  We consider a continuous system of classical particles confined in a finite
region $\Lambda$ of $\mathbb{R}^d$ interacting through a superstable and
tempered pair potential in presence of non free boundary conditions. We prove
that the thermodynamic limit of the pressure of the system at any fixed inverse
temperature $\beta$ and any fixed fugacity $\lambda$ does not depend on
boundary conditions produced by particles outside $\Lambda$ whose density may
increase sub-linearly with the distance from the origin at a rate which depends
on how fast the pair potential decays at large distances. In particular, if the
pair potential $v(x-y)$ is of Lennard-Jones type, i.e. it decays as
$C/\|x-y\|^{d+p}$ (with $p>0$) where $\|x-y\|$ is the Euclidean distance
between $x$ and $y$, then the existence of the thermodynamic limit of the
pressure is guaranteed in presence of boundary conditions generated by external
particles which may be distributed with a density increasing with the distance
$r$ from the origin as $\rho(1+ r^q)$, where $\rho$ is any positive constant
(even arbitrarily larger than the density $\rho_0(\beta,\lambda)$ of the system
evaluated with free boundary conditions) and $q\le {1\over 2}\min\{1, p\}$.
