  We investigate the problem of entire solutions for a class of fourth order,
dilation invariant, semilinear elliptic equations with power-type weights and
with subcritical or critical growth in the nonlinear term. These equations
define non compact variational problems and are characterized by the presence
of a term containing lower order derivatives, whose strength is ruled by a
parameter {\lambda}. We can prove existence of entire solutions found as
extremal functions for some Rellich-Sobolev type inequalities. Moreover, when
the nonlinearity is suitably close to the critical one and the parameter
{\lambda} is large, symmetry breaking phenomena occur and in some cases the
asymptotic behavior of radial and non radial ground states can be somehow
described.
