  It is known that the cosmological constant can be dynamically tuned to an
arbitrary small value in classes of scalar tensor theories. The trouble with
such schemes is that effective gravity itself vanishes. We explore the
possibility of avoiding this ``no-go'' with a spatially varying effective
gravity. We demonstrate this in principle with the non-minimally coupled scalar
field having an additional coupling to a fermionic field. The expectation value
of the scalar field gets anchored to a non-trivial value inside compact
domains. But for the non-minimal coupling to the scalar curvature, these
configurations are analogous to the non-topological solutions suggested by Lee
and Wick. With non-minimal coupling, this leads to a peculiar spatial variation
of effective gravity. As before, one can dynamically have the long distance
(global) gravitational constant $G$ and ${\Lambda}$, the cosmological constant,
tending to zero. However, inside compact domains, $G$ can be held to a
universal (non-vanishing) value. Long distance gravitational effects turn out
to be indistinguishable from those expected of general theory of relativity
(GTR). There are two ways in which the ensuing theory may lead to a viable
effective gravity theory: (a) the compact domains could be of microscopic
(sub-nuclear) size, or (b) the domains could be large enough to accommodate
structures as large as a typical galaxy. Aspects of effective gravity and
cosmology that follow are described. A toy Freidman - Robertson - Walker (FRW)
model free from several standard model pathologies and characteristic features
emerges.
