Existence and symmetry for elliptic equations in ^sup n^ with arbitrary growth in the gradient

We study the semilinear elliptic equation [Delta]u + g(x, u, Du) = 0 in n. The nonlinearities g can have arbitrary growth in u and Du, including, in particular, 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 which 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, such as the half-space and exterior domains. Finally, we give some examples.