, and the
other from the exponential expansion of the de Sitter space. Combining these
effects properly, one can make the instability found in the Minkowski
background never appear even for small-scale modes, provided that the IR limit
is sufficiently closed to the relativistic fixed point. At the fixed point, all
the modes become stabilized. We also show that the instability of Minkowski
spacetime can be cured by introducing mass to the spin-0 graviton. The strong
coupling problem is investigated following the effective field theory approach,
and found that it cannot be cured by the Blas-Pujolas-Sibiryakov mechanism,
initially designed for the case without projectability condition, but might be
circumvented by the Vainshtein mechanism, due to the non-linear effects. In
fact, we construct a class of exact solutions, and show explicitly that it
reduces smoothly to the de Sitter spacetime in the relativistic limit.
