  We study the semilinear elliptic equation $\Delta u + g(x,u,Du) = 0$ in
$\R^n$. The nonlinearities $g$ can have arbitrary growth in $u$ and $Du$,
including in particular the exponential behavior. No restriction is imposed on
the behavior of $g(x,z,p)$ at infinity except in the variable $x$. We obtain a
solution $u$ that is locally unique and inherits many of the symmetry
properties of $g$. Positivity and asymptotic behavior of the solution are also
addressed. Our results can be extended to other domains like half-space and
exterior domains. We give some examples.
