  The usual renormalization procedure for the variational approximation with a
trial Gaussian ansatz for the $\lap$ model in 3+1 dimensions is re-analysed as
a departing framework for the investigation of the parameters of the model. The
so-called asymmetric phase of the model (where $<vac|\phi |vac> \neq 0$) is
considered for the search of privileged values of these parameters (mass and
coupling constant) and possible conditions they may be expected to satisfy.
This also may yield a suitable approach for the investigation of the
reliability and stability of the approximation. The extremization of the
renormalized energy density with relation to the renormalized mass, coupling
and $\bphi$ is done. The minimizations of the renormalized energy with relation
to the mass and $\bphi$ provide different expressions from the ones obtained by
the usual variational procedure for the regularized theory. Sort of ``energy
scale'' invariances in expressions for the renormalized mass and coupling
constant are found. A different view on the restoration of symmetry issue is
discussed. The transcendental character of the GAP equation may be reduced or
even eliminated by placing some variables in the complex plane.
