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

Abstract

We study the semilinear elliptic equation $\Delta u+g(x,u,Du)=0$ in ${\mathbb{R}}^n$. 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.