  A variational approach is considered to calculate the free energy and the
conformational properties of a polyelectrolyte chain in $d$ dimensions. We
consider in detail the case of pure Coulombic interactions between the
monomers, when screening is not present, in order to compute the end-to-end
distance and the asymptotic properties of the chain as a function of the
polymer chain length $N$. We find $R \simeq N^{\nu}(\log N)^{\gamma}$ where
$\nu = \frac{3}{\lambda+2}$ and $\lambda$ is the exponent which characterize
the long-range interaction $U \propto 1/r^{\lambda}$. The exponent $\gamma$ is
shown to be non-universal, depending on the strength of the Coulomb
interaction. We check our findings, by a direct numerical minimization of the
variational energy for chains of increasing size $2^4<N<2^{15}$. The
electrostatic blob picture, expected for small enough values of the interaction
strength, is quantitatively described by the variational approach. We perform a
Monte Carlo simulation for chains of length $2^4<N<2^{10}$. The non universal
behavior of the exponent $ \gamma$ previously derived within the variational
method, is also confirmed by the simulation results. Non-universal behavior is
found for a polyelectrolyte chain in $d=3$ dimension. Particular attention is
devoted to the homopolymer chain problem, when short range contact interactions
are present.
