  We give applications of known and new Liouville type theorems to universal
singularity and decay estimates for non scale invariant elliptic problems,
including Lane-Emden and Schr\"odinger type systems. This applies to various
classes of nonlinearities with regular variation and possibly different
behaviors at $0$ and $\infty$. To this end, we adapt the method from [72] to
elliptic systems, which relies on a generalized rescaling technique and on
doubling arguments from [55].
  This is in particular facilitated by new Liouville type theorems in the whole
space and in a half-space, for elliptic problems without scale invariance, that
we obtain. Our results apply to some non-cooperative systems, for which maximum
principle based techniques such as moving planes do not apply. To prove these
Liouville type theorems, we employ two methods, respectively based on
Pohozaev-type identities combined with functional inequalities on the unit
sphere, and on reduction to a scalar equation by proportionality of components.
  In turn we will survey the existing methods for proving Liouville-type
theorems for superlinear elliptic equations and systems, and list some of the
typical existing results for (Sobolev subcritical) systems.
  In the case of scalar equations, we also revisit the classical Gidas-Spruck
integral Bernstein method, providing some improvements which turn out to be
efficient for certain nonlinearities, and we next compare the performances of
various methods on a benchmark example.
